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Page TOC-14

The chain rule for partial derivatives, 14-4

Total differential of a function z = z(x,y) , 14-5

Determining extrema in functions of two variables, 14-5

Using function HESS to analyze extrema, 14-6

Multiple integrals

, 14-8

Jacobian of coordinate transformation, 14-9

Double integral in polar coordinates, 14-9

Chapter 15 - Vector Analysis Applications

, 15-1

Definitions

, 15-1

Gradient and directional derivative

, 15-1

A program to calculate the gradient, 15-2

Using function HESS to obtain the gradient, 15-2

Potential of a gradient

, 15-3

Divergence

, 15-4

Laplacian

, 15-4

Curl

, 15-5

Irrotational fields and potential function, 15-5

Vector potential

, 15-6

Chapter 16 - Differential Equations

, 16-1

Basic operations with differential equations

, 16-1

Entering differential equations, 16-1

Checking solutions in the calculator, 16-2

Slope field visualization of solutions, 16-3

The CALC/DIFF menu

, 16-4

Solution to linear and non-linear equations

, 16-4

Function

LDEC,

16-4

Function

DESOLVE,

16-7

The variable ODETYPE, 16-8

Laplace transforms

, 16-10

Definitions,

16-10

Laplace transform and inverses in the calculator, 16-11

Laplace transform theorems, 16-12

Dirac’s delta function and Heaviside’s step function, 16-15

Applications of Laplace transform in the solution of ODEs, 16-17

Содержание 48GII

Страница 1: ...hp 48gII graphing calculator user s guide H Edition 4 HP part number F2226 90020 ...

Страница 2: ...ICULAR PURPOSE HEWLETT PACKARD CO SHALL NOT BE LIABLE FOR ANY ERRORS OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES IN CONNECTION WITH THE FURNISHING PERFORMANCE OR USE OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN Copyright 2003 Hewlett Packard Development Company L P Reproduction adaptation or translation of this manual is prohibited without prior written permission of Hewlett Packard Company exce...

Страница 3: ...expressions in paper Thus the operation 2 3 in ALG mode will be entered in the calculator by pressing the keys 2 and 3 in that order To complete the operation we use the ENTER key Examples of applications of the different functions and operations in this calculator are illustrated in this user s guide in both modes This guide contains examples that illustrate the use of the basic calculator functi...

Страница 4: ...nection through infrared or RS 232 allows the fast and efficient exchange of programs and data with other calculators or computers The calculator provides a flash memory card port to facilitate storage and exchange of data with other users The programming capabilities of the calculator allow you or other users to develop efficient applications for specific purposes Whether it is advanced mathemati...

Страница 5: ...calculator s mode 1 11 Operating mode 1 12 Number format and decimal dot or comma 1 16 Angle measure 1 21 Coordinate system 1 22 Beep Key Click and Last Stack 1 23 Selecting CAS settings 1 24 Selecting Display modes 1 24 Selecting the display font 1 25 Selecting properties of the line editor 1 26 Selecting properties of the Stack 1 26 Selecting properties of the Equation Writer EQW 1 27 Selecting ...

Страница 6: ...Creating sub directories 2 38 Moving among sub directories 2 42 Deleting sub directories 2 42 Variables 2 46 Creating variables 2 46 Checking variable contents 2 51 Replacing the contents of variables 2 53 Copying variables 2 54 Reordering variables in a directory 2 57 Moving variables using the FILES menu 2 58 Deleting variables 2 59 UNDO and CMD functions 2 61 Flags 2 62 Example of flag setting ...

Страница 7: ...tions in the MTH menu 3 7 Hyperbolic functions and their inverses 3 9 Real number functions 3 11 Special functions 3 14 Calculator constants 3 16 Operation with units 3 17 The UNITS menu 3 17 Available units 3 18 Converting to base units 3 21 Attaching units to numbers 3 22 Operations with units 3 25 Units manipulation tools 3 27 Physical constants in the calculator 3 28 Special physical functions...

Страница 8: ... complex numbers 4 8 Functions from the MTH menu 4 8 Function DROITE equation of a straight line 4 9 Chapter 5 Algebraic and arithmetic operations 5 1 Entering algebraic objects 5 1 Simple operations with algebraic objects 5 2 Functions in the ALG menu 5 3 COLLECT 5 5 EXPAND 5 5 FACTOR 5 5 LNCOLLECT 5 5 LIN 5 5 PARTFRAC 5 5 SOLVE 5 5 SUBST 5 5 TEXPAND 5 5 Other forms of substitution in algebraic e...

Страница 9: ...e HORNER function 5 20 The variable VX 5 20 The LAGRANGE function 5 21 The LCM function 5 21 The LEGENDRE function 5 22 The PCOEF function 5 22 The PROOT function 5 22 The PTAYL function 5 22 The QUOTIENT and REMAINDER functions 5 23 The EPSX0 function and the CAS variable EPS 5 23 The PEVAL function 5 23 The TCHEBYCHEFF function 5 24 Fractions 5 24 The SIMP2 function 5 24 The PROPFRAC function 5 ...

Страница 10: ...V 6 14 The SOLVE soft menu 6 27 The ROOT sub menu 6 27 Function ROOT 6 27 Variable EQ 6 28 The SOLVR sub menu 6 28 The DIFFE sub menu 6 30 The POLY sub menu 6 30 The SYS sub menu 6 31 The TVM sub menu 6 31 Chapter 7 Solving multiple equations 7 1 Rational equation systems 7 1 Example 1 Projectile motion 7 1 Example 2 Stresses in a thick wall cylinder 7 2 Example 3 System of polynomial equations 7 ...

Страница 11: ...that use two arguments 8 6 Lists of complex numbers 8 7 Lists of algebraic objects 8 8 The MTH LIST menu 8 8 Manipulating elements of a list 8 10 List size 8 10 Extracting and inserting elements 8 10 Element position in the list 8 11 HEAD and TAIL functions 8 11 The SEQ function 8 11 The MAP function 8 12 Defining functions that use lists 8 13 Applications of lists 8 14 Harmonic mean of a list 8 1...

Страница 12: ...ng coordinate system 9 13 Application of vector operations 9 16 Resultant of forces 9 16 Angle between vectors 9 16 Moment of a force 9 17 Equation of a plane in space 9 18 Row vectors column vectors and lists 9 19 Function OBJ 9 20 Function LIST 9 20 Function DROP 9 21 Transforming a row vector into a column vector 9 21 Transforming a column vector into a row vector 9 22 Transforming a list into ...

Страница 13: ... number of lists 10 15 Lists represent columns of the matrix 10 15 Lists represent rows of the matrix 10 17 Manipulating matrices by columns 10 18 Function COL 10 18 Function COL 10 19 Function COL 10 20 Function COL 10 20 Function CSWP 10 21 Manipulating matrices by rows 10 21 Function ROW 10 22 Function ROW 10 23 Function ROW 10 24 Function ROW 10 24 Function RSWP 10 25 Function RCI 10 25 Functi...

Страница 14: ...on LSQ 11 23 Solution with the inverse matrix 11 26 Solution by division of matrices 11 26 Solving multiple set of equations with the same coefficient matrix 11 27 Gaussian and Gauss Jordan elimination 11 28 Step by step calculator procedure for solving linear systems 11 37 Solution to linear systems using calculator functions 11 46 Residual errors in linear system solutions function RSD 11 43 Eig...

Страница 15: ...ponential function 12 10 The PPAR variable 12 11 Inverse functions and their graphs 12 12 Summary of FUNCTION plot operation 12 13 Plots of trigonometric and hyperbolic functions and their inverses 12 16 Generating a table of values for a function 12 17 The TPAR variable 12 18 Plots in polar coordinates 12 19 Plotting conic curves 12 21 Parametric plots 12 23 Generating a table of parametric equat...

Страница 16: ...MENU 12 48 SUB 12 48 REPL 12 48 PICT 12 48 X Y 12 48 Zooming in and out in the graphics display 12 49 ZFACT ZIN ZOUT and ZLAST 12 49 BOXZ 12 50 ZDFLT ZAUTO 12 50 HZIN HZOUT VZIN and VZOUT 12 50 CNTR 12 50 ZDECI 12 50 ZINTG 12 51 ZSQR 12 51 ZTRIG 12 51 The SYMBOLIC menu and graphs 12 51 The SYMB GRAPH menu 12 52 Function DRAW3DMATRIX 12 54 Chapter 13 Calculus Applications 13 1 The CALC Calculus men...

Страница 17: ...s and integrals 13 14 Functions INT INTVX RISCH SIGMA and SIGMAVX 13 14 Definite integrals 13 15 Step by step evaluation of derivatives and integrals 13 16 Integrating an equation 13 18 Techniques of integration 13 18 Substitution or change of variables 13 18 Integration by parts and differentials 13 19 Integration by partial fractions 13 20 Improper integrals 13 21 Integration with units 13 21 In...

Страница 18: ...nt 15 3 Divergence 15 4 Laplacian 15 4 Curl 15 5 Irrotational fields and potential function 15 5 Vector potential 15 6 Chapter 16 Differential Equations 16 1 Basic operations with differential equations 16 1 Entering differential equations 16 1 Checking solutions in the calculator 16 2 Slope field visualization of solutions 16 3 The CALC DIFF menu 16 4 Solution to linear and non linear equations 1...

Страница 19: ...l s equation 16 55 Chebyshev or Tchebycheff polynomials 16 57 Laguerre s equation 16 58 Weber s equation and Hermite polynomials 16 59 Numerical and graphical solutions to ODEs 16 60 Numerical solution of first order ODE 16 60 Graphical solution of first order ODE 16 62 Numerical solution of second order ODE 16 64 Graphical solution for a second order ODE 16 66 Numerical solution for stiff first o...

Страница 20: ...0 The Chi square distribution 17 11 The F distribution 17 12 Inverse cumulative distribution functions 17 13 Chapter 18 Statistical Applications 18 1 Pre programmed statistical features 18 1 Entering data 18 1 Calculating single variable statistics 18 2 Obtaining frequency distributions 18 5 Fitting data to a function y f x 18 10 Obtaining additional summary statistics 18 13 Calculation of percent...

Страница 21: ...ces concerning two means 18 39 Paired sample tests 18 40 Inferences concerning one proportion 18 41 Testing the difference between two proportions 18 41 Hypothesis testing using pre programmed features 18 43 Inferences concerning one variance 18 47 Inferences concerning two variances 18 48 Additional notes on linear regression 18 49 The method of least squares 18 49 Additional equations for linear...

Страница 22: ... 5 Recall current user defined key list 20 6 Assign an object to a user defined key 20 6 Operating user defined keys 20 6 Un assigning a user defined key 20 7 Assigning multiple user defined key 20 7 Chapter 21 Programming in User RPL language 21 1 An example of programming 21 1 Global and local variables and subprograms 21 2 Global Variable Scope 21 4 Local Variable Scope 21 5 The PRG menu 21 5 N...

Страница 23: ...ng a tagged quantity 21 33 Examples of tagged output 21 34 Using a message box 21 37 Relational and logical operators 21 43 Relational operators 21 43 Logical operators 21 44 Program branching 21 46 Branching with IF 21 46 The CASE construct 21 51 Program loops 21 53 The START construct 21 53 The FOR construct 21 59 The DO construct 21 61 The WHILE construct 21 62 Errors and error trapping 21 64 D...

Страница 24: ...amming examples using drawing functions 22 22 Pixel coordinates 22 25 Animating graphics 22 26 Animating a collection of graphics 22 27 More information on the ANIMATE function 22 29 Graphic objects GROBs 22 30 The GROB menu 22 31 A program with plotting and drawing functions 22 33 Modular programming 22 36 Running the program 22 36 A program to calculate principal stresses 22 38 Ordering the vari...

Страница 25: ...25 1 Browsing alarms 25 2 Setting time and date 25 2 TIME Tools 25 2 Calculations with dates 25 3 Calculations with times 25 4 Alarm functions 25 4 Chapter 26 Managing memory 26 1 Memory Structure 26 1 The HOME directory 26 2 Port memory 26 2 Checking objects in memory 26 2 Backup objects 26 3 Backing up objects in port memory 26 3 Backing up and restoring HOME 26 4 Storing deleting and restoring ...

Страница 26: ...acter set D 1 Appendix E The Selection Tree in the Equation Writer E 1 Appendix F The Applications APPS menu F 1 Appendix G Useful shortcuts G 1 Appendix H The CAS help facility H 1 Appendix I Command catalog list I 1 Appendix J The MATHS menu J 1 Appendix K The MAIN menu K 1 Appendix L Line editor commands L 1 Appendix M Index M 1 Limited Warranty W W 1 1 Service W 2 Regulatory information W 4 ...

Страница 27: ...tor output Most screenshots in this guide were generated using a computer based emulator a program that simulates the operation of the calculator in a computer and are missing the screen header lines Instead they will show additional screen output area in the location of the header lines as shown below This additional screen output area in many screen shots in this guide will not show when you try...

Страница 28: ...rations listed in the screenshot in the order shown your screen will show them occupying higher levels in the display as shown next The keystrokes required to complete these exercises are the following S2 5 R Ü5 5 2 5 The next operation 2 3 5 Ê 2 3 will force the lines corresponding to the operation SIN 2 5 to move upwards and be hidden by the header lines Many screenshots in this guide have also ...

Страница 29: ...ns of the screenshots are aimed at economizing output space in the guide Be aware of the differences between the guide s screenshots and the actual screen display and you should have no problem reproducing the exercises in this guide ...

Страница 30: ...ies The calculator uses 3 AAA LR03 batteries as main power and a CR2032 lithium battery for memory backup Before using the calculator please install the batteries according to the following procedure To install the main batteries a Make sure the calculator is off Slide up the battery compartment cover as illustrated b Insert 3 new AAA LR03 batteries into the main compartment Make sure each battery...

Страница 31: ...at the lower left corner of the keyboard Press it once to turn your calculator on To turn the calculator off press the red right shift key first key in the second row from the bottom of the keyboard followed by the key Notice that the key has a red OFF label printed in the upper right corner as a reminder of the OFF command Adjusting the display contrast You can adjust the display contrast by hold...

Страница 32: ...avigate through the file directory the second line of the display will change to reflect the proper file directory and sub directory At the bottom of the display you will find a number of labels namely EDIT VIEW RCL STO PURGE CLEAR associated with the six soft menu keys F1 through F6 ABCDEF The six labels displayed in the lower part of the screen will change depending on which menu is displayed Bu...

Страница 33: ...es clears up the contents of the screen To see this function in action type a number say 123 and then press the F key SOFT menus are typically used to select from among a number of related functions However SOFT menus are not the only way to access collections of related functions in the calculator The alternative way will be referred to as CHOOSE boxes To see an example of a choose box activate t...

Страница 34: ...lator system flags A system flag is a calculator variable that controls a certain calculator operation or mode For more information about flags see Chapter 24 System flag 117 can be set to produce either SOFT menus or CHOOSE boxes To access this flag use H FLAGS Your calculator will show the following screen highlighting the line starting with the number 117 By default the line will look as shown ...

Страница 35: ... menu you will get To revert to the CHOOSE boxes setting use H FLAGS CHK OK OK Notes 1 The TOOL menu obtained by pressing I will always produce a SOFT menu 2 Most of the examples in this user s manual are shown using both SOFT menus and CHOOSE boxes Programming applications Chapters 21 and 22 use exclusively SOFT menus 3 Additional information on SOFT menus vs CHOOSE boxes is presented in Chapter ...

Страница 36: ...a command from the CAS by selecting from a list HELP B HELP facility describing the commands available Pressing the L key will show the original TOOL menu Another way to recover the TOOL menu is to press the I key third key from the left in the second row of keys from the top of the keyboard Setting time and date The calculator has an internal real time clock This clock can be continuously display...

Страница 37: ...adjusting time and date is shown Setting the time of the day Using the number keys 1234567890 start by adjusting the hour of the day Suppose that we change the hour to 11 by pressing 11 as the hour field in the SET TIME AND DATE input form is highlighted This results in the number 11 being entered in the lower line of the input form Press the OK F soft menu key to effect the change The value of 11...

Страница 38: ...eld will change to either of the following options o AM indicates that displayed time is AM time o PM indicates that displayed time is PM time o 24 hr indicates that that the time displayed uses a 24 hour format where18 00 for example represents 6pm The last selected option will become the set option for the time format by using this procedure If using the CHOOS soft menu key the following options...

Страница 39: ...ght the date format as shown below Use the CHOOS soft menu key B to see the options for the date format Highlight your choice by using the up and down arrow keys and press the OK F soft menu key to make the selection Introducing the calculator s keyboard The figure below shows a diagram of the calculator s keyboard with the numbering of its rows and columns ...

Страница 40: ...ave 5 keys each There are 4 arrow keys located on the right hand side of the keyboard in the space occupied by rows 2 and 3 Each key has three four or five functions The main key function correspond to the most prominent label in the key Also the green left shift key key 8 1 the red right shift key key 9 1 and the blue ALPHA key key 7 1 can be ...

Страница 41: ...h the key only the first four are shown in the keyboard itself This is the way that the key looks in the keyboard Notice that the color and the position of the labels in the key namely SYMB MTH CAT and P indicate which is the main function SYMB and which of the other three functions is associated with the left shift MTH right shift CAT and P keys For detailed information on the calculator keyboard...

Страница 42: ...essing the H button The Operating Mode field will be highlighted Select the Algebraic or RPN operating mode by either using the key second from left in the fifth row from the keyboard bottom or pressing the CHOOS soft menu key B If using the latter approach use up and down arrow keys to select the mode and press the OK soft menu key to complete the operation To illustrate the difference between th...

Страница 43: ...Accept Approx mode on if asked by pressing OK Note The integer values used above e g 3 5 1 represent exact values The EXP 2 5 however cannot be expressed as an exact value therefore a switch to Approx mode is required You could also type the expression directly into the display without using the equation writer as follows R Ü3 Ü5 1 Ü3 3 23 Q3 2 5 to obtain the same result Change the operating mode...

Страница 44: ...ing the calculator to apply the operator or program to the objects occupying levels 1 and 2 The result 5 is then placed in level 1 A simpler way to calculate this operation is by using 3 2 Let s try some other simple operations before trying the more complicated expression used earlier for the algebraic operating mode 123 32 123 32 42 4 2Q 3 27 27 3 Notice the position of the y and the x in the la...

Страница 45: ...there are multiple advantages in using RPN For example in RPN mode you can see the equation unfolding step by step This is extremely useful to detect a possible input error Also as you become more efficient in this mode and learn more of the tricks you will be able to calculate expression faster and will much less keystrokes Consider for example the calculation of 4 6 5 1 4 6 5 In RPN mode you can...

Страница 46: ...e calculator You will find this feature extremely useful in operations with powers of tens or to limit the number of decimals in a result To select a number format first open the CALCULATOR MODES input form by pressing the H button Then use the down arrow key to select the option Number format The default value is Std or Standard format In the standard format the calculator will show floating poin...

Страница 47: ...he option Number format Press the CHOOS soft menu key B and select the option Fixed with the arrow down key Notice that the Number Format mode is set to Fix followed by a zero 0 This number indicates the number of decimals to be shown after the decimal point in the calculator s display Press the OK soft menu key to return to the calculator display The number now is shown as This setting will force...

Страница 48: ... soft menu key B and select the option Fixed with the arrow down key Press the right arrow key to highlight the zero in front of the option Fix Press the CHOOS soft menu key and using the up and down arrow keys select say 3 decimals Press the OK soft menu key to complete the selection Press the OK soft menu key return to the calculator display The number now is shown as Notice how the number is ro...

Страница 49: ... 23E2 is the calculator s version of powers of ten notation i e 1 235 102 In this so called scientific notation the number 3 in front of the Sci number format shown earlier represents the number of significant figures after the decimal point Scientific notation always includes one integer figure as shown above For this case therefore the number of significant figures is four Engineering format The...

Страница 50: ...nt numbers can be replaced by commas if the user is more familiar with such notation To replace decimal points for commas change the FM option in the CALCULATOR MODES input form to commas as follows Notice that we have changed the Number Format to Std Press the H button Next use the down arrow key once and the right arrow key highlighting the option __FM To select commas press the CHK soft menu ke...

Страница 51: ...alculator Grades There are 400 grades 400 g in a complete circumference or 100 grades 100 g in a right angle This notation is similar to the degree mode and was introduced in order to simplify the degree notation but is seldom used now The angle measure affects the trig functions like SIN COS TAN and associated functions To change the angle measure mode use the following procedure Press the H butt...

Страница 52: ...red from the origin on the xy plane θ is the angle that the radial distance r forms with the positive x axis measured as positive in a counterclockwise direction and z is the same as the z coordinate in a Cartesian system in 2 d mode z is assumed to be 0 The Rectangular and Polar systems are related by the following quantities 2 2 cos y x r r x θ x y r y 1 tan sin θ θ z z In a Spherical coordinate...

Страница 53: ...he corresponding option is activated These options are described next _Beep When selected the calculator beeper is active This operation mainly applies to error messages but also some user functions like BEEP _Key Click When selected each keystroke produces a click sound _Last Stack Keeps the contents of the last stack entry for use with the functions UNDO and ANS see Chapter 2 The _Beep option ca...

Страница 54: ...mmed and performed The CAS offers a number of settings can be adjusted according to the type of operation of interest These are The default independent variable Numeric vs symbolic mode Approximate vs Exact mode Verbose vs Non verbose mode Step by step mode for operations Increasing power format for polynomials Rigorous mode Simplification of non rational expressions Details on the selection of CA...

Страница 55: ... DISPLAY MODES input form press the OK soft menu key This will take you back to the CALCULATOR MODES input form To return to normal calculator display at this point press the OK soft menu key once more Selecting the display font Changing the font display allows you to have the calculator look and feel changed to your own liking By using a 6 pixel font for example you can display up to 9 stack leve...

Страница 56: ...be modified When these properties are selected checked the following effects are activated _Small Changes font size to small _Full page Allows to place the cursor after the end of the line _Indent Auto intend cursor when entering a carriage return Detailed instructions on the use of the line editor are presented in Chapter 2 in this guide Selecting properties of the Stack First press the H button ...

Страница 57: ...result Selecting properties of the equation writer EQW First press the H button to activate the CALCULATOR MODES input form Within the CALCULATOR MODES input form press the DISP soft menu key D to display the DISPLAY MODES input form Press the down arrow key three times to get to the EQW Equation Writer line This line shows two properties that can be modified When these properties are selected che...

Страница 58: ...he user can select to change this setting to 1 or 0 to reduce the number of header lines in the display Selecting the clock display First press the H button to activate the CALCULATOR MODES input form Within the CALCULATOR MODES input form press the DISP soft menu key D to display the DISPLAY MODES input form Press the down arrow key four times to get to the Header line The Header field will be hi...

Страница 59: ...ter the exponent and the key to change the sign of the exponent or mantissa Note that real must be entered with a decimal point even if the number does not have a fractional part Otherwise the number is taken as an integer number which is a different calculator objects Reals behave as you would expect a number to when used in a mathematical operation Integers These objects represent integer number...

Страница 60: ...ematics functions work on complex numbers There is no need to use a special complex function to add complex numbers you can use the same function that on reals or integers Vector and matrix operations utilize objects of type 3 real arrays and if needed type 4 complex arrays Objects type 2 strings are simply lines of text enclosed between quotes produced with the alphanumeric keyboard A list is jus...

Страница 61: ...jects of type 15 are memory locations used to organize your variables in a similar fashion as folders are used in a personal computer Libraries objects of type 16 are programs residing in memory ports that are accessible within any directory or sub directory in your calculator They resemble built in functions objects of type 18 and built in commands objects of type 19 in the way they are used Edit...

Страница 62: ...g result shown in Fix decimal mode with three decimal places see Chapter 1 In this case when the expression is entered directly into the stack as soon as you press the calculator will attempt to calculate a value for the expression If the expression is entered between quotes however the calculator will reproduce the expression as entered In the following example we enter the same expression as abo...

Страница 63: ...r is set to the RPN operating mode We also set the CAS to Exact and the display to Textbook The keystrokes to enter the expression between quotes are the same used earlier i e 5 Ü1 1 7 5 ÜR3 2Q3 Resulting in the output Press once more to keep two copies of the expression available in the stack for evaluation We first evaluate the expression using the function EVAL and next using the function NUM F...

Страница 64: ...merical solution of equation statistics applications etc the APPROX mode see Appendix C works better For mathematical applications e g calculus vector analysis algebra etc the EXACT mode is preferred Become acquainted with operations in both modes and learn how to switch from one to the other for different types of operations see Appendix C Editing arithmetic expressions Suppose that we entered th...

Страница 65: ...ess the delete key ƒ twice to erase the characters 1 Press the right arrow key once to move the cursor to the right of the 7 Type a decimal point with Press the right arrow key until the cursor is immediately to the right of the 5 Press the delete key ƒ once to erase the character 5 Type a 3 with 3 Press to return to the stack The edited expression is now available in the stack Editing of a line o...

Страница 66: ...hen the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise Editing algebraic expressions Editing of an algebraic expression with the line editor is very similar to that of an arithmetic expression see exercise above Suppose that we want to modify the expression entered above to read b L x R R x L 2 1 2 2 To edit this algebraic expression using the line editor use...

Страница 67: ...he characters y Type x to enter an x Press the right arrow key 4 times to move the cursor to the right of the Type R to enter a square root symbol Type Ü to enter a set of parentheses they come in pairs Press the right arrow key once and the delete key ƒ once to delete the right parenthesis of the set inserted above Press the right arrow key 4 times to move the cursor to the right of the b Type Ü ...

Страница 68: ...s in algebraic expressions use the CHARS menu This menu is activated by the keystroke combination Details are presented in Appendix D Using the Equation Writer EQW to create expressions The equation writer is an extremely powerful tool that not only let you enter or see an equation but also allows you to modify and work apply functions on all or part of the equation The equation writer EQW therefo...

Страница 69: ...if factoring is possible SIMP lets you simplify an expression highlighted in the equation writer screen as much as it can be simplified according to the algebraic rules of the CAS If you press the L key two more soft menu options show up as shown below The six soft menu keys for the Equation Writer activate the following functions CMDS allows access to the collection of CAS commands listed in alph...

Страница 70: ...eration will enter the corresponding character or characters in the cursor location For example for the cursor in the location indicated above type now Ü5 1 3 The edited expression looks as follows Suppose that you want to replace the quantity between parentheses in the denominator i e 5 1 3 with 5 π2 2 First we use the delete key ƒ delete the current 1 3 expression and then we replace that fracti...

Страница 71: ...s highlighted i e seven times producing NOTE Alternatively from the original position of the cursor to the right of the 2 in the denominator of π2 2 we can use the keystroke combination interpreted as Once the expression is highlighted as shown above type 1 3 to add the fraction 1 3 Resulting in Showing the expression in smaller size To show the expression in a smaller size font which could be use...

Страница 72: ...e first highlight the entire expression by pressing Then press the EVAL D soft menu key If your calculator is set to Exact CAS mode i e the _Approx CAS mode is not checked then you will get the following symbolic result If you want to recover the unevaluated expression at this time use the function UNDO i e the first key in the third row of keys from the top of the keyboard The recovered expressio...

Страница 73: ...irst fraction Highlights denominator of the first fraction Highlights first term in denominator of first fraction Highlights second term in denominator of first fraction Highlights first factor in second term in denominator of first fraction Highlights expression in parentheses in denominator of first fraction Since this is the sub expression we want evaluated we can now press the EVAL D soft menu...

Страница 74: ...rical evaluation of that term too and show the sum of these two decimal values in small font format by using ï C we get To highlight and evaluate the expression in the Equation Writer we use D resulting in Editing arithmetic expressions We will show some of the editing features in the Equation Writer as an exercise We start by entering the following expression used in the previous exercises ...

Страница 75: ...shown here Press the down arrow key to trigger the clear editing cursor The screen now looks like this By using the left arrow key š you can move the cursor in the general left direction but stopping at each individual component of the expression For example suppose that we will first will transform the expression π 2 2 into the expression LN π5 3 With the clear cursor active as shown above press ...

Страница 76: ...s like this Next we ll change the 5 within the parentheses to a by using these keystrokes šƒƒ1 2 Next we highlight the entire expression in parentheses an insert the square root symbol by using R Next we ll convert the 2 in front of the parentheses in the denominator into a 2 3 by using šƒƒ2 3 At this point the expression looks as follows The final step is to remove the 1 3 in the right hand side ...

Страница 77: ...e of the alphabetic keyboard is included To illustrate the use of the Equation Writer to enter an algebraic equation we will use the following example Suppose that we want to enter the expression 3 1 2 3 2 θ µ λ µ y x LN e Use the following keystrokes 2 R3 n m x 2 m c y t Q1 3 This results in the output In this example we used several lower case English letters e g x x several Greek letters e g λ ...

Страница 78: ...will be shown framed in the graphics display After selecting a sub expression you can press to show the selected sub expression highlighted in the Equation writer The following figures show different sub expressions selected with and the corresponding Equation Writer screen after pressing Editing algebraic expressions The editing of algebraic equations follows the same rules as the editing of alge...

Страница 79: ... of selection of the clear editing cursor in this example is the following press the left arrow key š repeatedly 1 The 1 in the 1 3 exponent 2 θ 3 y 4 µ 5 2 6 x 7 µ in the exponential function 8 λ 9 3 in the 3 term 10 the 2 in the 2 3 fraction At any point we can change the clear editing cursor into the insertion cursor by pressing the delete key ƒ Let s use these two cursors the clear editing cur...

Страница 80: ...dy have the sub expression 3 1 θ SIN highlighted let s press the EVAL D soft menu key to evaluate this sub expression The result is Some algebraic expressions cannot be simplified anymore Try the following keystrokes D You will notice that nothing happens other than the highlighting of the entire argument of the LN function This is because this expression cannot be evaluated or simplified any more...

Страница 81: ...erm of the numerator Then press the right arrow key to navigate through the expression Simplifying an expression Press the BIG C soft menu key to get the screen to look as in the previous figure see above Now press the SIMP C soft menu key to see if it is possible to simplify this expression as it is shown in the Equation Writer The result is the following screen This screen shows the argument of ...

Страница 82: ... select the first 3 terms in the expression and attempt a factoring of this sub expression This produces Now press the FACTO soft menu key to get Press to recover the original expression Next enter the following keystrokes to select the last two terms in the expression i e press the soft menu key to get Press to recover the original expression Now let s select the entire expression by pressing the...

Страница 83: ...cise still selected press the L key to show the CMDS and HELP soft menu keys These two commands belong to the second part of the soft menu available with the Equation Writer Let s try this example as an application of the CMDS soft menu key Press the CMDS soft menu key to get the list of CAS commands Next select the command DERVX the derivative with respect to the variable X the current CAS indepe...

Страница 84: ...IN END COPY CUT and PASTE To facilitate editing whether with the Equation Writer or on the stack the calculator provides five editing functions BEGIN END COPY CUT and PASTE activated by combining the right shift key with keys 2 1 2 2 3 1 3 2 and 3 3 respectively These keys are located in the leftmost part of rows 2 and 3 The action of these editing functions are as follows BEGIN marks the beginnin...

Страница 85: ...he numerator of the argument for the LN function Try the following keystrokes šš ššš The resulting screen is as follows The functions BEGIN and END are not necessary when operating in the Equation Writer since we can select strings of characters by using the arrow keys Functions BEGIN and END are more useful when editing an expression with the line editor For example let s select the expression x ...

Страница 86: ...s the expression in the Equation Writer in small font format press the BIG C soft menu key Press to exit the Equation Writer Creating and editing summations derivatives and integrals Summations derivatives and integrals are commonly used for calculus probability and statistics applications In this section we show some examples of such operations created with the equation writer Use ALG mode Summat...

Страница 87: ...tor press and the A soft menu key to show This expression shows the general form of a summation typed directly in the stack or line editor Σ index starting_value ending_value summation expression Press to return to the Equation Writer The resulting screen shows the value of the summation To recover the unevaluated summation use To evaluate the summation again you can use the D soft menu key This s...

Страница 88: ...nput locations use the following keystrokes t a tQ2 b t d The resulting screen is the following To see the corresponding expression in the line editor press and the A soft menu key to show This indicates that the general expression for a derivative in the line editor or in the stack is variable function of variables Press to return to the Equation Writer The resulting screen is not the derivative ...

Страница 89: ...ls We will use the Equation Writer to enter the following definite integral τ 0 sin dt t t Press O to activate the Equation Writer Then press Á to enter the integral sign Notice that the sign when entered into the Equation Writer screen provides input locations for the limits of integration the integrand and the variable of integration To fill these input locations use the following keystrokes 0 u...

Страница 90: ...tive again you can use the D soft menu key This shows again that cos sin sin 0 τ τ τ τ dt t t Double integrals are also possible For example which evaluates to 36 Partial evaluation is possible for example This integral evaluates to 36 Organizing data in the calculator You can organize data in your calculator by storing variables in a directory tree To understand the calculator s memory we first t...

Страница 91: ...For example to change directory to the CASDIR press the down arrow key and press CHDIR A This action closes the File Manager window and returns us to normal calculator display You will notice that the second line from the top in the display now starts with the characters HOME CASDIR indicating that the current directory is CASDIR within the HOME directory Functions for manipulation of variables Th...

Страница 92: ...o send variable with X modem protocol CHDIR To change directory To move between the different soft menu commands you can use not only the NEXT key L but also the PREV key The user is invited to try these functions on his or her own Their applications are straightforward The HOME directory The HOME directory as pointed out earlier is the base directory for memory operation for the calculator To get...

Страница 93: ...K F soft menu key or to get the following screen The screen shows a table describing the variables contained in the CASDIR directory These are variables pre defined in the calculator memory that establish certain parameters for CAS operation see appendix C The table above contains 4 columns The first column indicate the type of variable e g EQ means an equation type variable R indicates a real val...

Страница 94: ...key shows one more variable stored in this directory To see the contents of the variable EPS for example use EPS This shows the value of EPS to be 0000000001 To see the value of a numerical variable we need to press only the soft menu key for the variable For example pressing cz followed by shows the same value of the variable in the stack if the calculator is set to Algebraic If the calculator is...

Страница 95: ...etic keyboard temporarily and enter a full name before unlocking it again The following combinations of keystrokes will lock the alphabetic keyboard locks the alphabetic keyboard in upper case When locked in this fashion pressing the before a letter key produces a lower case letter while pressing the key before a letter key produces a special character If the alphabetic keyboard is already locked ...

Страница 96: ...ing the FILES environment or by using the command CRDIR The two approaches for creating sub directories are presented next Using the FILES menu Regardless of the mode of operation of the calculator Algebraic or RPN we can create a directory tree based on the HOME directory by using the functions activated in the FILES menu Press to activate the FILES menu If the HOME directory is not already highl...

Страница 97: ...tents of a new variable that is being created Since we have no contents for the new sub directory at this point we simply skip this input field by pressing the down arrow key once The Name input field is now highlighted This is where we enter the name of the new sub directory or variable as the case may be as follows mans The cursor moves to the _Directory check field Press the C soft menu key to ...

Страница 98: ...hown below because there are no variables defined within this directory Let s create the sub directory INTRO by using OK L NEW intro CHK OK Press the key followed by the J key to see the contents of the MANS directory as follows Press the INTRO soft menu key to move into the INTRO sub directory This will show an empty sub directory Later on we will do some exercises in creating variables Using the...

Страница 99: ...hen press OK This will produce the following pull down menu Use the down arrow key to select the 5 CRDIR option and press OK Command CRDIR in Algebraic mode Once you have selected the CRDIR through one of the means shown above the command will be available in your stack as follows At this point you need to type a directory name say chap1 chap1 The name of the new directory will be shown in the sof...

Страница 100: ...ressing the J VARiables key To move up in the directory tree use the function UPDIR i e enter Alternatively you can use the FILES menu i e press Use the up and down arrow keys to select the sub directory you want to move to and then press the CHDIR CHange DIRectory or A soft menu key This will show the contents of the sub directory you moved to in the soft menu key labels Deleting subdirectories T...

Страница 101: ... F Do not delete sub directory or variable After selecting one of these four commands you will be returned to the screen listing the contents of the sub directory The ABORT command however will show an error message and you will have to press OK before returning to the variable listing Using the command PGDIR The command PGDIR can be used to purge directories Like the command CRDIR the PGDIR comma...

Страница 102: ...row key to select the 6 PGDIR option and press OK Command PGDIR in Algebraic mode Once you have selected the PGDIR through one of the means shown above the command will be available in your stack as follows At this point you need to type the name of an existing directory say S4 s4 As a result sub directory S4 is deleted Instead of typing the name of the directory you can simply press the correspon...

Страница 103: ...mode To use the PGDIR in RPN mode you need to have the name of the directory between quotes already available in the stack before accessing the command For example s2 Then access the PGDIR command by either of the means shown above e g through the N key Press the OK soft menu key to activate the command and delete the sub directory ...

Страница 104: ...be any combination of alphabetic and numerical characters starting with a letter either English or Greek Some non alphabetic characters such as the arrow can be used in a variable name if combined with an alphabetical character Thus A is a valid variable name but is not Valid examples of variable names are A B a b α β A1 AB12 A12 Vel Z0 z1 etc A variable can not have the same name than a function ...

Страница 105: ...nu We will use the FILES menu to enter the variable A We assume that we are in the sub directory HOME MANS INTRO To get to this sub directory use the following and select the INTRO sub directory as shown in this screen Press OK to enter the directory You will get a files listing with no entries the INTRO sub directory is empty at this point Press the L key to move to the next set of soft menu keys...

Страница 106: ...y To see the contents of the variable in this screen press L VIEW Press the GRAPH A soft menu key to see the contents in a graphical format Press the TEXT A soft menu key to see the contents in text format Press OK to return to the variable list Press once more to return to normal display Variable A should now be featured in the soft menu key labels Using the STO command A simpler way to create a ...

Страница 107: ...reen will look as follows This expression means that the value 0 25 is being stored into α the symbol suggests the operation Press to create the variable The variable is now shown in the soft menu key labels The following are the keystrokes required to enter the remaining variables A12 3V5K a12 Q r Ü m r K q R Ô3 í2 í1 K r z1 3 5 K z1 if needed accept change to Complex mode p1 å é r ì rQ2 K p1 The...

Страница 108: ...abels To enter the value 3 105 into A12 we can use a shorter version of the procedure 3V5 a12 K Here is a way to enter the contents of Q Q r Ü m r q K To enter the value of R we can use an even shorter version of the procedure R Ô3 2 1 r K Notice that to separate the elements of a vector in RPN mode we can use the space key rather than the comma í used above in Algebraic mode z1 3 5 z1 K if needed...

Страница 109: ...a variable Pressing the soft menu key label for the variable This procedure will show the contents of a variable as long as the variable contains a numerical value or an algebraic value or an array For example for the variables listed above press the following keys to see the contents of the variables Algebraic mode Type these keystrokes J z1 R Q At this point the screen looks as follows Next type...

Страница 110: ...c π r 2 In the example shown above r took the value of 5 and thus the value of πr2 π 25 is returned This program therefore calculates the area of a circle given its radius r RPN mode In RPN mode you only need to press the corresponding soft menu key label to get the contents of a numerical or algebraic variable For the case under consideration we can try peeking into the variables z1 R Q A12 α and...

Страница 111: ...e the keystroke combination to list the contents of all variables in the screen For example Press to return to normal calculator display Replacing the contents of variables Replacing the contents of a variable can be thought of as storing a different value in the same variable name Thus the examples for creating variables shown above can be used to illustrate the replacement of a variable s conten...

Страница 112: ...he algebraic expression a b i in level 1 in the stack To enter this result into variable z1 use J z1 To check the new contents of z1 use z1 An equivalent way to do this in Algebraic mode is the following a b K z1 To check the new contents of z1 use z1 Using the ANS 1 variable Algebraic mode In Algebraic mode one can use the ANS 1 variable to replace the contents of a variable For example the proce...

Страница 113: ...abeled PICK DESTINATION Use the up arrow key to select the sub directory MANS and press OK If you now press the screen will show the contents of sub directory MANS notice that variable A is shown in this list as expected Press INTRO Algebraic mode or INTRO RPN mode to return to the INTRO directory Press OK to produce the list of variables in HOME MANS INTRO Use the down arrow key to select variabl...

Страница 114: ...sing the stack in RPN mode To demonstrate the use of the stack in RPN mode to copy a variable from one sub directory to another we assume you are within sub directory HOME MANS INTRO and that we will copy the contents of variable z1 into the HOME directory Use the following procedure z1 z1 This procedure lists the contents and the name of the variable in the stack The calculator screen will look l...

Страница 115: ...contents of the variables use R and Q This procedure can be generalized to the copying of three or more variables Reordering variables in a directory In this section we illustrate the use of the ORDER command to reorder the variables in a directory We assume we start within the sub directory HOME MANS containing the variables A12 R Q z1 A and the sub directory INTRO as shown below Copy A12 from IN...

Страница 116: ...ming menu OK Select DIRECTORY from the MEMORY menu OK Select ORDER from the DIRECTORY menu The result is the following screen Moving variables using the FILES menu To move a variable from one directory to another you can use the FILES menu For example within the sub directory HOME MANS INTRO we have variables p1 z1 R Q A12 α and A Suppose that we want to move variable A12 to sub directory HOME MAN...

Страница 117: ... can be accessed directly by using the TOOLS menu I or by using the FILES menu OK Using the FILES command The FILES command can be used to purge one variable at a time To delete a variable from a given directory you can use the FILES menu For example within the sub directory HOME MANS INTRO we have variables p1 z1 R Q α and A left Suppose that we delete variable A Here is how to do it Press OK to ...

Страница 118: ...following exercise Press I PURGE ä J R í J Q At this point the screen will show the following command ready to be executed To finish deleting the variables press The screen will now show the remaining variables Using function PURGE in the stack in RPN mode We start again at subdirectory HOME MANS INTRO containing variables p1 z1 Q R and α We will use command PURGE to delete variable p1 Press p1 I ...

Страница 119: ... will undo the most recent operation 20 3 leaving the original terms back in the stack To illustrate the use of CMD let s enter the following entries in ALG mode Press after each entry Next use the CMD function to show the four most recent commands entered by the user i e You can use the up and down arrow keys to navigate through these commands and highlight any of them that you want to entry anew...

Страница 120: ...r operates To see the current system flag setting press the H button and then the FLAGS soft menu key i e F1 You will get a screen labeled SYSTEM FLAGS listing flag numbers and the corresponding setting Note In this screen as only system flags are present only the absolute value of the flag number Is displayed A flag is said to be set if you see a check mark in front of the flag number Otherwise t...

Страница 121: ...flag 01 i e select Principal Value Press OK twice to return to normal calculator display We will try solving a quadratic equation solution say t2 5t 6 0 with command QUAD Algebraic mode Use the following keystroke sequence N q use the up and down arrow keys to select command QUAD press OK To enter the equation as the first argument of function QUAD use the following keystrokes O t Q2 5 t 6 Å0 í t ...

Страница 122: ...e the current flag setting by pressing the H button and then the FLAGS soft menu key Make sure to clear system flag 01 which was left set from the previous exercise Use the up and down arrow keys to move about the system flag list Some flags of interest and their preferred value for the purpose of the exercises that follow in this manual are 02 Constant symb Constant values e g π are kept as symbo...

Страница 123: ...tory we used Show PROG menu list and select MEMORY OK Show the MEMORY menu list and select DIRECTORY OK Show the DIRECTORY menu list and select ORDER OK activate the ORDER command There is an alternative way to access these menus as soft MENU keys by setting flag 117 To set this flag try the following H FLAGS The screen shows flag 117 not set CHOOSE boxes as shown here Press the soft menu key to s...

Страница 124: ... The ORDER command is not shown in this screen To find it we use the L key to find it To activate the ORDER command we press the C ORDER soft menu key Although not applied to a specific example this exercise shows the two options for menus in the calculator CHOOSE boxes and soft MENUs Selected CHOOSE boxes Some menus will only produce CHOOSE boxes e g The APPS APPlicationS menu activated with the ...

Страница 125: ...Page 2 67 The HELP menu activated with I L HELP The CMDS CoMmanDS menu activated within the Equation Writer i e O L CMDS ...

Страница 126: ...ple you may see the following setting RAD XYZ DEC R X This stands for RADians for angular measurements XYZ for Rectangular Cartesian coordinates DECimal number base Real numbers preferred means exact results and X is the value of the default independent variable Another possible listing of options could be DEG R Z HEX C t This stands for DEGrees as angular measurements R Z for Polar coordinates HE...

Страница 127: ... the ALGEBRAIC mode is selected there are no numbered stack levels and the word ALG is listed in the top line of the display towards the right hand side The difference between these operating modes was described in detail in Chapter 1 Real number calculations To perform real number calculations it is preferred to have the CAS set to Real as opposite to Complex mode In some cases a complex result m...

Страница 128: ... number or algebraic expression e g Y2 Result or 0 5 In RPN mode enter the number first then use the key e g 4 Y Result or 0 25 Addition subtraction multiplication division Use the proper operation key namely In ALG mode press an operand then an operator then an operand followed by an to obtain a result Examples 3 7 5 2 6 3 8 5 4 2 2 5 2 3 4 5 The first three operations above are shown in the foll...

Страница 129: ...ou to enter the expression like in algebraic mode Ü5 3 2 Ü7 2 2 µ For both ALG and RPN modes using the Equation Writer O5 3 2 7 2 2 The expression can be evaluated within the Equation writer by using EVAL or EVAL Absolute value function The absolute value function ABS is available through the keystroke combination Ê When calculating in the stack in ALG mode enter the function before the argument e...

Страница 130: ...tion XROOT y x is available through the keystroke combination When calculating in the stack in ALG mode enter the function XROOT followed by the arguments y x separated by commas e g 3 í 27 In RPN mode enter the argument y first then x and finally the function call e g 27 3 Base 10 logarithms and powers of 10 Logarithms of base 10 are calculated by the keystroke combination Ã function LOG while it...

Страница 131: ...ne T and tangent U The arguments of these functions are angles therefore they can be entered in any system of angular measure degrees radians grades For example with the DEG option selected we can calculate the following trigonometric functions In ALG mode S30 T45 U135 In RPN mode 30 S 45 T 135 U Inverse trigonometric functions The inverse trigonometric functions available in the keyboard are the ...

Страница 132: ... ABS x Some functions like XROOT require two arguments e g XROOT x y This function has the equivalent keystroke sequence Operators on the other hand are placed after a single argument or between two arguments The factorial operator for example is placed after a number e g 5 2 Since this operator requires a single argument it is referred to as a unary operator Operators that require two arguments s...

Страница 133: ...ed in the next chapter Option 10 CONSTANTS provides access to the constants in the calculator This option will be presented later in this section Finally option 11 SPECIAL FUNCTIONS includes functions of advanced mathematics that will be discussed in this section also In general to apply any of these functions you need to be aware of the number and order of the arguments required and keep in mind ...

Страница 134: ...and its inverse ATANH or tanh 1 This menu contains also the functions EXPM x exp x 1 LNP1 x ln x 1 Finally option 9 MATH returns the user to the MTH menu For example in ALG mode the keystroke sequence to calculate say tanh 2 5 is the following Select MTH menu 4 OK Select the 4 HYPERBOLIC menu 5 OK Select the 5 TANH function 2 5 Evaluate tanh 2 5 The screen shows the following output In the RPN mod...

Страница 135: ... mode right hand side in RPN mode Pressing L shows the remaining options Note Pressing will return to the first set of MTH options Also using the combination will list all menu functions in the screen e g Thus to select for example the hyperbolic functions menu with this menu format press HYP to produce Finally in order to select for example the hyperbolic tangent tanh function simply press TANH N...

Страница 136: ...ect the HYPERBOLIC menu TANH Select the TANH function As an exercise of applications of hyperbolic functions verify the following values SINH 2 5 6 05020 ASINH 2 0 1 4436 COSH 2 5 6 13228 ACOSH 2 0 1 3169 TANH 2 5 0 98661 ATANH 0 2 0 2027 EXPM 2 0 6 38905 LNP1 1 0 0 69314 Once again the general procedure shown in this section can be applied for selecting options in any calculator menu Real number ...

Страница 137: ...alues as follows y x calculates the x percentage of y CH y x calculates 100 y x x i e the percentage change the difference between two numbers T y x calculates 100 x y i e the percentage total the portion that one number x is of another y These functions require two arguments we illustrate the calculation of T 15 45 i e calculation 15 of 45 next We assume that the calculator is set to ALG mode and...

Страница 138: ...lect the 5 T function Note The exercises in this section illustrate the general use of calculator functions having 2 arguments The operation of functions having 3 or more arguments can be generalized from these examples As an exercise for percentage related functions verify the following values 5 20 1 CH 22 25 13 6363 T 500 20 4 Minimum and maximum Use these functions to determine the minimum or m...

Страница 139: ...e fractional part of a real number As an exercise verify that ABS 3 3 3 SIGN 5 1 MANT 2540 2 540 XPON 2540 3 IP 2 35 2 FP 2 35 0 35 Rounding truncating floor and ceiling functions RND x y rounds up y to x decimal places TRNC x y truncate y to x decimal places FLOOR x closest integer that is less than or equal to x CEIL x closest integer that is greater than or equal to x As an exercise verify that...

Страница 140: ...s enter the number first followed by the sequence 2 Example 5 2 The Gamma function defined above has the property that Γ α α 1 Γ α 1 for α 1 Therefore it can be related to the factorial of a number i e Γ α α 1 when α is a positive integer We can also use the factorial function to calculate the Gamma function and vice versa For example Γ 5 4 or 4 2 The factorial function is available in the MTH men...

Страница 141: ...y your calculator e the base of natural logarithms i the imaginary unit ii 2 1 π the ratio of the length of the circle to its diameter MINR the minimum real number available to the calculator MAXR the maximum real number available to the calculator To have access to these constants select option 11 CONSTANTS in the MTH menu The constants are listed as follows Selecting any of these entries will pl...

Страница 142: ...TS menu The units menu is launched by the keystroke combination Û associated with the 6 key With system flag 117 set to CHOOSE boxes the result is the following menu Option 1 Tools contains functions used to operate on units discussed later Options 3 Length through 17 Viscosity contain menus with a number of units for each of the quantities described For example selecting option 8 Force shows the ...

Страница 143: ...ess L to move to the next menu page Pressing on the appropriate soft menu key will open the sub menu of units for that particular selection For example for the SPEED sub menu the following units are available Pressing the soft menu key UNITS will take you back to the UNITS menu Recall that you can always list the full menu labels in the screen by using e g for the ENRG set of units the following l...

Страница 144: ...ard ft 3 cubic feet in 3 cubic inch l liter galUK UK gallon galC Canadian gallon gal US gallon qt quart pt pint ml mililiter cu US cup ozfl US fluid ounce ozUK UK fluid ounce tbsp tablespoon tsp teaspoon bbl barrel bu bushel pk peck fbm board foot TIME yr year d day h hour min minute s second Hz hertz SPEED m s meter per second cm s centimeter per second ft s feet per second kph kilometer per hour...

Страница 145: ...gree Celsius o F degree Fahrenheit K Kelvin o R degree Rankine ELECTRIC CURRENT Electric measurements V volt A ampere C coulomb Ω ohm F farad W watt Fdy faraday H henry mho mho S siemens T tesla Wb weber ANGLE planar and solid angle measurements o sexagesimal degree r radian grad grade arcmin minute of arc arcs second of arc sr steradian LIGHT Illumination measurements fc footcandle flam footlambe...

Страница 146: ...hese units are also accessible through the catalog for example gmol N g lbmol N l rpm N r dB N d Converting to base units To convert any of these units to the default units in the SI system use the function UBASE For example to find out what is the value of 1 poise unit of viscosity in the SI units use the following In ALG mode system flag 117 set to CHOOSE boxes Û Select the UNITS menu OK Select ...

Страница 147: ... Select the TOOLS menu UBASE Select the UBASE function 1 Ý Enter 1 and underline Û Select the UNITS menu VISC Select the VISCOSITY option P Select the unit P poise Convert the units In RPN mode system flag 117 set to SOFT menus 1 Enter 1 no underline Û Select the UNITS menu VISC Select the VISCOSITY option P Select the unit P poise Û Select the UNITS menu TOOLS Select the TOOLS menu UBASE Select t...

Страница 148: ...r in RPN mode use the following keystrokes 5 Enter number do not enter underscore Û Access the UNITS menu 8 OK Select units of force 8 Force OK Select Newtons N Notice that the underscore is entered automatically when the RPN mode is active The result is the following screen As indicated earlier if system flag 117 is set to SOFT menus then the UNITS menu will show up as labels for the soft menu ke...

Страница 149: ...according to the following table of prefixes from the SI system The prefix abbreviation is shown first followed by its name and by the exponent x in the factor 10x corresponding to each prefix ___________________________________________________ Prefix Name x Prefix Name x ____________________________________________________ Y yotta 24 d deci 1 Z zetta 21 c centi 2 E exa 18 m milli 3 P peta 15 µ mi...

Страница 150: ...nts of functions say SQ or SIN Thus attempting to calculate LN 10_m will produce an error message Error Bad Argument Type Here are some calculation examples using the ALG operating mode Be warned that when multiplying or dividing quantities with units you must enclosed each quantity with its units between parentheses Thus to enter for example the product 12 5m 5 2_yd type it to read 12 5_m 5 2_yd ...

Страница 151: ...can be performed in ALG mode without using parentheses e g 5 m 3200 mm can be entered simply as 5_m 3200_mm More complicated expression require the use of parentheses e g 12_mm 1_cm 2 2_s Stack calculations in the RPN mode do not require you to enclose the different terms in parentheses e g 12_m 1 5_yd 3250_mi 50_h These operations produce the following output ...

Страница 152: ...to SI units UVAL x extract the value from unit object x UFACT x y factors a unit y from unit object x UNIT x y combines value of x with units of y The UBASE function was discussed in detail in an earlier section in this chapter To access any of these functions follow the examples provided earlier for UBASE Notice that while function UVAL requires only one argument functions CONVERT UFACT and UNIT ...

Страница 153: ...o convert 33 watts to btu s CONVERT 33_W 1_hp CONVERT 33_W 11_hp These operations are shown in the screen as Examples of UVAL UVAL 25_ft s UVAL 0 021_cm 3 Examples of UFACT UFACT 1_ha 18_km 2 UFACT 1_mm 15 1_cm Examples of UNIT UNIT 25 1_m UNIT 11 3 1_mph ...

Страница 154: ...ry activated with the command CONLIB To launch this command you could simply type it in the stack conlib or you can select the command CONLIB from the command catalog as follows First launch the catalog by using N c Next use the up and down arrow keys to select CONLIB Finally press the F OK soft menu key Press if needed The constants library screen will look like the following use the down arrow k...

Страница 155: ...UE when selected constants are shown without units STK copies value with or without units to the stack QUIT exit constants library Active only if the function VALUE is active This is the way the top of the CONSTANTS LIBRARY screen looks when the option VALUE is selected units in the SI system To see the values of the constants in the English or Imperial system press the ENGL option If we de select...

Страница 156: ...ame operation in RPN mode will require the following keystrokes after the value of Vm was extracted from the constants library 2 Special physical functions Menu 117 triggered by using MENU 117 in ALG mode or 117 MENU in RPN mode produces the following menu labels listed in the display by using The functions include ZFACTOR gas compressibility Z factor function FANNING Fanning friction factor for f...

Страница 157: ... Function ZFACTOR Function ZFACTOR calculates the gas compressibility correction factor for nonideal behavior of hydrocarbon gas The function is called by using ZFACTOR xT yP where xT is the reduced temperature i e the ratio of actual temperature to pseudo critical temperature and yP is the reduced pressure i e the ratio of the actual pressure to the pseudo critical pressure The value of xT must b...

Страница 158: ...ame units as T0 if any Otherwise it returns simply the difference in numbers For example The purpose of this function is to facilitate the calculation of temperature differences given temperatures in different units Otherwise it s simply calculates a subtraction e g Function TINC Function TINC T0 T calculates T0 DT The operation of this function is similar to that of function TDELTA in the sense t...

Страница 159: ...pe the expression in the right hand side for each separate value In the following example we assume you have set your calculator to ALG mode Enter the following sequence of keystrokes à h Ü x Å x 1 x The screen will look like this Press the J key and you will notice that there is a new variable in your soft menu key H To see the contents of this variable press H The screen will show now Thus the v...

Страница 160: ...e to activate the function enter the argument first then press the soft menu key corresponding to the variable name H For example you could try 2 H The other examples shown above can be entered by using 1 2 H 2 3 H Functions can have more than 2 arguments For example the screen below shows the definition of the function K α β α β and its evaluation with the arguments K 2 π and K 1 2 2 3 The conten...

Страница 161: ...he command DEF f x IFTE x 0 x 2 1 2 x 1 then press In RPN mode type the function definition between apostrophes f x IFTE x 0 x 2 1 2 x 1 then press à Press J to recover your variable menu The function f should be available in your soft key menu Press f to see the resulting program x IFTE x 0 x 2 1 2 x 1 To evaluate the function in ALG mode type the function name f followed by the number at which y...

Страница 162: ...Page 3 37 Define this function by any of the means presented above and check that g 3 3 g 1 0 g 1 0 g 3 9 ...

Страница 163: ...plex number can also be represented in polar coordinates polar representation as z re iθ r cosθ i r sinθ where r z 2 2 y x is the magnitude of the complex number z and θ Arg z arctan y x is the argument of the complex number z The relationship between the Cartesian and polar representation of complex numbers is given by the Euler formula e iθ cos θ i sin θ The complex conjugate of a complex number...

Страница 164: ...een results after entering these complex numbers In RPN mode these numbers will be entered using the following keystrokes Ü3 5 í1 2 Notice that the change sign keystroke is entered after the number 1 2 has been entered in the opposite order as the ALG mode exercise and 3 5 1 2 Notice the need to enter an apostrophe before typing the number 3 5 1 2i in RPN mode The resulting RPN screen will be Noti...

Страница 165: ... the catalog N A complex number in polar representation is written as z r eiθ You can enter this complex number into the calculator by using an ordered pair of the form r θ The angle symbol can be entered as 6 For example the complex number z 5 2e1 5i can be entered as follows the figures show the stack before and after entering the number Because the coordinate system is set to rectangular or Car...

Страница 166: ... imaginary part 2 Try the following operations on your own 5 2i 3 4i 2 6 3 i 2 4i 2 14 5 2i 3 4i 0 28 1 04 1 3 4i 0 12 0 16 Notes The product of two numbers is represented by x1 iy1 x2 iy2 x1x2 y1y2 i x1y2 x2y1 The division of two complex numbers is accomplished by multiplying both numerator and denominator by the complex conjugate of the denominator i e 2 2 2 2 2 1 1 2 2 2 2 2 2 1 2 1 2 2 2 2 2 2...

Страница 167: ...rough the CMPLX menus detailed later The CMPLX menus There are two CMPLX CoMPLeX numbers menus available in the calculator One is available through the MTH menu introduced in Chapter 3 and one directly into the keyboard ß The two CMPLX menus are presented next CMPLX menu through the MTH menu Assuming that system flag 117 is set to CHOOSE boxes see Chapter 2 the CMPLX sub menu within the MTH menu i...

Страница 168: ...z NEG Changes the sign of z CONJ z Produces the complex conjugate of z Examples of applications of these functions are shown next Recall that for ALG mode the function must precede the argument while in RPN mode you enter the argument first and then select the function Also recall that you can get these functions as soft menus by changing the setting of system flag 117 See Chapter 3 This first scr...

Страница 169: ...h the 1 key i e ß With system flag 117 set to CHOOSE boxes the keyboard CMPLX menu shows up as the following screens The resulting menu include some of the functions already introduced in the previous section namely ARG ABS CONJ IM NEG RE and SIGN It also includes function i which serves the same purpose as the keystroke combination i e to enter the unit imaginary number i in an expression The key...

Страница 170: ...x numbers the arguments are no longer angles Therefore the angular measure selected for the calculator has no bearing in the calculation of these functions with complex arguments To understand the way that trigonometric functions and other functions are defined for complex numbers consult a book on complex variables Functions from the MTH menu The hyperbolic functions and their inverses as well as...

Страница 171: ... a straight line Function DROITE takes as argument two complex numbers say x1 iy1 and x2 iy2 and returns the equation of the straight line say y a bx that contains the points x1 y1 and x2 y2 For example the line between points A 5 3 and B 6 2 can be found as follows example in Algebraic mode Function DROITE is found in the command catalog N Using EVAL ANS 1 simplifies the result to ...

Страница 172: ...gebraic objects can be created by typing the object between single quotes directly into stack level 1 or by using the equation writer O For example to enter the algebraic object π D 2 4 directly into stack level 1 use ì dQ2 4 The resulting screen is shown next for both the ALG mode left hand side and the RPN mode right hand side An algebraic object can also be built in the Equation Writer and then...

Страница 173: ... a couple of objects say π R 2 and g t 2 4 and store them in variables A1 and A2 See Chapter 2 to learn how to create variables and store values in them Here are the keystrokes for storing variables A1 in ALG mode ì rQ2 K a1 resulting in The keystrokes corresponding to RPN mode are ì r Q2 a1 K After storing the variable A2 and pressing the key the screen will show the variables as follows In ALG m...

Страница 174: ...OSE boxes the ALG menu shows the following functions Rather than listing the description of each function in this manual the user is invited to look up the description using the calculator s help facility I L HELP To locate a particular function type the first letter of the function For example for function COLLECT we type c then use the up and down arrow keys to locate COLLECT within the help win...

Страница 175: ...ing information for EXPAND The help facility provides not only information on each command but also provides an example of its application This example can be copied onto your stack by pressing the ECHO soft menu key For example for the EXPAND entry shown above press the ECHO soft menu key to get the following example copied to the stack press to execute the command Thus we leave for the user to e...

Страница 176: ...Page 5 5 The help facility will show the following information on the commands COLLECT EXPAND FACTOR LNCOLLECT LIN PARTFRAC SOLVE SUBST TEXPAND ...

Страница 177: ...lished by using the associated with the I key For example in ALG mode the following entry will substitute the value x 2 in the expression x x2 The figure to the left shows the way to enter the expression the substituted value x 2 must be enclosed in parentheses before pressing After the key is pressed the result is shown in the right hand figure In RPN mode this can be accomplished by entering fir...

Страница 178: ...fferent approach to substitution consists in defining the substitution expressions in calculator variables and placing the name of the variables in the original expression For example in ALG mode store the following variables Then enter the expression A B The last expression entered is automatically evaluated after pressing the key producing the result shown above Operations with transcendental fu...

Страница 179: ...us in more detail Expansion and factoring using log exp functions The Ð produces the following menu Information and examples on these commands are available in the help facility of the calculator Some of the command listed in the EXP LN menu i e LIN LNCOLLECT and TEXPAND are also contained in the ALG menu presented earlier Functions LNP1 and EXPM were introduced in menu HYPERBOLIC under the MTH me...

Страница 180: ...menu is the HYPERBOLIC menu whose functions were introduced in Chapter 2 Functions in the ARITHMETIC menu The ARITHMETIC menu contains a number of sub menus for specific applications in number theory integers polynomials etc as well as a number of functions that apply to general arithmetic operations The ARITHMETIC menu is triggered through the keystroke combination Þ associated with the 1 key Wit...

Страница 181: ...ivating the ARITHMETIC menu Þ under these circumstances produces Following we present the help facility entries for the functions of options 5 through 9 in the ARITHMETIC menu DIVIS FACTORS LGCD Greatest Common Denominator PROPFRAC proper fraction SIMP2 The functions associated with the ARITHMETIC submenus INTEGER POLYNOMIAL MODULO and PERMUTATION are the following INTEGER menu EULER Number of int...

Страница 182: ...s EGDC Returns u v from au bv gcd a b FACTOR Factorizes an integer number or a polynomial FCOEF Generates fraction given roots and multiplicity FROOTS Returns roots and multiplicity given a fraction GCD Greatest common divisor of 2 numbers or polynomials HERMITE n th degree Hermite polynomial HORNER Horner evaluation of a polynomial LAGRANGELagrange polynomial interpolation LCM Lowest common multi...

Страница 183: ...ome of the background necessary for application of the ARITHMETIC menu functions Definitions are presented next regarding the subjects of polynomials polynomial fractions and modular arithmetic The examples presented below are presented independently of the calculator setting ALG or RPN Modular arithmetic Consider a counting system of integer numbers that periodically cycles back on itself and sta...

Страница 184: ...The rule for subtraction will be such that if j k 0 then j k is defined as j k n Therefore 8 10 2 mod 12 is read eight minus ten is congruent to two modulus twelve Other examples of subtraction in modulus 12 arithmetic would be 10 5 5 mod 12 6 9 9 mod 12 5 8 9 mod 12 5 10 7 mod 12 etcetera Multiplication follows the rule that if j k n so that j k m n r where m and r are nonnegative integers both l...

Страница 185: ...ollow the rules presented earlier For example 17 5 mod 6 and 21 3 mod 6 Using these rules we can write 17 21 5 3 mod 6 38 8 mod 6 38 2 mod 6 17 21 5 3 mod 6 4 2 mod 6 17 21 5 3 mod 6 357 15 mod 6 357 3 mod 6 Notice that whenever a result in the right hand side of the congruence symbol produces a result that is larger than the modulo in this case n 6 you can always subtract a multiple of the modulo...

Страница 186: ...nd SUBTMOD Brief descriptions of these functions were provided in an earlier section Next we present some applications of these functions Setting the modulus or MODULO The calculator contains a variable called MODULO that is placed in the HOME CASDIR directory and will store the magnitude of the modulus to be used in modular arithmetic The default value of MODULO is 13 To change the value of MODUL...

Страница 187: ...exist 26 12 mod 12 does not exist 125 17 mod 12 1 with remainder 0 68 7 4 mod 12 with remainder 0 7 5 1 mod 12 with remainder 0 Note DIVMOD provides the quotient of the modular division j k mod n while DIMV2MOD provides no only the quotient but also the remainder of the modular division j k mod n POWMOD examples 23 4 mod 12 35 3 mod 12 510 1 mod 12 118 1 mod 12 62 0 mod 12 99 3 mod 12 In the examp...

Страница 188: ...d 12 1 7 5 mod 12 1 3 mod 12 does not exist 1 11 1 mod 12 The MOD operator The MOD operator is used to obtain the ring number of a given modulus corresponding to a given integer number On paper this operation is written as m mod n p and is read as m modulo n is equal to p For example to calculate 15 mod 8 enter ALG mode 15 MOD 8 RPN mode 15 8 MOD The result is 7 i e 15 mod 8 7 Try the following ex...

Страница 189: ...s of a polynomial values of X for which P X 0 Poles of a fraction roots of the denominator Multiplicity of roots or poles the number of times a root shows up e g P X X 1 2 X 3 has roots 1 3 with multiplicities 2 1 Cyclotomic polynomial Pn X a polynomial of order EULER n whose roots are the primitive n th roots of unity e g P2 X X 1 P4 X X2 1 Bézout s polynomial equation A X U X B X V X C X Specifi...

Страница 190: ...y if x a is any solution then all other solutions are congruent to a modulo equal to the product m1 m2 mr The EGCD function EGCD stands for Extended Greatest Common Divisor Given two polynomials A X and B X function EGCD produces the polynomials C X U X and V X so that C X U X A X V X B X For example for A X X 2 1 B X X 2 1 EGCD A X B X 2 1 1 i e 2 1 X 2 1 1 X 2 1 Also EGCD X 3 2 X 5 X 5 X 2 2 1 i...

Страница 191: ...re the polynomial P X and the number a The function returns the quotient polynomial Q X that results from dividing P X by X a the value of a and the value of P a in that order In other words P X Q X X a P a For example HORNER X 3 2 X 2 3 X 1 2 X 2 4 X 5 2 11 We could therefore write X3 2X2 3X 1 X2 4X 5 X 2 11 A second example HORNER X 6 1 5 X 5 5 X 4 25 X 3 125 X 2 625 X 3125 5 15624 i e X6 1 X5 5...

Страница 192: ... x y x y x y y y x x x x y x x x x x p Check this result with your calculator LAGRANGE x1 x2 y1 y2 y1 y2 X y2 x1 y1 x2 x1 x2 Other examples LAGRANGE 1 2 3 2 8 15 X 2 9 X 6 2 LAGRANGE 0 5 1 5 2 5 3 5 4 5 12 2 13 5 19 2 27 3 32 5 1375 X 4 7666666666667 X 3 74375 X 2 1 991666666667 X 12 92265625 Note Matrices are introduced in Chapter 10 The LCM function The function LCM Least Common Multiple obtains...

Страница 193: ... 0 which represents the polynomial X6 X5 5X4 5X3 4X2 4X The PROOT function Given an array containing the coefficients of a polynomial in decreasing order the function PROOT provides the roots of the polynomial Example from X2 5X 6 0 PROOT 1 5 6 2 3 The PTAYL function Given a polynomial P X and a number a the function PTAYL is used to obtain an expression Q X a P X i e to develop a polynomial in po...

Страница 194: ... variable EPS with default value 0 0000000001 10 10 when you use the EPSX0 function You can change this value once created if you prefer a different value for EPS The function EPSX0 when applied to a polynomial will replace all coefficients whose absolute value is less than EPS with a zero Function EPSX0 is not available in the ARITHMETIC menu it must be accessed from the function catalog N Exampl...

Страница 195: ... X Y X 2 4 X 4 EXPAND X X Y X 2 1 X 2 Y X X 2 1 EXPAND 4 2 X 1 3 X 2 X 3 5 X 2 2 X 5 4 X 4 10 X 3 14 X 2 5 X X 4 X 3 6 X 2 FACTOR 3 X 3 2 X 2 X 2 5 X 6 X 2 3 X 2 X 2 X 3 FACTOR X 3 9 X X 2 5 X 6 X X 3 X 2 FACTOR X 2 1 X 3 Y Y X 1 X 2 X 1 Y The SIMP2 function Functions SIMP2 and PROPFRAC are used to simplify a fraction and to produce a proper fraction respectively Function SIMP2 takes as arguments ...

Страница 196: ...on The function FCOEF is used to obtain a rational fraction given the roots and poles of the fraction Note If a rational fraction is given as F X N X D X the roots of the fraction result from solving the equation N X 0 while the poles result from solving the equation D X 0 The input for the function is a vector listing the roots followed by their multiplicity i e how many times a given root is rep...

Страница 197: ...es 2 5 respectively and the roots are 0 2 5 with multiplicities 3 1 2 respectively Another example is FROOTS X 2 5 X 6 X 5 X 2 0 2 1 1 3 1 2 1 i e poles 0 2 1 1 and roots 3 1 2 1 If you have had Complex mode selected then the results would be 0 2 1 1 1 i 3 2 1 1 i 3 2 1 Step by step operations with polynomials and fractions By setting the CAS modes to Step step the calculator will show simplificat...

Страница 198: ... menu summarizes all conversion menus in the calculator The list of these menus is shown next The functions available in each of the sub menus are shown next UNITS convert menu Option 1 This menu is the same as the UNITS menu obtained by using Û The applications of this menu are discussed in detail in Chapter 3 ...

Страница 199: ...f this menu are discussed in detail in this Chapter MATRICES convert menu Option 5 This menu contains the following functions These functions are discussed in detail in Chapter 10 REWRITE convert menu Option 4 This menu contains the following functions Functions I R and R I are used to convert a number from integer I to real R or vice versa Integer numbers are shown without trailing decimal points...

Страница 200: ... of π if a fraction of π can be found for the number otherwise it converts the number to a fraction Examples are of these three functions are shown next Out of the functions in the REWRITE menu functions DISTRIB EXPLN EXP2POW FDISTRIB LIN LNCOLLECT POWEREXPAND and SIMPLIFY apply to algebraic expressions Many of these functions are presented in this Chapter However for the sake of completeness we p...

Страница 201: ...Page 5 30 LIN LNCOLLECT POWEREXPAND SIMPLIFY ...

Страница 202: ...ns DESOLVE and LDEC are used for the solution of differential equations the subject of a different chapter and therefore will not be presented here Similarly function LINSOLVE relates to the solution of multiple linear equations and it will be presented in a different chapter Functions ISOL and SOLVE can be used to solve for any unknown in a polynomial equation Function SOLVEVX solves a polynomial...

Страница 203: ...SOL can be an expression as shown above or an equation For example in ALG mode try Note To type the equal sign in an equation use Å associated with the key The same problem can be solved in RPN mode as illustrated below figures show the RPN stack before and after the application of function ISOL Function SOLVE Function SOLVE has the same syntax as function ISOL except that SOLVE can also be used t...

Страница 204: ...olution is not visible because the result occupies more characters than the width of the calculator s screen However you can still see all the solutions by using the down arrow key which triggers the line editor this operation can be used to access any output line that is wider than the calculator s screen The corresponding RPN screens for these two examples before and after the application of fun...

Страница 205: ...e following screens show the RPN stack for solving the two examples shown above before and after application of SOLVEVX The equation used as argument for function SOLVEVX must be reducible to a rational expression For example the following equation will not processed by SOLVEVX Function ZEROS The function ZEROS finds the solutions of a polynomial equation without showing their multiplicity The fun...

Страница 206: ...ver menu The calculator provides a very powerful environment for the solution of single algebraic or transcendental equations To access this environment we start the numerical solver NUM SLV by using Ï This produces a drop down menu that includes the following options Item 2 Solve diff eq is to be discussed in a later chapter on differential equations Item 4 Solve lin sys will be discussed in a la...

Страница 207: ...in an algebraic expression for the polynomial as a function of X Finding the solutions to a polynomial equation A polynomial equation is an equation of the form anxn an 1xn 1 a1x a0 0 The fundamental theorem of algebra indicates that there are n solutions to any polynomial equation of order n Some of the solutions could be complex numbers nevertheless As an example solve the equation 3s4 2s3 s 1 0...

Страница 208: ...e The fundamental theorem of algebra indicates that there are n solutions for any polynomial equation of order n There is another theorem of algebra that indicates that if one of the solutions to a polynomial equation with real coefficients is a complex number then the conjugate of that number is also a solution In other words complex solutions to a polynomial equation with real coefficients come ...

Страница 209: ...omplex coefficients Generating an algebraic expression for the polynomial You can use the calculator to generate an algebraic expression for a polynomial given the coefficients or the roots of the polynomial The resulting expression will be given in terms of the default CAS variable X The examples below shows how you can replace X with any other variable by using the function To generate the algeb...

Страница 210: ...icients highlighted For example for this case try Ï OK Select Solve poly Ô1 í3 Enter vector of roots í2 í 1 OK SOLVE Solve for coefficients SYMB Generate symbolic expression Return to stack The expression thus generated is shown in the stack as X 4 3 X 3 3 X 2 11 X 6 X 0 The coefficients are listed in stack level 2 Financial calculations The calculations in item 5 Solve finance in the Numerical So...

Страница 211: ...be worth at the end of n periods Typically payment occurs at the end of each period so that the borrower starts paying at the end of the first period and pays the same fixed amount at the end of the second third etc up to the end of the n th period Example 1 Calculating payment on a loan If 2 million are borrowed at an annual interest rate of 6 5 to be repaid in 60 monthly payments what should be ...

Страница 212: ...7 79 in the 5 years that his money is used to finance the borrower s project Example 2 Calculating amortization of a loan The same solution to the problem in Example 1 can be found by pressing AMOR which is stands for AMORTIZATION This option is used to calculate how much of the loan has been amortized at the end of a certain number of payments Suppose that we use 24 periods in the first line of t...

Страница 213: ...ill be solving for it 0 OK Enter FV 0 the option End is highlighted CHOOS OK Change payment option to Begin š SOLVE Highlight PMT and solve for it The screen now shows the value of PMT as 38 921 47 i e the borrower must pay the lender US 38 921 48 at the beginning of each month for the next 60 months to repay the entire amount Notice that the amount the borrower pays monthly if paying at the begin...

Страница 214: ...these variables for future use or use the PURGE function to erase them from your directory To erase all of the variables at once if using ALG mode try the following I PURGE J ä Enter PURGE prepare list of variables n Enter name of variable N í Enter a comma I YR Enter name of variable I YR í Enter a comma PV Enter name of variable PV í Enter a comma PMT Enter name of variable PMT í Enter a comma P...

Страница 215: ...V menu provides item 1 Solve equation solve different types of equations in a single variable including non linear algebraic and transcendental equations For example let s solve the equation ex sin πx 3 0 Simply enter the expression as an algebraic object and store it into variable EQ The required keystrokes in ALG mode are the following x S ì x 3 Å 0 K e q Function STEQ Function STEQ available th...

Страница 216: ...ss SOLVE The solution shown is X 4 5006E 2 This however is not the only possible solution for this equation To obtain a negative solution for example enter a negative number in the X field before solving the equation Try 3 OK SOLVE The solution is now X 3 045 Solution procedure for Equation Solve The numerical solver for single unknown equations works as follows It lets the user type in or CHOOS a...

Страница 217: ...nknown input field Examples of the equations solutions are shown following Example 1 Hooke s law for stress and strain The equation to use is Hooke s law for the normal strain in the x direction for a solid particle subjected to a state of stress given by zz zy zx yz yy yx xz xy xx σ σ σ σ σ σ σ σ σ The equation is 1 T n E e zz yy xx xx α σ σ σ here exx is the unit strain in the x direction σxx σy...

Страница 218: ...variables i e exx is translated as ex etc this is done to save typing time Use the following shortcuts for special characters σ s α a c and recall that lower case letters are entered by using before the letter key thus x is typed as x Press to return to the solver screen Enter the values proposed above into the corresponding fields so that the solver screen looks like this With the ex field highli...

Страница 219: ... results of the calculations performed within the numerical solver environment have been copied to the stack Also you will see in your soft menu key labels variables corresponding to those variables in the equation stored in EQ press L to see all variables in your directory i e variables ex T α σz σy n σx and E Example 2 Specific energy in open channel flow Specific energy in an open channel is de...

Страница 220: ... Q g m and b as follows First create a sub directory called SPEN SPecific ENergy and work within that sub directory Next define the following variables Launch the numerical solver for solving equations Ï OK Notice that the input form contains entries for the variables y Q b m and g Try the following input data E 10 ft Q 10 cfs cubic feet per second b 2 5 ft m 1 0 g 32 2 ft s2 Solve for y ...

Страница 221: ...n NUM SLV is activated the substitutions implied by the auxiliary variables are implemented and the input screen for the equation provides input field for the primitive or fundamental variables resulting from the substitutions The example also illustrates an equation that has more than one solution and how choosing the initial guess for the solution may produce those different solutions In the nex...

Страница 222: ...ulate the friction factor f The function DARCY can be found through the command catalog For example for ε D 0 0001 Re 1000000 you can find the friction factor by using DARCY 0 0001 1000000 In the following screen the function NUM was used to obtain a numerical value of the function The result is f DARCY 0 0001 1000000 0 01341 The function FANNING ε D Re In aerodynamics applications a different fri...

Страница 223: ...in the following equation into EQ Also enter the following variables f A V Re In this case we stored the main equation Darcy Weisbach equation into EQ and then replaced several of its variables by other expressions through the definition of variables f A V and Re To see the combined equation use EVAL EQ In this example we changed the display setting so that we can see the entire equation in the sc...

Страница 224: ... 0 00001 m Q 0 05 m3 s Nu 0 000001 m2 s L 20 m and g 9 806 m s2 find the diameter D Enter the input values and solve for D The solution is 0 12 i e D 0 12 m If the equation is dimensionally consistent you can add units to the input values as shown in the figure below However you must add those units to the initial guess in the solution Thus in the example below we place 0_m in the D field before s...

Страница 225: ...ough the use of the function CONST in the calculator by using We can solve for any term in the equation except G by entering the equation as This equation is then stored in EQ Launching the numerical solver for this equation results in an input form containing input fields for F G m1 m2 and r Let s solve this problem using units with the following values for the known variables m1 1 0 106 kg m2 1 ...

Страница 226: ...quation to be solved directly into variable EQ before activating the numerical solver You can actually type the equation to be solved directly into the solver after activating it by editing the contents of the EQ field in the numerical solver input form If variable EQ has not been defined previously when you launch the numerical solver Ï OK the EQ field will be highlighted At this point you can ei...

Страница 227: ... to be entered into EQ This means that your equation would have to have been stored in a variable name previously to activating the numerical solver For example suppose that we have entered the following equations into variables EQ1 and EQ2 Now launch the numerical solver Ï OK and highlight the EQ field At this point press the CHOOS soft menu key Use the up and down arrow keys to select say variab...

Страница 228: ...ng functions and sub menus Function ROOT Function ROOT is used to solve an equation for a given variable with a starting guess value In RPN mode the equation will be in stack level 3 while the variable name will be located in level 2 and the initial guess in level 1 The following figure shows the RPN stack before and after activating function ROOT In ALG mode you would use ROOT TAN θ θ θ 5 to acti...

Страница 229: ...s t The result is now t 4 0000000003 To verify this result press the soft menu key labeled EXPR which evaluates the expression in EQ for the current value of t The results in this case are To exit the SOLVR environment press J The access to the SOLVE menu is lost at this point so you have to activate it once more as indicated earlier to continue with the exercises below Example 2 Solving the equat...

Страница 230: ...ill be listed in the top part of the display You can enter values for the variables a b and c say 2 a 5 b 19 c Also since we can only solve one equation at a time let s enter a guess value for Y say 0 Y and solve for X by using X This gives the value X 9 4999 To check the value of the equation at this point press EXPR The results are Left 19 Right 19 To solve the next equation press L NEXQ The scr...

Страница 231: ... list as the new guess i e use the format number A list of numbers can be given as a guess for a variable In this case the units takes the units used belong to the last number in the list For example entering 1 41_ft 1_cm 1_m indicates that meters m will be used for that variable The expression used in the solution must have consistent units or an error will result when trying to solve for a value...

Страница 232: ...ector whose roots are given by 1 2 2 1 0 will produce the following coefficients 1 4 3 4 4 0 The polynomial is x5 4x4 3x3 4x2 4x Function PEVAL This function evaluates a polynomial given a vector of its coefficients an an 1 a2 a1 a0 and a value x0 i e PEVAL calculates anx0 n an 1x0 n 1 a2x0 2 a1x0 a0 For example for coefficients 2 3 1 2 and a value of 2 PEVAL returns the value 28 The SYS sub menu ...

Страница 233: ...he other functions available Function TVMROOT This function requires as argument the name of one of the variables in the TVM problem The function returns the solution for that variable given that the other variables exist and have values stored previously For example having solved a TVM problem above we can solve for say N as follows n TVMRO The result is 10 Function AMORT This function takes a va...

Страница 234: ...Page 6 33 Function BEG If selected the TMV calculations use payments at the beginning of each period If deselected the TMV calculations use payments at the end of each period ...

Страница 235: ...or The list of variables to solve for must also be provided as a vector Make sure that the CAS is set to mode Exact before attempting a solution using this procedure Also the more complicated the expressions the longer the CAS takes in solving a particular system of equations Examples of this application follow Example 1 Projectile motion Use function SOLVE with the following vector arguments the ...

Страница 236: ... equations listed separately in the stack Note This method worked fine in this example because the unknowns t and y0 were algebraic terms in the equations This method would not work for solving for θ0 since θ0 belongs to a transcendental term Example 2 Stresses in a thick wall cylinder Consider a thick wall cylinder for inner and outer radius a and b respectively subject to an inner pressure Pi an...

Страница 237: ... stack and adding and subtracting them Here is how to do it with the equation writer Enter and store term T1 Enter and store term T2 Notice that we are using the RPN mode in this example however the procedure in the ALG mode should be very similar Create the equation for σθθ J T1 T2 s t Å Create the equation for σrr J T1 T2 s r Å Put together a vector with the two equations using function ARRY fin...

Страница 238: ...ion OBJ The result is These two examples constitute systems of linear equations that can be handled equally well with function LINSOLVE see Chapter 11 The following example shows function SOLVE applied to a system of polynomial equations Example 3 System of polynomial equations The following screen shot shows the solution of the system X2 XY 10 X2 Y2 5 using function SOLVE Solution to simultaneous...

Страница 239: ...s example In ALG mode press ECHO to copy the example to the stack press to run the example To see all the elements in the solution you need to activate the line editor by pressing the down arrow key In RPN mode the solution for this example is produced by using Activating function MSLV results in the following screen You may have noticed that while producing the solution the screen shows intermedi...

Страница 240: ...12 P is the wetted perimeter of the cross section m or ft So is the slope of the channel bed expressed as a decimal fraction For a trapezoidal channel as shown below the area is given by y my b A while the wetted perimeter is given by 2 1 2 m y b P where b is the bottom width m or ft and m is the side slope 1V mH of the cross section Typically one has to solve the equations of energy and Manning s...

Страница 241: ...ont option selected We can see that these equations are indeed given in terms of the primitive variables b m y g So n Cu Q and Ho In order to solve for y and Q we need to give values to the other variables Suppose we use H0 5 ft b 1 5 ft m 1 n 0 012 S0 0 00001 g 32 2 and Cu 1 486 Before being able to use MSLV for the solution we need to enter these values into the corresponding variable names This...

Страница 242: ...o the value of Ho which is the maximum value that y can take and Q 10 this is a guess To obtain the solution we select function MSLV from the NUM SLV menu e g Ï6 OK to place the command in the screen Next we ll enter variable EQS LL EQS followed by vector y Q í Ô y í q and by the initial guesses í Ô5 í 10 Before pressing the screen will look like this Press to solve the system of equations You may...

Страница 243: ...alue close to zero At that point a numerical solution would have been found The screen after MSLV finds a solution will look like this The result is a list of three vectors The first vector in the list will be the equations solved The second vector is the list of unknowns The third vector represents the solution To be able to see these vectors press the down arrow key to activate the line editor T...

Страница 244: ...he RPN mode Application 1 Solution of triangles In this section we use one important application of trigonometric functions calculating the dimensions of a triangle The solution is implemented in the calculator using the Multiple Equation Solver or MES Consider the triangle ABC shown in the figure below A b B a C c α β ｙ The sum of the interior angles of any triangle is always 180o i e α β γ 180o ...

Страница 245: ... Then it searches for another equation that can be solved for the next unknowns and so on until all unknowns have been solved for Creating a working directory We will use the MES to solve for triangles by creating a list of equations corresponding to the sine and cosine laws the law of the sum of interior angles and Heron s formula for the area First create a sub directory within HOME that we will...

Страница 246: ...riangle Enter text Triangle_ solution Enter text Solution Enter string Triangle Solution in stack Open single quotes in stack title Enter variable name TITLE K Store string into TITLE Creating a list of variables Next create a list of variable names in the stack that will look like this a b c α β γ A s and store it in variable LVARI List of VARIables The list of variables represents the order in w...

Страница 247: ... menu keys in the order indicated by the list In the present exercise we already have a title Triangle Solution and a list of variables a b c α β γ A s in stack levels 2 and 1 respectively ready to activate MITM MSOLVR MES SOLVER activates the Multiple Equation Solver MES and waits for input by the user Running the MES interactively To get the MES started with the variables TITLE and LVARI listed ...

Страница 248: ... is β 34 9152062474 γ The result is γ 72 5423968763 You should have the values of the three angles listed in stack levels 3 through 1 Press twice to check that they add indeed to 180o Press L to move to the next variables menu To calculate the area use A The calculator first solves for all the other variables and then finds the area as A 7 15454401063 Note When a solution is found the calculator r...

Страница 249: ... If you use Mpar to see the contents of the variable Mpar You will get the cryptic message Library Data The meaning of this is that the MES parameters are coded in a binary file which cannot be accessed by the editor Next we want to place them in the menu labels in a different order than the one listed above by following these steps 1 Create a list containing EQ Mpar LVARI TITLE by using ä EQ Mpar...

Страница 250: ... in MITM_ msolvr Type in MSOLVR Enter program in stack Store the program in a variable called TRISOL for TRIangle SOLution by using trisol K Press J if needed to recover your list of variables A soft key label TRISO should be available in your menu Running the program solution examples To run the program press the TRISO soft menu key You will now have the MES menu corresponding to the triangle sol...

Страница 251: ... rather than the equations from which they were solved are shown in the display To see the equations used in the solution of each variable press the EQNS soft menu key The display will now look like this The soft menu key PRINT is used to print the screen in a printer if available And EXIT returns you to the MES environment for a new solution if needed To return to normal calculator display press ...

Страница 252: ...SO to get a triangle solution started You may want to type in the following program Press TRISO to start MSGBOX and store it in a variable called INFO As a result the first variable in your directory will be the INFO button Application 2 Velocity and acceleration in polar coordinates Two dimensional particle motion in polar coordinates often involves determining the radial and transverse component...

Страница 253: ...l acc polar coord LIST a list of the variable used in the calculations placed in the order we want them to show up in the multiple equation solver environment PEQ list of equations to be solved corresponding to the radial and transverse components of velocity vr vθ and acceleration ar a in polar coordinates as well as equations to calculate the magnitude of the velocity v and the acceleration a wh...

Страница 254: ...s value in the upper left corner of the display We have now entered the known variables To calculate the unknowns we can proceed in two ways a Solve for individual variables for example vr gives vr 0 500 Press L vθ to get vθ 5 750 and so on The remaining results are v 5 77169819031 ar 14 725 aθ 13 95 and a 20 2836911089 or b Solve for all variables at once by pressing ALL The calculator will flash...

Страница 255: ...age 7 21 To use a new set of values press either EXIT ALL LL or J SOLVE Let s try another example using r 2 5 vr rD 0 5 rDD 1 5 v 3 0 a 25 0 Find θD θDD vθ ar and aθ You should get the following results ...

Страница 256: ...e t 1 BETA h2 4 1 1 5 2 0 a a a a 1 2 3 3 2 1 1 2 3 In the examples shown below we will limit ourselves to numerical lists Creating and storing lists To create a list in ALG mode first enter the braces key ä associated with the key then type or enter the elements of the list separating them with commas í The following keystrokes will enter the list 1 2 3 4 and store it into variable L1 ä 1 í 2 í 3...

Страница 257: ...ith a small list before and after application of function OBJ Notice that after applying OBJ the elements of the list occupy levels 4 through 2 while level 1 shows the number of elements in the list To compose a list in RPN mode place the elements of the list in the stack enter the list size and apply function LIST select it from the function catalog as follows N é then use the up and down arrow k...

Страница 258: ... shows the three lists and their names ready to be stored To store the lists in this case you need to press K three times Changing sign The sign change key when applied to a list of numbers will change the sign of all elements in the list For example Addition subtraction multiplication division Multiplication and division of a list by a single number is distributed across the list for example Subt...

Страница 259: ...finity entry because one of the elements in L3 is zero If the lists involved in the operation have different lengths an error message is produced Error Invalid Dimension The plus sign when applied to lists acts a concatenation operator putting together the two lists rather than adding them term by term For example In order to produce term by term addition of two lists of the same length we need to...

Страница 260: ...d ANTILOG SQ and square root SIN ASIN COS ACOS TAN ATAN INVERSE 1 x Real number functions from the MTH menu Functions of interest from the MTH menu include from the HYPERBOLIC menu SINH ASINH COSH ACOSH TANH ATANH and from the REAL menu CH T MIN MAX MOD SIGN MANT XPON IP FP RND TRNC FLOOR CEIL D R R D Some of the functions that take a single argument are illustrated below applied to lists of real ...

Страница 261: ...hat use two arguments The screen shots below show applications of the function to list arguments Function requires two arguments The first two examples show cases in which only one of the two arguments is a list The results are lists with the function distributed according to the list argument For example ...

Страница 262: ...ation of any function with two arguments when one or both arguments are lists Examples of applications of function RND are shown next Lists of complex numbers The following exercise shows how to create a list of complex numbers given two lists of the same length one representing the real parts and one the imaginary parts of the complex numbers Use L1 ADD i L2 The screen also shows that the resulti...

Страница 263: ...d ARG argument of complex numbers The results are lists of real numbers Lists of algebraic objects The following are examples of lists of algebraic objects with the function SIN applied to them The MTH LIST menu The MTH menu provides a number of functions that exclusively to lists With flag 117 set to CHOOSE boxes ...

Страница 264: ...ements in the list ΠLIST Calculate product of elements in the list SORT Sorts elements in increasing order REVLIST Reverses order of list ADD Operator for term by term addition of two lists of the same length examples of this operator were shown above Examples of application of these functions in ALG mode are shown next SORT and REVLIST can be combined to sort a list in decreasing order ...

Страница 265: ...ain the size also known as length of the list e g Extracting and inserting elements in a list To extract elements of a list we use function GET available in the PRG LIST ELEMENTS sub menu The arguments of function GET are the list and the number of the element you want to extract To insert an element into a list use function PUT also available in the PRG LST ELEMENTS sub menu The arguments of func...

Страница 266: ...mine the position of an element in a list use function POS having the list and the element of interest as arguments For example HEAD and TAIL functions The HEAD function extracts the first element in the list The TAIL function removes the first element of a list returning the remaining list Some examples are shown next The SEQ function Item 2 PROCEDURES in the PRG LIST menu contains the following ...

Страница 267: ... ALG mode we identify expression n2 index n start 1 end 4 and increment 1 The list produced corresponds to the values 12 22 32 42 In RPN mode you can list the different arguments of the function as follows before applying function SEQ The MAP function The MAP function available through the command catalog N takes as arguments a list of numbers and a function f X or a program of the form a and prod...

Страница 268: ... if we define the function F X Y X 5 Y 2 shown here in ALG mode we can use lists e g variables L1 and L2 defined earlier in this Chapter to evaluate the function resulting in Since the function statement includes no additions the application of the function to list arguments is straightforward However if we define the function G X Y X 3 Y an attempt to evaluate this function with list arguments L1...

Страница 269: ...ADD rather than the plus sign from the start i e use DEFINE G X Y X ADD 3 Y You can also define the function as G X Y X 3 Y Applications of lists This section shows a couple of applications of lists to the calculation of statistics of a sample By a sample we understand a list of values say s1 s2 sn Suppose that the sample of interest is the list 1 5 3 1 2 1 3 4 2 1 ...

Страница 270: ...ting the function Harmonic mean of a list This is a small enough sample that we can count on the screen the number of elements n 10 For a larger list we can use function SIZE to obtain that number e g Suppose that we want to calculate the harmonic mean of the sample defined as n n k n h s s s n s n s 1 1 1 1 1 1 1 1 2 1 1 L To calculate this value we can follow this procedure 1 Apply function INV ...

Страница 271: ...348 Geometric mean of a list The geometric mean of a sample is defined as n n n n k k g x x x x x L 2 1 1 To find the geometric mean of the list stored in S we can use the following procedure 1 Apply function ΠLIST to list S 2 Apply function XROOT x y i e keystrokes to the result in 1 Thus the geometric mean of list S is sg 1 003203 ...

Страница 272: ...ined by wk k Thus we can use function SEQ to generate this list and then store it into variable W as follows Given the data list s1 s2 sn and the weight list w1 w2 wn the weighted average of the data in S is defined as n k k n k k k w w s w s 1 1 To calculate the weighted average of the data in list S with the weights in list W we can use th e following steps 1 Multiply lists S and W 2 Use functio...

Страница 273: ...previous to last result 121 Statistics of grouped data Grouped data is typically given by a table showing the frequency w of data in data classes or bins Each class or bin is represented by a class mark s typically the midpoint of the class An example of grouped data is shown next Class Frequency Class mark count boundaries sk wk 0 2 1 5 2 4 3 12 4 6 5 18 6 8 7 1 8 10 9 3 The class mark data can b...

Страница 274: ...all s in this context N s w w s w s n k k k n k k n k k k 1 1 1 where n k k w N 1 represents the total frequency count The mean value for the data in lists S and W therefore can be calculated using the procedure outlined above for the weighted average i e We ll store this value into a variable called XBAR The variance of this grouped data is defined as N s s w w s s w V n k k k n k k n k k k 1 2 1...

Страница 275: ...Page 8 20 To calculate this last result we can use the following The standard deviation of the grouped data is the square root of the variance ...

Страница 276: ... that a physical vector A can be written in terms of its components Ax Ay Az as A Axi Ayj Azk Alternative notation for this vector are A Ax Ay Az A Ax Ay Az or A Ax Ay Az A two dimensional version of this vector will be written as A Axi Ayj A Ax Ay A Ax Ay or A Ax Ay Since in the calculator vectors are written between brackets we will choose the notation A Ax Ay Az or A Ax Ay Az to refer to two an...

Страница 277: ...s A and B are perpendicular θ 900 π 2rad A B 0 Entering vectors In the calculator vectors are represented by a sequence of numbers enclosed between brackets and typically entered as row vectors The brackets are generated in the calculator by the keystroke combination Ô associated with the key The following are examples of vectors in the calculator 3 5 2 2 1 3 5 6 2 3 A general row vector 1 5 2 2 A...

Страница 278: ...ter MTRW to enter vectors Vectors can also be entered by using the Matrix Writer third key in the fourth row of keys from the top of the keyboard This command generates a species of spreadsheet corresponding to rows and columns of a matrix Details on using the Matrix Writer to enter matrices will be presented in a subsequent chapter For a vector we are interested in filling only elements in the to...

Страница 279: ...while using the Matrix Writer The WID key is used to decrease the width of the columns in the spreadsheet Press this key a couple of times to see the column width decrease in your Matrix Writer The WID key is used to increase the width of the columns in the spreadsheet Press this key a couple of times to see the column width increase in your Matrix Writer The GO key when selected automatically sel...

Страница 280: ...sheet The STK key will place the contents of the selected cell on the stack The GOTO key when pressed will request that the user indicate the number of the row and column where he or she wants to position the cursor Pressing L once more produces the last menu which contains only one function DEL delete The function DEL will delete the contents of the selected cell and replace it with a zero To see...

Страница 281: ...ector using the Matrix Writer simply activate the writer and place the elements of the vector pressing after each of them Then press Make sure that the VEC and GO keys are selected Example xQ2 2 5 produces x 2 2 5 Building a vector with ARRY The function ARRY available in the function catalog N é use to locate the function can also be used to build a vector or array in the following way In ALG mod...

Страница 282: ... Function ARRY is also available in the PRG TYPE menu Identifying extracting and inserting vector elements If you store a vector into a variable name say A you can identify elements of the vector by using A i where i is an integer number less than or equal to the vector size For example create the following array and store it in variable A 1 2 3 4 5 To recall the third element of A for example you...

Страница 283: ...e an element in an array use function PUT you can find it in the function catalog N or in the PRG LIST ELEMENTS sub menu the later was introduced in Chapter 8 In ALG mode you need to use function PUT with the following arguments PUT array location to be replaced new value For example to change the contents of A 3 to 4 5 use In RPN mode you can change the value of an element of A by storing a new v...

Страница 284: ...through the command catalog N or through the PRG LIST ELEMENTS sub menu Some examples based on the arrays or vectors stored previously are shown below Simple operations with vectors To illustrate operations with vectors we will use the vectors A u2 u3 v2 and v3 stored in an earlier exercise Changing sign To change the sign of a vector use the key e g Addition subtraction Addition and subtraction o...

Страница 285: ...absolute value function ABS when applied to a vector produces the magnitude of the vector For a vector A A1 A2 An the magnitude is defined as 2 2 2 z y x A A A A L In the ALG mode enter the function name followed by the vector argument For example ABS 1 2 6 ABS A ABS u3 will show in the screen as follows The MTH VECTOR menu The MTH menu contains a menu of functions that specifically to vector obje...

Страница 286: ...tored earlier are shown next in ALG mode Attempts to calculate the dot product of two vectors of different length produce an error message Cross product Function CROSS is used to calculate the cross product of two 2 D vectors of two 3 D vectors or of one 2 D and one 3 D vector For the purpose of calculating a cross product a 2 D vector of the form Ax Ay is treated as the 3 D vector Ax Ay 0 Example...

Страница 287: ...r into its elements or components If used in the ALG mode V will provide the elements of the vector in a list e g In the RPN mode application of function V will list the components of a vector in the stack e g V A will produce the following output in the RPN stack vector A is listed in stack level 6 Building a two dimensional vector Function V2 is used in the RPN mode to build a vector with the va...

Страница 288: ...ts When the rectangular or Cartesian coordinate system is selected the top line of the display will show an XYZ field and any 2 D or 3 D vector entered in the calculator is reproduced as the x y z components of the vector Thus to enter the vector A 3i 2j 5k we use 3 2 5 and the vector is shown as If instead of entering Cartesian components of a vector we enter cylindrical polar components we need ...

Страница 289: ...is the angle that the xy projection of the vector forms with the positive side of the x axis and φ is the angle that ρ forms with the positive side of the z axis with ρ 5 θ 25o and φ 45o We will use Ô5 í 6 25 í 6 45 The figure below shows the transformation of the vector from spherical to Cartesian coordinates with x ρ sin φ cos θ y ρ sin φ cos θ z ρ cos φ For this case x 3 204 y 1 494 and z 3 536...

Страница 290: ...to polar coordinates enter the vector components as real numbers i e add a decimal point e g 2 3 5 With the cylindrical coordinate system selected if we enter a vector in spherical coordinates it will be automatically transformed to its cylindrical polar equivalent r θ z with r ρ sin φ θ θ z ρ cos φ For example the following figure shows the vector entered in spherical coordinates and transformed ...

Страница 291: ... 2k F2 2i 3j 5k and F3 2i 3k To determine the resultant i e the sum of all these forces you can use the following approach in ALG mode Thus the resultant is R F1 F2 F3 3i 8j 6k N RPN mode use 3 5 2 2 3 5 2 0 3 Angle between vectors The angle between two vectors A B can be found as θ cos 1 A B A B Suppose that you want to find the angle between vectors A 3i 5j 6k B 2i j 3k you could try the followi...

Страница 292: ... of application of the force Suppose that a force F 2i 5j 6k N has an arm r 3i 5j 4k m To determine the moment exerted by the force with that arm we use function CROSS as shown next Thus M 10i 26j 25k m N We know that the magnitude of M is such that M r F sin θ where θ is the angle between r and F We can find this angle as θ sin 1 M r F by the following operations 1 ABS ANS 1 ABS ANS 2 ABS ANS 3 c...

Страница 293: ...ne Thus this vector r P0P x x0 i y y0 j z z0 k is perpendicular to the normal vector N since r is contained entirely in the plane We learned that for two normal vectors N and r N r 0 Thus we can use this result to determine the equation of the plane To illustrate the use of this approach consider the point P0 2 3 1 and the normal vector N 4i 6j 2k we can enter vector N and point P0 as two vectors ...

Страница 294: ...s by enclosing each vector element within brackets all contained within an external set of brackets For example enter 1 2 2 5 3 2 4 5 6 2 This is represented as the following column vector In this section we will showing you ways to transform a column vector into a row vector a row vector into a column vector a list into a vector and a vector or matrix into a list We first demonstrate these transf...

Страница 295: ...r example 1 2 3 TYPE OBJ results in When function OBJ is applied to a vector it will list the elements of the vector in the stack with the number of elements in level 1 enclosed in braces a list The following example illustrates this application 1 2 3 TYPE OBJ results in If we now apply function OBJ once more the list in stack level 1 3 will be decomposed as follows Function LIST This function is ...

Страница 296: ...k and in stack level 1 enter the list n 1 For example 1 2 3 ä 1 í3 TYPE ARRY Function DROP This function has the same effect as the delete key ƒ Transforming a row vector into a column vector We illustrate the transformation with vector 1 2 3 Enter this vector into the RPN stack to follow the exercise To transform a row vector into a column vector we need to carry on the following operations in th...

Страница 297: ... After having defined this variable we can use it in ALG mode to transform a row vector into a column vector Thus change your calculator s mode to ALG and try the following procedure 1 2 3 J RXC Ü î resulting in Transforming a column vector into a row vector To illustrate this transformation we ll enter the column vector 1 2 3 in RPN mode Then follow the next exercise to transform a row vector int...

Страница 298: ...cxr K A new variable CXR will be available in the soft menu labels after pressing J Press CXR to see the program contained in the variable CXR OBJ OBJ DROP ARRY This variable CXR can now be used to directly transform a column vector to a row vector In RPN mode enter the column vector and then press CXR Try for example 1 2 3 CXR After having defined variable CXR we can use it in ALG mode to transfo...

Страница 299: ... a list in stack level 1 3 Use function ARRY to create the vector These three steps can be put together into a UserRPL program entered as follows in RPN mode å TYPE OBJ 1 LIST ARRY lxv K A new variable LXV will be available in the soft menu labels after pressing J Press LXV to see the program contained in the variable LXV OBJ 1 LIST ARRY This variable LXV can now be used to directly transform a li...

Страница 300: ...XV Ü î resulting in Transforming a vector or matrix into a list To transform a vector into a list the calculator provides function AXL You can find this function through the command catalog as follows N axl OK As an example apply function AXL to the vector 1 2 3 in RPN mode by using 1 2 3 AXL The following screen shot shows the application of function AXL to the same vector in ALG mode ...

Страница 301: ...tion we can write matrix A as A aij n m The full matrix is shown next 2 1 2 22 21 1 12 11 nm n n m m m n ij a a a a a a a a a a L O M M L L A A matrix is square if m n The transpose of a matrix is constructed by swapping rows for columns and vice versa Thus the transpose of matrix A is AT aT ij m n aji m n The main diagonal of a square matrix is the collection of elements aii An identity matrix In...

Страница 302: ...ed in Chapter 9 matrices can be entered into the stack by using the Matrix Writer For example to enter the matrix first start the matrix writer by using Make sure that the option GO is selected Then use the following keystrokes 2 5 4 2 2 ššš 3 1 9 2 8 2 1 5 At this point the Matrix Writer screen may look like this Press once more to place the matrix on the stack The ALG mode stack is shown next be...

Страница 303: ...Ô 2 í 1 í 5 Thus to enter a matrix directly into the stack open a set of brackets Ô and enclose each row of the matrix with an additional set of brackets Ô Commas í should separate the elements of each row as well as the brackets between rows Note In RPN mode you can omit the inner brackets after the first set has been entered thus instead of typing for example 1 2 3 4 5 6 7 8 9 type 1 2 3 4 5 6 7...

Страница 304: ...ATRICES CREATE menu available through Ø The MTH MATRIX MAKE sub menu let s call it the MAKE menu contains the following functions while the MATRICES CREATE sub menu let s call it the CREATE menu has the following functions ...

Страница 305: ...ll show how to access functions through use of the matrix MAKE menu At the end of this section we present a table with the keystrokes required to obtain the same functions with the CREATE menu when system flag 117 is set to SOFT menus If you have set that system flag flag 117 to SOFT menu the MAKE menu will be available through the keystroke sequence MATRX MAKE The functions available will be show...

Страница 306: ... Notice that we achieve the same result by simply typing A 2 3 and pressing In RPN mode this exercise is performed by entering A 3 GET or by using A 2 3 Suppose that we want to place the value π into element a31 of the matrix We can use function PUT for that purpose e g In RPN mode you can use J A 3 1 ì PUT Alternatively in RPN mode you can use ì A 2 3 K To see the contents of variable A after thi...

Страница 307: ...UTI The screen shots below show the RPN stack before and after the application of function PUTI In this case the 2 was replaced in position 3 1 i e now A 3 1 2 and the index list was increased by 1 by column first i e from 3 1 to 3 2 The matrix is in level 2 and the incremented index list is in level 1 Function SIZE Function SIZE provides a list showing the number of rows and columns of the matrix...

Страница 308: ...pter 1 If the argument is a real matrix TRN simply produces the transpose of the real matrix Try for example TRN A and compare it with TRAN A In RPN mode the transconjugate of matrix A is calculated by using A TRN Note The calculator also includes Function TRAN in the MATRICES OPERATIONS sub menu For example in ALG mode Function CON The function takes as argument a list of two elements correspondi...

Страница 309: ...t an identity matrix has to be a square matrix therefore only one value is required to describe it completely For example to create a 4 4 identity matrix in ALG mode use You can also use an existing square matrix as the argument of function IDN e g The resulting identity matrix will have the same dimensions as the argument matrix Be aware that an attempt to use a rectangular i e non square matrix ...

Страница 310: ...ng a vector into a matrix The following example shows how to re dimension a vector of 6 elements into a matrix of 2 rows and 3 columns in ALG mode In RPN mode we can use 1 2 3 4 5 6 2 3 RDM to produce the matrix shown above Re dimensioning a matrix into another matrix In ALG mode we now use the matrix created above and re dimension it into a matrix of 3 rows and 2 columns In RPN mode we simply use...

Страница 311: ... random elements are produced by using the same command namely RANM 2 3 In RPN mode use 2 3 RANM Obviously the results you will get in your calculator will most certainly be different than those shown above The random numbers generated are integer numbers uniformly distributed in the range 10 10 i e each one of those 21 numbers has the same probability of being selected Function RANM is useful for...

Страница 312: ...to be inserted For example keeping the matrix that we inherited from the previous example enter the matrix 1 2 3 4 5 6 7 8 9 In ALG mode the following screen shot to the left shows the new matrix before pressing The screen shot to the right shows the application of function RPL to replace the matrix in ANS 2 the 2 2 matrix into the 3 3 matrix currently located in ANS 1 starting at position 2 2 If ...

Страница 313: ...th the main diagonal replaced with the proper vector elements For example the command DIAG 1 1 2 3 3 3 produces a diagonal matrix with the first 3 elements of the vector argument In RPN mode we can use 1 1 2 3 3 3 DIAG to obtain the same result as above Another example of application of the DIAG function follows in ALG mode In RPN mode use 1 2 3 4 5 3 2 DIAG In this case a 3 2 matrix was to be cre...

Страница 314: ...se the length of the list If the input list consists of objects x1 x2 xn then a Vandermonde matrix in the calculator is a matrix made of the following elements 1 2 1 3 2 3 3 1 2 2 2 2 1 1 2 1 1 1 1 1 1 n n n n n n n x x x x x x x x x x x x L M O M M M L L L For example the following command in ALG mode for the list 1 2 3 4 In RPN mode enter 1 2 3 4 VANDERMONDE Function HILBERT Function HILBERT cre...

Страница 315: ...you to practice accessing programming functions in the calculator The programs are listed below showing in the left hand side the keystrokes necessary to enter the program steps and in the right hand side the characters entered in the display as you perform those keystrokes First we present the steps necessary to produce program CRMC Lists represent columns of the matrix The program CRMC allows yo...

Страница 316: ...rom any other sub directory you use To see the contents of the program use J CRMC The program listing is the following DUP n 1 SWAP FOR j OBJ ARRY IF j n THEN j 1 ROLL END NEXT IF n 1 THEN 1 n 1 FOR j j 1 ROLL NEXT END n COL To use this program in RPN mode enter the n lists in the order that you want them as columns of the matrix enter the value of n and press CRMC As an example try the following ...

Страница 317: ... easily modified to create a matrix when the input lists will become the rows of the resulting matrix The only change to be performed is to change COL for ROW in the program listing To perform this change use CRMC List program CRMC in stack ššš Move to end of program ƒƒƒ Delete COL row Type in ROW enter program To store the program use crmr K 1 2 3 4 1 4 9 16 1 8 27 64 3 CRMR The following screen ...

Страница 318: ...he same functions When system flag 117 is set to SOFT menus the COL menu is accessible through MATRX MAKE COL or through Ø CREAT COL Both approaches will show the same set of functions The operation of these functions is presented below Function COL Function COL takes as argument a matrix and decomposes it into vectors corresponding to its columns An application of function COL in ALG mode is show...

Страница 319: ...1 is occupied by the number of columns of the original matrix The matrix does not survive decomposition i e it is no longer available in the stack Function COL Function COL has the opposite effect of Function COL i e given n vectors of the same length and the number n function COL builds a matrix by placing the input vectors as columns of the resulting matrix Here is an example in ALG mode The com...

Страница 320: ...For example in ALG mode we ll insert the second column in matrix A with the vector 1 2 3 i e In RPN mode enter the matrix first then the vector and the column number before applying function COL The figure below shows the RPN stack before and after applying function COL Function COL Function COL takes as argument a matrix and an integer number representing the position of a column in the matrix Fu...

Страница 321: ...ample This matrix is listed first In RPN mode function CSWP lets you swap the columns of a matrix listed in stack level 3 whose indices are listed in stack levels 1 and 2 For example the following figure shows the RPN stack before and after applying function CSWP to matrix A in order to swap columns 2 and 3 As you can see the columns that originally occupied positions 2 and 3 have been swapped Swa...

Страница 322: ...T ROW Both approaches will show the same set of functions The operation of these functions is presented below Function ROW Function ROW takes as argument a matrix and decomposes it into vectors corresponding to its rows An application of function ROW in ALG mode is shown below The matrix used has been stored earlier in variable A The matrix is shown in the figure to the left The figure to the righ...

Страница 323: ...n i e it is no longer available in the stack Function ROW Function ROW has the opposite effect of the function ROW i e given n vectors of the same length and the number n function ROW builds a matrix by placing the input vectors as rows of the resulting matrix Here is an example in ALG mode The command used was ROW 1 2 3 4 5 6 7 8 9 3 In RPN mode place the n vectors in stack levels n 1 n n 1 2 and...

Страница 324: ...er before applying function ROW The figure below shows the RPN stack before and after applying function ROW Function ROW Function ROW takes as argument a matrix and an integer number representing the position of a row in the matrix The function returns the original matrix minus a row as well as the extracted row shown as a vector Here is an example in the ALG mode using the matrix stored in A In R...

Страница 325: ...rows of a matrix listed in stack level 3 whose indices are listed in stack levels 1 and 2 For example the following figure shows the RPN stack before and after applying function CSWP to matrix A in order to swap rows 2 and 3 As you can see the columns that originally occupied positions 2 and 3 have been swapped Function RCI Function RCI stands for multiplying Row I by a Constant value and replace ...

Страница 326: ...nation more details on this procedure are presented in a subsequent Chapter The arguments of the function are 1 the matrix 2 the constant value 3 the row to be multiplied by the constant in 2 and 4 the row to be replaced by the resulting sum as described above For example taking the matrix stored in variable A we are going to multiply column 3 times 1 5 and add it to column 2 The following example...

Страница 327: ...e operations are presented next To illustrate the operations we will create a number of matrices that we will store in variables The generic name of the matrices will be Aij and Bij where i represents the number of rows and j the number of columns of the matrices The matrices to be used are generated by using function RANM random matrices If you try this exercise in your calculator you will get di...

Страница 328: ...22 B22 A22 B22 A23 B23 A23 B23 A32 B32 A32 B32 Translating the ALG examples to RPN is straightforward as illustrated here The remaining examples of matrix operations will be performed in ALG mode only Multiplication There are different multiplication operations that involve matrices These are described next Multiplication by a scalar Multiplication of the matrix A aij m n by a scalar k results in ...

Страница 329: ... we can multiply a matrix by an imaginary number to obtain a matrix of complex numbers e g Matrix vector multiplication Matrix vector multiplication is possible only if the number of columns of the matrix is equal to the length of the vector This operation follows the rules of matrix multiplication as shown in the next section A couple of examples of matrix vector multiplication follow ...

Страница 330: ...nt in the i th row and j th column of the product C results from multiplying term by term the i th row of A with the j th column of B and adding the products together Matrix multiplication is not commutative i e in general A B B A Furthermore one of the multiplications may not even exist The following screen shots show the results of multiplications of the matrices that we stored earlier The matri...

Страница 331: ... of function HADAMARD The result is of course another matrix of the same dimensions This function is available through Function catalog N or through the MATRICES OPERATIONS sub menu Ø Applications of function HADAMARD are presented next The identity matrix In Chapter 9 we introduce the identity matrix as the matrix I δij n n where δij is the Kronecker s delta function Identity matrices can be obta...

Страница 332: ...r by using the inverse function INV i e the Y key An example of the inverse of one of the matrices stored earlier is presented next To verify the properties of the inverse matrix we present the following multiplications Characterizing a matrix The matrix NORM menu The matrix NORM NORMALIZE menu is accessed through the keystroke sequence system flag 117 set to CHOOSE boxes This menu contains the fo...

Страница 333: ...ij m n the Frobenius norm of the matrix is defined as n i m j ij F a A 1 1 2 If the matrix under consideration in a row vector or a column vector then the Frobenius norm A F is simply the vector s magnitude Function ABS is accessible directly in the keyboard as Ê Try the following exercises in ALG mode using the matrices stored earlier for matrix operations Function SNRM Function SNRM calculates t...

Страница 334: ... for i 1 2 n 1 The columns uj of U and vj of V are the corresponding singular vectors Orthogonal matrices are such that U UT I A diagonal matrix has non zero elements only along its main diagonal The rank of a matrix can be determined from its SVD by counting the number of non singular values Examples of SVD will be presented in a subsequent section Functions RNRM and CNRM Function RNRM returns th...

Страница 335: ... l are known as the eigenvectors of the matrix Further details on calculating eigenvalues and eigenvectors are presented later in the chapter Function COND Function COND determines the condition number of a matrix Examples Condition number of a matrix The condition number of a square non singular matrix is defined as the product of the matrix norm times the norm of its inverse i e cond A A A 1 We ...

Страница 336: ...nverse matrix INV A33 are shown to the right Since RNRM A33 CNRM A33 then we take A33 RNRM A33 21 Also since CNRM INV A33 RNRM INV A33 then we take INV A33 CNRM INV A33 0 261044 Thus the condition number is also calculated as CNRM A33 CNRM INV A33 COND A33 6 7871485 Function RANK Function RANK determines the rank of a square matrix Try the following examples The rank of a matrix The rank of a squa...

Страница 337: ...we find that rank A n then the matrix has an inverse and it is a non singular matrix If on the other hand rank A n then the matrix is singular and no inverse exist For example try finding the rank for the matrix You will find that the rank is 2 That is because the second row 2 4 6 is equal to the first row 1 2 3 multiplied by 2 thus row two is linearly dependent of row 1 and the maximum number of ...

Страница 338: ... or negative sign as indicated in the diagram shown below The 2 2 determinant is therefore 21 12 22 11 22 21 12 11 a a a a a a a a A 3 3 determinant is calculated by augmenting the determinant an operation that consists on copying the first two columns of the determinant and placing them to the right of column 3 as shown in the diagram below The diagram also shows the elements to be multiplied wit...

Страница 339: ...then calculated through the method shown above The method of calculating a determinant by cofactor expansion is very inefficient in the sense that it involves a number of operations that grows very fast as the size of the determinant increases A more efficient method and the one preferred in numerical applications is to use a result from Gaussian elimination The method of Gaussian elimination is u...

Страница 340: ...117 set to CHOOSE boxes The OPERATIONS menu includes the following functions Functions ABS CNRM COND DET RANK RNRM SNRM TRACE and TRAN are also found in the MTH MATRIX NORM menu the subject of the previous section Function SIZE was presented in Chapter 10 Function HADAMARD was presented earlier in the context of matrix multiplication Functions LSQ MAD and RSD are related to the solution of systems...

Страница 341: ... to generate matrices such that the element aij is a function of i and j The input to this function consists of two integers n and m representing the number of rows and columns of the matrix to be generated and a program that takes i and j as input The numbers n m and the program occupy stack levels 3 2 and 1 respectively Function LCXM is accessible through the command catalog N For example to gen...

Страница 342: ...n linear equations in m variables can be written as a11 x1 a12 x2 a13 x3 a1 m 1 x m 1 a1 m x m b1 a21 x1 a22 x2 a23 x3 a2 m 1 x m 1 a2 m x m b2 a31 x1 a32 x2 a33 x3 a3 m 1 x m 1 a3 m x m b3 an 1 1 x1 an 1 2 x2 an 1 3 x3 an 1 m 1 x m 1 an 1 m x m bn 1 an1 x1 an2 x2 an3 x3 an m 1 x m 1 an m x m bn This system of linear equations can be written as a matrix equation An m xm 1 bn 1 if we define the fol...

Страница 343: ...he X field is highlighted press SOLVE If a solution is available the solution vector x will be shown in the X field The solution is also copied to stack level 1 Some examples follow A square system The system of linear equations 2x1 3x2 5x3 13 x1 3x2 8x3 13 2x1 2x2 4x3 6 can be written as the matrix equation A x b if 6 13 13 4 2 2 8 3 1 5 3 2 3 2 1 b x A and x x x This system has the same number o...

Страница 344: ...ss to select the B field The vector b can be entered as a row vector with a single set of brackets i e 13 13 6 OK After entering matrix A and vector b and with the X field highlighted we can press SOLVE to attempt a solution to this system of equations A solution was found as shown next To see the solution in the stack press The solution is x 1 2 1 To check that the solution is correct enter the m...

Страница 345: ...nate system x1 x2 x3 The solution to the system of equations shown above will be the intersection of two planes in space We know however that the intersection of two non parallel planes is a straight line and not a single point Therefore there is more than one point that satisfy the system In that sense the system is not uniquely determined Let s use the numerical solver to attempt a solution to t...

Страница 346: ...ronment press The procedure that we describe next can be used to copy the matrix A and the solution vector X into the stack To check that the solution is correct try the following Press to highlight the A field Press L CALC to copy matrix A onto the stack Press OK to return to the numerical solver environment Press CALC to copy solution vector X onto the stack Press OK to return to the numerical s...

Страница 347: ...e This result indicates that x 15 10 3 10 is also a solution to the system confirming our observation that a system with more unknowns than equations is not uniquely determined under determined How does the calculator came up with the solution x 15 37 2 46 9 62 shown earlier Actually the calculator minimizes the distance from a point which will constitute the solution to each of the planes represe...

Страница 348: ...s not unique Some numerical algorithms can be used to force a solution to the system by minimizing the distance from the presumptive solution point to each of the lines in the system Such is the approach followed by the HP 49 G numerical solver Let s use the numerical solver to attempt a solution to this system of equations Ï OK Enter matrix A and vector b as illustrated in the previous example an...

Страница 349: ...esult in a variable X and the matrix into variable A as follows Press K x to store the solution vector into variable X Press ƒ ƒ ƒ to clear three levels of the stack Press K a to store the matrix into variable A Now let s verify the solution by using A X which results in the vector 8 6917 3 4109 1 1301 which is not equal to 15 5 22 the original vector b The solution is simply the point that is clo...

Страница 350: ...f equations LSQ returns the solution with the minimum residual value e A x b The system of equations may not have a solution therefore the value returned is not a real solution to the system just the one with the smallest residual Function LSQ takes as input vector b and matrix A in that order Function LSQ can be found in Function catalog N Next we use function LSQ to repeat the solutions found ea...

Страница 351: ... 3x2 5x3 10 x1 3x2 8x3 85 with 85 10 8 3 1 5 3 2 3 2 1 b x A and x x x The solution using LSQ is shown next Over determined system Consider the system x1 3x2 15 2x1 5x2 5 x1 x2 22 with The solution using LSQ is shown next 22 5 15 1 1 5 2 3 1 2 1 b x A and x x ...

Страница 352: ...x2 5x3 13 x1 3x2 8x3 13 2x1 2x2 4x3 6 we can find the solution in the calculator as follows which is the same result found earlier Solution by division of matrices While the operation of division is not defined for matrices we can use the calculator s key to divide vector b by matrix A to solve for x in the matrix equation A x b This is an arbitrary extension of the algebraic division operation to...

Страница 353: ...und above with the inverse matrix Solving multiple set of equations with the same coefficient matrix Suppose that you want to solve the following three sets of equations X 2Y 3Z 14 2X 4Y 6Z 9 2X 4Y 6Z 2 3X 2Y Z 2 3X 2Y Z 5 3X 2Y Z 2 4X 2Y Z 5 4X 2Y Z 19 4X 2Y Z 12 We can write the three systems of equations as a single matrix equation A X B where 1 2 4 1 2 3 3 2 1 3 2 1 3 2 1 3 2 1 Z Z Z Y Y Y X X...

Страница 354: ...edure is known as forward elimination The reduction of the coefficient matrix to an upper triangular form allows for the solution of all n unknowns utilizing only one equation at a time in a procedure known as backward substitution Example of Gaussian elimination using equations To illustrate the Gaussian elimination procedure we will use the following system of 3 equations in 3 unknowns 2X 4Y 6Z ...

Страница 355: ...third by equation 3 4 equation 1 to get Next divide the second equation by 8 to get Next replace the third equation E3 with equation 3 6 equation 2 i e E2 6 E3 to get Notice that when we perform a linear combination of equations the calculator modifies the result to an expression on the left hand side of the equal sign i e an expression 0 Thus the last set of equations is interpreted to be the fol...

Страница 356: ...nd solve E2 for Y Next we substitute Z 2 and Y 1 into E1 and solve E1 for X The solution is therefore X 1 Y 1 Z 2 Example of Gaussian elimination using matrices The system of equations used in the example above can be written as a matrix equation A x b if we use 4 3 14 1 2 4 1 2 3 6 4 2 b x A Z Y X To obtain a solution to the system matrix equation using Gaussian elimination we first create what i...

Страница 357: ...ed matrix and make an extra copy of the same in the stack This step is not necessary except as an insurance that you have an extra copy of the augmented matrix saved in case you make a mistake in the forward elimination procedure that we are about to undertake 2 4 6 14 3 2 1 3 4 2 1 4 Save augmented matrix in variable AAUG aaug K With a copy of the augmented matrix in the stack press MATRX ROW to ...

Страница 358: ...triangular and equivalent to the set of equations X 2Y 3Z 7 Y Z 3 7Z 14 which can now be solved one equation at a time by backward substitution as in the previous example Gauss Jordan elimination using matrices Gauss Jordan elimination consists in continuing the row operations in the upper triangular matrix resulting from the forward elimination process until an identity matrix results in place of...

Страница 359: ...lement become zero in which case we cannot divide the row by its pivot Also to improve the numerical solution of a system of equations using Gaussian or Gauss Jordan elimination it is recommended that the pivot be the element with the largest absolute value in a given column In such cases we exchange rows before performing row operations This exchange of rows is called partial pivoting To follow t...

Страница 360: ...r words the final solution is given by P x b where b is the last column of the augmented matrix after the solution has been found Example of Gauss Jordan elimination with full pivoting Let s illustrate full pivoting with an example Solve the following system of equations using full pivoting and the Gauss Jordan elimination procedure X 2Y 3Z 2 2X 3Z 1 8X 16Y Z 41 The augmented matrix and the permut...

Страница 361: ...ation matrix now are 16 8 1 41 0 0 1 0 2 3 1 1 0 0 2 1 3 2 0 1 0 Now we have the largest possible value in position 1 1 i e we performed full pivoting at 1 1 Next we proceed to divide by the pivot 16Y1L RCI The permutation matrix does not change but the augmented matrix is now 1 1 2 1 16 41 16 0 0 1 0 2 3 1 1 0 0 2 1 3 2 0 1 0 The next step is to eliminate the 2 from position 3 2 by using 2 1 3 RC...

Страница 362: ...0 1 0 0 1 0 3 2 1 1 0 0 Next we eliminate the 3 from position 3 2 by using 3 2 3 RCIJ 1 1 16 1 2 41 16 0 1 0 0 1 0 1 0 0 1 0 0 2 2 1 0 0 Having filled with zeroes the position below the pivot we proceed to check the pivot at position 3 3 The current value of 2 is larger than or 0 thus we keep it unchanged We do divide the whole third row by 2 to convert the pivot to 1 by using 2Y3 RCI 1 1 16 1 2 4...

Страница 363: ...he modified independent vector b and the permutation matrix P as The solution is given by P x b or Which results in Step by step calculator procedure for solving linear systems The example we just worked is of course the step by step user driven procedure to use full pivoting for Gauss Jordan elimination solution of linear equation systems You can see the step by step procedure used by the calcula...

Страница 364: ...2 1 1 RCIJ Press OK and follow the operations in your calculator s screen You will see the following operations performed L3 L3 8 L1 L1 2 L1 1 L2 L1 25 L1 3 L3 L2 25 L2 3 L3 and finally a message indicating Reduction result showing When you press OK the calculator returns the final result 1 2 1 Calculating the inverse matrix step by step The calculation of an inverse matrix can be considered as ca...

Страница 365: ...lator showed you the steps up to the point in which the left hand half of the augmented matrix has been converted to a diagonal matrix From there the final step is to divide each row by the corresponding main diagonal pivot In other words the calculator has transformed Aaug n n A n n In n into I A 1 Inverse matrices and determinants Notice that all the elements in the inverse matrix calculated abo...

Страница 366: ...ther possibilities for solving linear systems of equations by using Functions included in the MATRICES LINEAR SYSTEMS menu accessible through Ø Set system flag 117 to CHOOSE boxes The functions included are LINSOLVE REF rref RREF and SYST2MAT Function LINSOLVE Function LINSOLVE takes as arguments an array of equations and a vector containing the names of the unknowns and produces the solution to t...

Страница 367: ...own as an echelon form Function REF Reduce to Echelon Form produces such a matrix given the augmented matrix in stack level 1 Consider the augmented matrix 12 3 0 1 2 5 2 1 2 1 2 1 aug A Representing a linear system of equations A x b where A 1 2 1 2 1 2 5 2 1 and b 0 3 12 Enter the augmented matrix and save it into variable AAUG in ALG mode 1 2 1 0 2 1 2 3 5 2 1 12 AAUG Application of function RE...

Страница 368: ...ix resulting from a Gauss Jordan elimination without pivoting A row reduced echelon form for an augmented matrix can be obtained by using function rref This function produces a list of the pivots and an equivalent matrix in row reduced echelon form so that the matrix of coefficients is reduced to a diagonal matrix For example for matrix AAUG function rref produces the following result The second s...

Страница 369: ... a numerical method we produce as a first approximation the solution x 0 Evaluating f x 0 b A x 0 e 0 Thus e is a vector of residuals of Function for the vector x x 0 To use Function RSD you need the terms b A and x 0 as arguments The vector returned is e b A x 0 For example using A 2 1 0 2 x 0 1 8 2 7 and b 1 6 we can find the vector of residuals as follows The result is e b A x 0 0 1 0 6 Note If...

Страница 370: ...raic equation involving a polynomial of order n for a square matrix An n The resulting equation is known as the characteristic polynomial of matrix A Solving the characteristic polynomial produces the eigenvalues of the matrix The calculator provides a number of functions that provide information regarding the eigenvalues and eigenvectors of a square matrix Some of these functions are located unde...

Страница 371: ... matrix shown below are calculated in ALG mode using function EGVL The eigenvalues λ 10 10 Note In some cases you may not be able to find an exact solution to the characteristic polynomial and you will get an empty list as a result when using Function EGVL If that were to happen to you change the calculation mode to Approx in the CAS and repeat the calculation For example in exact mode the followi...

Страница 372: ...the eigenvectors and eigenvalues of the matrix listed below are found by applying function EGV The result shows the eigenvalues as the columns of the matrix in the result list To see the eigenvalues we can use GET ANS 1 2 i e get the second element in the list in the previous result The eigenvalues are In summary λ1 0 29 x1 1 00 0 79 0 91 T λ2 3 16 x2 1 00 0 51 0 65 T λ3 7 54 x1 0 03 1 00 0 84 T N...

Страница 373: ...each eigenvalue of matrix A stack level 2 A vector with the eigenvectors of matrix A stack level 4 For example try this exercise in RPN mode 4 1 2 1 2 1 2 1 0 JORDAN The output is the following 4 X 3 6 x 2 2 X 8 3 X 3 6 x 2 2 X 8 2 1 The same exercise in ALG mode looks as in the following screen shots Function MAD This function although not available in the EIGEN menu also provides information rel...

Страница 374: ...milar in form to the eigenvalue equation A x λ x As an example in RPN mode try 4 1 2 1 2 1 2 1 0 MAD The result is 4 8 3 0 13 0 25 0 38 0 25 0 50 0 25 0 38 0 25 0 88 2 1 0 0 0 1 0 0 0 1 2 1 2 1 4 1 2 1 6 1 2 3 2 4 2 3 2 7 1 X 3 6 x 2 2 X 8 The same exercise in ALG mode will look as follows Matrix factorization Matrix factorization or decomposition consists of obtaining matrices that when multiplie...

Страница 375: ...osition of A using partial pivoting For example in RPN mode 1 2 5 3 1 2 7 6 5 LU produces 3 7 0 0 1 2 86 0 3 1 57 1 2 1 0 86 0 71 0 1 2 0 0 1 1 0 0 1 1 0 0 0 1 0 In ALG mode the same exercise will be shown as follows Orthogonal matrices and singular value decomposition A square matrix is said to be orthogonal if its columns represent unit vectors that are mutually orthogonal Thus if we let matrix ...

Страница 376: ... and V are as defined earlier for singular value decomposition while the vector s represents the main diagonal of the matrix S used earlier For example in RPN mode 5 4 1 2 3 5 7 2 8 SVD 3 0 27 0 81 0 53 0 37 0 59 0 72 0 89 3 09E 3 0 46 2 0 68 0 14 0 72 0 42 0 73 0 54 0 60 0 67 0 44 1 12 15 6 88 1 42 Function SVL Function SVL Singular VaLues returns the singular values of a matrix An m as a vector ...

Страница 377: ...tion QR produces the QR factorization of a matrix An m returning a Qn n orthogonal matrix a Rn m upper trapezoidal matrix and a Pm m permutation matrix in stack levels 3 2 and 1 The matrices A P Q and R are related by A P Q R For example 1 2 1 2 1 2 5 2 1 QR produces 3 0 18 0 39 0 90 0 37 0 88 0 30 0 91 0 28 0 30 2 5 48 0 37 1 83 0 2 42 2 20 0 0 0 90 1 1 0 0 0 0 1 0 1 0 Note Examples and definitio...

Страница 378: ... accessed through Ø This menu includes functions AXQ CHOLESKY GAUSS QXA and SYLVESTER Function AXQ In RPN mode function AXQ produces the quadratic form corresponding to a matrix An n in stack level 2 using the n variables in a vector placed in stack level 1 Function returns the quadratic form in stack level 1 and the vector of variables in stack level 1 For example 2 1 1 5 4 2 3 5 1 X Y Z AXQ retu...

Страница 379: ...adratic form Q so that it only contains square terms from a variable y such that x P y by using Q x A xT P y A P y T y PT A P yT y D yT Function SYLVESTER Function SYLVESTER takes as argument a symmetric square matrix A and returns a vector containing the diagonal terms of a diagonal matrix D and a matrix P so that PT A P D For example 2 1 1 1 4 2 1 2 1 SYLVESTER produces 2 1 2 2 7 23 7 1 2 1 1 0 ...

Страница 380: ... 0 0 1 2 61 3 Z 2 1 3 16 Z 3 Y 2 8 z 2 Y X 2 1 X Y Z Linear Applications The LINEAR APPLICATIONS menu is available through the Ø Information on the functions listed in this menu is presented below by using the calculator s own help facility The figures show the help facility entry and the attached examples Function IMAGE Function ISOM ...

Страница 381: ...Page 11 55 Function KER Function MKISOM ...

Страница 382: ...uence ô D Please notice that if you are using the RPN mode these two keys must be pressed simultaneously to activate any of the graph functions After activating the 2D 3D function the calculator will produce the PLOT SETUP window which includes the TYPE field as illustrated below Right in front of the TYPE field you will most likely see the option Function highlighted This is the default type of g...

Страница 383: ...g curved surfaces in space showing wireframe grids Ps Contour for plotting contour plots of surfaces Y Slice for plotting a slicing view of a function f x y Gridmap for plotting real and imaginary part traces of a complex function Pr Surface for parametric surfaces given by x x u v y y u v z z u v Plotting an expression of the form y f x In this section we present an example of a plot of a functio...

Страница 384: ... to get you into the equation writer You will be prompted to fill the right hand side of an equation Y1 x Type the function to be plotted so that the Equation Writer shows the following Press to return to the PLOT SETUP window The expression Y1 X EXP X 2 2 2 π will be highlighted Press L OK to return to normal calculator display Note Two new variables show up in your soft menu key labels namely EQ...

Страница 385: ...h ERASE DRAW wait till the calculator finishes the graphs To see labels EDIT L LABEL MENU To recover the first graphics menu LL PICT To trace the curve TRACE X Y Then use the right and left arrow keys š to move about the curve The coordinates of the points you trace will be shown at the bottom of the screen Check that for x 1 05 y 0 231 Also check that for x 1 48 y 0 134 Here is picture of the gra...

Страница 386: ...Y1 if in RPN mode or îK Y1 in ALG mode The function to be plotted is now 1 0 2 exp 2 1 2 x x f π Enter the PLOT WINDOW environment by entering ò press them simultaneously if in RPN mode Keep the range of 4 to 4 for H VIEW press AUTO to generate the V VIEW To plot the graph press ERASE DRAW Once the graph is plotted press FCN to access the function menu With this menu you can obtain additional info...

Страница 387: ...the negative root SLOPE 0 16670 Press L to recover the menu To determine the highest point in the curve place the cursor near the vertex and press EXTR The result is EXTRM 0 Press L to recover the menu Other buttons available in the first menu are AREA to calculate the area under the curve and SHADE to shade an area under the curve Press L to see more options The second menu includes one button ca...

Страница 388: ...ontains a list instead of a single expression The list has as elements an expression for the derivative of Y1 X and Y1 X itself Originally EQ contained only Y1 x After we pressed F in the FCN environment the calculator automatically added the derivative of Y1 x to the list of equations in EQ Saving a graph for future use If you want to save your graph to a variable get into the PICTURE environment...

Страница 389: ... key and the ô D key to produce the PLOT SETUP window The field labeled Type will be highlighted If the option Function is not already selected press the soft key labeled CHOOS use the up and down keys to select Function and press OK to complete the selection Check that the field labeled Indep contains the variable X If that is not so press the down arrow key twice until the Indep field is highlig...

Страница 390: ...press EDITL LABEL Press MENU to remove the menu labels and get a full view of the graph Press L to recover the current graphic menu Press L PICT to recover the original graphical menu To determine the coordinates of points on the curve press TRACE the cursor moves on top of the curve at a point located near the center of the horizontal range Next press X Y to see the coordinates of the current cur...

Страница 391: ... at a time in the variable X as the graph is drawn For the horizontal range 1 10 the increment used seems to be 0 275 When the value of X becomes larger than the maximum value in the range in this case when X 10 275 the drawing of the graph stops The last value of X for the graphic under consideration is kept in variable X Delete X and Y1 before continuing Graph of the exponential function First l...

Страница 392: ... left corner and the upper right corner of the plot respectively Next PPAR lists the name of the independent variable X followed by a number that specifies the increment of the independent variable in the generation of the plot The value shown here is the default value zero 0 which specifies increments in X corresponding to 1 pixel in the graphics display The next element in PPAR is a list contain...

Страница 393: ... X and EXP X by following this procedure Press simultaneously if in RPN mode ñ The function Y1 X EXP X should be available in the PLOT FUNCTION window from the previous exercise Press ADD and type the function Y2 X LN X Also load the function Y3 X X Press L OK to return to normal calculator display Press simultaneously if in RPN mode ò and change the H View range to read H View 8 8 Press AUTO to g...

Страница 394: ...Based on the graphing examples presented above the procedure to follow to produce a FUNCTION plot i e one that plots one or more functions of the form Y F X is the following ô simultaneously if in RPN mode Access to the PLOT SETUP window If needed change TYPE to FUNCTION and enter the name of the independent variable Settings A check on _Simult means that if you have two or more plots in the same ...

Страница 395: ...e Use CANCL to cancel any changes to the PLOT SETUP window and return to normal calculator display Press OK to save changes to the options in the PLOT SETUP window and return to normal calculator display ñ simultaneously if in RPN mode Access to the PLOT window in this case it will be called PLOT FUNCTION window Soft menu key options Use EDIT to edit the highlighted equation Use ADD to add new equ...

Страница 396: ...n one of the V View fields to generate the vertical view V View range automatically Or Enter lower and upper limits for vertical view V View and press AUTO while the cursor is in one of the H View fields to generate the horizontal view H View range automatically The calculator will use the horizontal view H View range to generate data values for the graph unless you change the options Indep Low In...

Страница 397: ... field Use CANCL to cancel any changes to the PLOT WINDOW screen and return to normal calculator display Press OK to accept changes to the PLOT WINDOW screen and return to normal calculator display ó simultaneously if in RPN mode Plots the graph based on the settings stored in variable PPAR and the current functions defined in the PLOT FUNCTION screen If a graph different from the one you are plot...

Страница 398: ... 2 AUTO ACOSH X 1 5 AUTO COS ACOS 5 5 1 5 TANH X 5 5 AUTO ATANH X 1 2 1 2 AUTO TAN ATAN 5 5 2 5 2 5 Generating a table of values for a function The combinations õ E and ö F pressed simultaneously if in RPN mode let s the user produce a table of values of functions For example we will produce a table of the function Y X X X 10 in the range 5 X 5 following these instructions We will generate values ...

Страница 399: ...en press OK This will return you to normal calculator display The TPAR variable After finishing the table set up your calculator will create a variable called TPAR Table PARameters that store information relevant to the table that is to be generated To see the contents of this variable press TPAR To see the table press ö i e soft menu key F simultaneously if in RPN mode This will produce a table o...

Страница 400: ...ecover the original x increment of 0 5 you can do an un zoom again or use the option zoom out by pressing ZOOM OK The option Decimal in ZOOM produces x increments of 0 10 The option Integer in ZOOM produces x increments of 1 The option Trig in produces increments related to fractions of π thus being useful when plotting trigonometric functions To return to normal calculator display press Plots in ...

Страница 401: ...RASE DRAW to plot the function in polar coordinates The result is a curve shaped like a hearth This curve is known as a cardiod cardios Greek for heart Press EDIT L LABEL MENU to see the graph with labels Press L to recover the menu Press L PICT to recover the original graphics menu Press TRACE x y to trace the curve The data shown at the bottom of the display is the angle θ and the radius r altho...

Страница 402: ...iven in the canonical form for the following figures circle x xo 2 y yo 2 r2 ellipse x xo 2 a2 y yo 2 b2 1 parabola y b 2 K x a or x a 2 K y b hyperbola x xo 2 a2 y yo 2 b2 1 or xy K where xo yo a b and K are constant The name conic curves follows because these figures circles ellipses parabolas or hyperbolas result from the intersection of a plane with a cone For example a circle is the intersect...

Страница 403: ...multaneously if in RPN mode Change the range for H VIEW to 3 to 3 by using 3 OK 3 OK Also change the V VIEW range to 1 5 to 2 by using 1 5 OK 2 OK Change the Indep Low and High fields to Default by using L RESET while each of those fields is highlighted Select the option Reset value after pressing RESET Press OK to complete the resetting of values Press L to return to the main menu Plot the graph ...

Страница 404: ...the right intersection is near 1 89 0 5 To recover the menu and return to the PLOT environment press L CANCL To return to normal calculator display press L OK Parametric plots Parametric plots in the plane are those plots whose coordinates are generated through the system of equations x x t and y y t where t is known as the parameter An example of such graph is the trajectory of a projectile x t x...

Страница 405: ...to normal calculator display Press ò simultaneously if in RPN mode to access the PLOT window in this case it will be called PLOT PARAMETRIC window Instead of modifying the horizontal and vertical views first as done for other types of plot we will set the lower and upper values of the independent variable first as follows Select the Indep Low field by pressing Change this value to 0 OK Then change...

Страница 406: ...return to normal calculator display A review of your soft menu key labels shows that you now have the following variables t EQ PPAR Y X g θ0 V0 Y0 X0 Variables t EQ and PPAR are generated by the calculator to store the current values of the parameter t of the equation to be plotted EQ which contains X t I Y t and the plot parameters The other variables contain the values of constants used in the d...

Страница 407: ...e Starting value to 0 0 and the Step value to 0 1 Press OK Generate the table by pressing simultaneously if in RPN mode ö The resulting table has three columns representing the parameter t and the coordinates of the corresponding points For this table the coordinates are labeled X1 and Y1 Use the arrow keys š to move about the table Press to return to normal calculator display This procedure for c...

Страница 408: ...L OK to return to normal calculator display Press ò simultaneously if in RPN mode to access the PLOT window in this case it will be called PLOT WINDOW DIFF EQ Change the H VIEW and V VIEW parameters to read H VIEW 1 5 V VIEW 1 1 5 Change the Init value to 0 and the Final value to 5 by using 0 OK 5 OK The values Step and Tol represent the step in the independent variable and the tolerance for conve...

Страница 409: ...labels for the axes are shown as 0 horizontal and 1 vertical These are the definitions for the axes as given in the PLOT WINDOW screen see above i e H VAR t 0 and V VAR x 1 Press LL PICT to recover menu and return to PICT environment Press X Y to determine coordinates of any point on the graph Use and š to move the cursor in the plot area At the bottom of the screen you will see the coordinates of...

Страница 410: ... 3 2 To reset them use L RESET select Reset all OK L Note if the window s ranges are not set to default values the quickest way to reset them is by using L RESET select Reset all OK L Press ERASE DRAW to draw the truth plot Because the calculator samples the entire plotting domain point by point it takes a few minutes to produce a truth plot The present plot should produce a shaded ellipse of semi...

Страница 411: ...s and scatter plots are used to plot discrete data stored in the reserved variable ΣDAT This variable is used not only for these types of plots but also for all kind of statistical applications as will be shown in Chapter 18 As a matter of fact the use of histogram plots is postponed until we get to that chapter for the plotting of a histogram requires to perform a grouping of data and a frequency...

Страница 412: ...ce the graph Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Change TYPE to Bar A matrix will be shown at the ΣDAT field This is the matrix we stored earlier into ΣDAT Highlight the Col field This field lets you choose the column of ΣDAT that is to be plotted The default value is 1 Keep it to plot column 1 in ΣDAT Press L OK to return to normal calculator display Press ò s...

Страница 413: ...in column 2 of the ΣDAT matrix Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Press to highlight the Col field and type 2 OK followed by L OK Press ò simultaneously if in RPN mode to access to the PLOT SETUP window Change V View to read V View 0 6 Press ERASE DRAW Press CANCL to return to the PLOT WINDOW screen then to return to normal calculator display Scatter plots We ...

Страница 414: ...t however Press LL PICT to leave the EDIT environment Press CANCL to return to the PLOT WINDOW environment Then press or L OK to return to normal calculator display To plot y vs z use Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Press to highlight the Cols field Enter 3 OK 2 OK to select column 3 as X and column 2 as Y in the Y vs X scatter plot Press L OK to return to ...

Страница 415: ...example to visualize the solution to the differential equation y f x y x y use the following Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Change TYPE to Slopefield Press and type X Y OK Make sure that X is selected as the Indep and Y as the Depnd variables Press L OK to return to normal calculator display Press ò simultaneously if in RPN mode to access the PLOT WINDOW s...

Страница 416: ...ly if in RPN mode to access to the PLOT SETUP window Change TYPE to Slopefield Press and type Y X 2 OK Press ERASE DRAW to draw the slope field plot Press EDIT L LABEL MENU to see the plot unencumbered by the menu and with identifying labels Press LL PICT to leave the EDIT environment Press CANCL to return to the PLOT WINDOW environment Then press or L OK to return to normal calculator display Fas...

Страница 417: ...ation are relatively fast For the time being we ll keep the default values of 10 and 8 for the Step data Press ERASE DRAW to draw the three dimensional surface The result is a wireframe picture of the surface with the reference coordinate system shown at the lower left corner of the screen By using the arrow keys š you can change the orientation of the surface The orientation of the reference coor...

Страница 418: ...described by z f x y Unlike Fast 3D plots wireframe plots are static plots The user can choose the viewpoint for the plot i e the point from which the surface is seen For example to produce a wireframe plot for the surface z x 2y 3 use the following Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Change TYPE to Wireframe Press and type X 2 Y 3 OK Make sure that X is select...

Страница 419: ...s EDIT L LABEL MENU to see the graph with labels and ranges This particular version of the graph is limited to the lower part of the display We can change the viewpoint to see a different version of the graph Press LL PICT CANCL to return to the PLOT WINDOW environment Change the eye coordinate data to read XE 0 YE 3 ZE 3 Press ERASE DRAW to see the surface plot Press EDIT L LABEL MENU to see the ...

Страница 420: ... type X 2 Y 2 OK Press ERASE DRAW to draw the slope field plot Press EDIT L MENU LABEL to see the plot unencumbered by the menu and with identifying labels Press LL PICT to leave the EDIT environment Press CANCL to return to the PLOT WINDOW environment Then press or L OK to return to normal calculator display Ps Contour plots Ps Contour plots are contour plots of three dimensional surfaces describ...

Страница 421: ... will take some time so be patient The result is a contour plot of the surface Notice that the contour are not necessarily continuous however they do provide a good picture of the level surfaces of the function Press EDITL LABEL MENU to see the graph with labels and ranges Press LL PICT CANCL to return to the PLOT WINDOW environment Press or L OK to return to normal calculator display Try also a P...

Страница 422: ... Press L OK to return to normal calculator display Press ò simultaneously if in RPN mode to access the PLOT WINDOW screen Keep the default plot window ranges to read X Left 1 X Right 1 Y Near 1 Y Far 1 Z Low 1 Z High 1 Step Indep 10 Depnd 8 Press ERASE DRAW to draw the three dimensional surface You will see the calculator produce a series of curves on the screen that will immediately disappear Whe...

Страница 423: ...t For example to produce a Gridmap plot for the function w sin z use the following Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Change TYPE to Gridmap Press and type SIN X i Y OK Make sure that X is selected as the Indep and Y as the Depnd variables Press L OK to return to normal calculator display Press ò simultaneously if in RPN mode to access the PLOT WINDOW screen K...

Страница 424: ...ndent parameters Most textbooks will use u v as the parameters rather than X Y Thus the parametric description of a surface is given as x x u v y y u v z z u v For example to produce a Pr Surface plot for the surface x x X Y X sin Y y y X Y x cos Y z z X Y X use the following Press ô simultaneously if in RPN mode to access to the PLOT SETUP window Change TYPE to Pr Surface Press and type X SIN Y X...

Страница 425: ...enever you create a three dimensional plot such as Fast3D Wireframe or Pr Surface Interactive drawing Whenever we produce a two dimensional graph we find in the graphics screen a soft menu key labeled EDIT Pressing EDIT produces a menu that include the following options press L to see additional functions Through the examples above you have the opportunity to try out functions LABEL MENU PICT and ...

Страница 426: ...nction Press EDIT L LABEL to add labels to the graph Press LL or to recover the original EDIT menu Next we illustrate the use of the different drawing functions on the resulting graphics screen They require use of the cursor and the arrow keys š to move the cursor about the graphics screen DOT and DOT When DOT is selected pixels will be activated wherever the cursor moves leaving behind a trace of...

Страница 427: ... is still active indicating that the calculator is ready to plot a line starting at that point Press to move the cursor downwards say about another cm and press LINE again Now you should have a straight angle traced by a horizontal and a vertical segments The cursor is still active To deactivate it without moving it at all press LINE The cursor returns to its normal shape a cross and the LINE func...

Страница 428: ...es a circle Mark the center of the circle with a MARK command then move the cursor to a point that will be part of the periphery of the circle and press CIRCL To deactivate CIRCL return the cursor to the MARK position and press LINE Try this command by moving the cursor to a clear part of the graph press MARK Move the cursor to another point then press CIRCL A circle centered at the MARK and passi...

Страница 429: ...letely fill the graphic window make sure that the cursor is placed at the upper left corner of the display PICT This command places a copy of the graph currently in the graphics window on to the stack as a graphic object The graphic object placed in the stack can be saved into a variable name for storage or other type of manipulation X Y This command copies the coordinates of the current cursor po...

Страница 430: ... and Y Factors relate the horizontal and vertical user defined unit ranges to their corresponding pixel ranges Change the H Factor to read 8 and press OK then change the V Factor to read 2 and press OK Check off the option Recenter on cursor and press OK Back in the graphics display press ZIN The graphic is re drawn with the new vertical and horizontal scale factors centered at the position where ...

Страница 431: ...rs which may not recover the graph view from which you started the zoom box operation ZDFLT ZAUTO Pressing ZDFLT re draws the current plot using the default x and y ranges i e 6 5 to 6 5 in x and 3 1 to 3 1 in y The command ZAUTO on the other hand creates a zoom window using the current independent variable x range but adjusting the dependent variable y range to fit the curve as when you use the f...

Страница 432: ...o 3π the preferred range for trigonometric functions Note None of these functions are programmable They are only useful in an interactive way Do not confuse the command ZFACT in the ZOOM menu with the function ZFACTOR which is used for gas dynamic and chemistry applications see Chapter 3 The SYMBOLIC menu and graphs The SYMBOLIC menu is activated by pressing the P key fourth key from the left in f...

Страница 433: ... GROBs first over the second See Chapter 22 PLOT function plots a function similar to ô PLOTADD function adds this function to the list of functions to plot similar to ô Plot setup same as ô SIGNTAB function sign table of given function showing intervals of positive and negative variation zero points and infinite asymptotes TABVAL table of values for a function TABVAR variation table of a function...

Страница 434: ...1 1 3 produces a list of min max values of the function in the interval 1 3 while SIGNTAB X 2 1 shows the sign of the function in the interval with f x 0 in 1 f x 0 in 1 1 and f x 0 in 1 TABVAR LN X X produces the following table of variation A detailed interpretation of the table of variation is easier to follow in RPN mode ...

Страница 435: ...ncreases before reaching this value as indicated by the upward arrow and decreases after this value X e becoming slightly larger than zero 0 as X goes to infinity A plot of the graph is shown below to illustrate these observations Function DRAW3DMATRIX This function takes as argument a n m matrix Z zij and minimum and maximum values for the plot You want to select the values of vmin and vmax so th...

Страница 436: ...hics The functions in entries 1 and 2 will be presented in this Chapter Differential equations the subject of item 3 are presented in Chapter 16 Graphic functions the subject of item 4 were presented at the end of Chapter 12 Finally entries 5 DERVX and 6 INTVX are the functions to obtain a derivative and a indefinite integral for a function of the default CAS variable typically X Functions DERVX a...

Страница 437: ...ed to divide two polynomials producing a series expansion Functions DIVPC SERIES TAYLOR0 and TAYLOR are used in series expansions of functions and discussed in more detail in this Chapter Function lim is entered in ALG mode as lim f x x a to calculate the limit lim x f a x In RPN mode enter the function first then the expression x a and finally function lim Examples in ALG mode are shown next incl...

Страница 438: ...kes derivatives with respect to the CAS default variable VX typically X While function DERVX is available directly in the CALC menu both functions are available in the DERIV INTEG sub menu within the CALCL menu Ö Function DERIV requires a function say f t and an independent variable say t while function DERVX requires only a function of VX Examples are shown next in ALG mode Recall that in RPN mod...

Страница 439: ...ng derivatives with The symbol is available as the T key This symbol can be used to enter a derivative in the stack or in the Equation Writer see Chapter 2 If you use the symbol to write a derivative into the stack follow it immediately with the independent variable then by a pair of parentheses enclosing the function to be differentiated Thus to calculate the derivative d sin r r use in ALG mode ...

Страница 440: ...o be differentiated say s ln s To evaluate the derivative in the Equation Writer press the up arrow key four times to select the entire expression then press EVAL The derivative will be evaluated in the Equation Writer as Note The symbol is used formally in mathematics to indicate a partial derivative i e the derivative of a function with more than one variable However the calculator does not dist...

Страница 441: ...a results in The terms d1 in front of g x and f g x in the expression above are abbreviations the calculator uses to indicate a first derivative when the independent variable in this case x is clearly defined Thus the latter result is interpreted as in the formula for the chain rule shown above Here is another example of a chain rule application Derivatives of equations You can use the calculator ...

Страница 442: ...e graphs of functions and for optimizing functions of one variable i e finding maxima and minima Some applications of derivatives are shown next Analyzing graphics of functions In Chapter 11 we presented some functions that are available in the graphics screen for analyzing graphics of functions of the form y f x These functions include X Y and TRACE for determining points on the graph as well as ...

Страница 443: ...ultaneously to access the PLOT window Change H VIEW range to 2 to 2 and V VIEW range to 5 to 5 Press ERASE DRAW to plot the function in polar coordinates The resulting plot looks as follows Notice that there are vertical lines that represent asymptotes These are not part of the graph but show points where TAN X goes to at certain values of X Press TRACE X Y and move the cursor to the point X 1 08E...

Страница 444: ...fined while from 0 to the function is defined On the other hand indicates that the function is not defined between and 1 nor between 1 and The domain of this function is therefore 1 X 1 Function TABVAL This function is accessed through the command catalog or through the GRAPH sub menu in the CALC menu Function TABVAL takes as arguments a function of the CAS variable f X and a list of two numbers r...

Страница 445: ...π 2 For this case SIGNTAB does not provide information in the intervals between and π 2 nor between π 2 and Thus SIGNTAB for this case provides information only on the main domain of TAN X namely π 2 X π 2 A second example of function SIGNTAB is shown below For this case the function is negative for X 1 and positive for X 1 Function TABVAR This function is accessed through the command catalog or t...

Страница 446: ...on TABVAR Use the following keystrokes in RPN mode X 3 4 X 2 11 X 30 N t select TABVAR OK This is what the calculator shows in stack level 1 This is a graphic object To be able to the result in its entirety press The variation table of the function is shown as follows Press to recover normal calculator display Press ƒ to drop this last result from the stack Two lists corresponding to the top and b...

Страница 447: ...at point then values of x for which f x 0 represent points where the graph of the function reaches a maximum or minimum Furthermore the value of the second derivative of the function f x at those points determines whether the point is a relative or local maximum f x 0 or minimum f x 0 These ideas are illustrated in the figure below In this figure we limit ourselves to determining extreme points of...

Страница 448: ...evaluate the second derivative at each point use The last screen shows that f 11 3 14 thus x 11 3 is a relative minimum For x 1 we have the following This result indicates that f 1 14 thus x 1 is a relative maximum Evaluate the function at those points to verify that indeed f 1 f 11 3 Higher order derivatives Higher order derivatives can be calculated by applying a derivative function several time...

Страница 449: ... functions INT INTVX RISCH SIGMA and SIGMAVX to calculate anti derivatives of functions Functions INT RISCH and SIGMA work with functions of any variable while functions INTVX and SIGMAVX utilize functions of the CAS variable VX typically x Functions INT and RISCH require therefore not only the expression for the function being integrated but also the independent variable name Function INT require...

Страница 450: ...r also provides the integral symbol as the keystroke combination Á associated with the U key The simplest way to build an integral is by using the Equation Writer see Chapter 2 for an example Within the Equation Writer the symbol Á produces the integral sign and provides placeholders for the integration limits a b for the function f x and for the variable of integration x The following screen shot...

Страница 451: ...egration Pressing at this point will evaluate the integral in the stack The integral can be evaluated also in the Equation Writer by selecting the entire expression an using the soft menu key EVAL Step by step evaluation of derivatives and integrals With the Step Step option in the CAS MODES windows selected see Chapter 1 the evaluation of derivatives and integrals will be shown step by step For e...

Страница 452: ...int where further application of function EVAL produce no more changes in the expression The following example shows the evaluation of a definite integral in the Equation Writer step by step Notice that the step by step process provides information on the intermediate steps followed by the CAS to solve this integral First CAS identifies a square root integral next a rational fraction and a second ...

Страница 453: ...tegration Several techniques of integration can be implemented in the calculators as shown in the following examples Substitution or change of variables Suppose we want to calculate the integral dx x x 2 0 2 1 If we use step by step calculation in the Equation Writer this is the sequence of variable substitutions This second step shows the proper substitution to use u x2 1 ...

Страница 454: ... is written as vdu uv d udv Since by the definition of a differential dy y we write the previous expression as vdu uv udv This formulation known as integration by parts can be used to find an integral if dv is easily integrable For example the integral xex dx can be solved by integration by parts if we use u x dv ex dx since v ex With du dx the integral becomes xex dx udv uv vdu xex ex dx xex ex T...

Страница 455: ...tion PARTFRAC presented in Chapter 5 provides the decomposition of a fraction into partial fractions This technique is useful to reduce a complicated fraction into a sum of simple fractions that can then be integrated term by term For example to integrate dX X X X X 3 4 5 2 5 we can decompose the fraction into its partial component fractions as follows The direct integration produces the same resu...

Страница 456: ...culator we proceed as follows Alternatively you can evaluate the integral to infinity from the start e g Integration with units An integral can be calculated with units incorporated into the limits of integration as in the example shown below that uses ALG mode with the CAS set to Approx mode The left hand side figure shows the integral typed in the line editor before pressing The right hand figur...

Страница 457: ...output Some notes in the use of units in the limits of integrations 1 The units of the lower limit of integration will be the ones used in the final result as illustrated in the two examples below 2 Upper limit units must be consistent with lower limit units Otherwise the calculator simply returns the unevaluated integral For example 3 The integrand may have units too For example 4 If both the lim...

Страница 458: ...nfinite series around a point x x0 by using a Taylor s series namely 0 n n o o n x x n x f x f where f n x represents the n th derivative of f x with respect to x f 0 x f x If the value x0 is zero the series is referred to as a Maclaurin s series i e 0 0 n n n x n f x f Taylor polynomial and reminder In practice we cannot evaluate all terms in an infinite series instead we approximate the series b...

Страница 459: ...hn 1 or R O hk 1 If h is a small number say h 1 then hk 1 will be typically very small i e hk 1 hk h 1 Thus for x close to x0 the larger the number of elements in the Taylor polynomial the smaller the order of the residual Functions TAYLR TAYLR0 and SERIES Functions TAYLR TAYLR0 and SERIES are used to generate Taylor polynomials as well as Taylor series with residuals These functions are available...

Страница 460: ...roduced Function SERIES returns two output items a list with four items and an expression for h x a if the second argument in the function call is x a i e an expression for the increment h The list returned as the first output object includes the following items 1 Bi directional limit of the function at point of expansion i e lim x f a x 2 An equivalent value of the function near x a 3 Expression ...

Страница 461: ...Page 13 26 In the right hand side figure above we are using the line editor to see the series expansion in detail ...

Страница 462: ...he DEFINE function à To illustrate the concept of partial derivative we will define a couple of multi variate functions f x y x cos y and g x y z x2 y2 1 2 sin z as follows We can evaluate the functions as we would evaluate any other calculator function e g Graphics of two dimensional functions are possible using Fast3D Wireframe Ps Contour Y Slice Gridmap and Pr Surface plots as described in Chap...

Страница 463: ...atives of multi variate functions use the rules of ordinary derivatives with respect to the variable of interest while considering all other variables as constant Thus for example sin cos cos cos y x y x y y y x x which are the same results as found with the limits calculated earlier Consider another example xy yx y yx x 2 0 2 2 2 In this calculation we treat y as a constant and take derivatives o...

Страница 464: ... 2 2 y f y y f x f x x f y f x y x f x f y x y f 2 2 The last two expressions represent cross derivatives the partial derivatives signs in the denominator shows the order of derivation In the left hand side the derivation is taking first with respect to x and then with respect to y and in the right hand side the opposite is true It is important to indicate that if a function is continuous and diff...

Страница 465: ...hain rule for the derivative dz dt for this case is written as v y y z v x x z v z To see the expression that the calculator produces for this version of the chain rule use The result is given by d1y t d2z x t y t d1x t d1z x y y t The term d1y t is to be interpreted as the derivative of y t with respect to the 1st independent variable i e t or d1y t dy dt Similarly d1x t dx dt On the other hand d...

Страница 466: ...sh at that point These are necessary conditions The sufficient conditions for the function to have an extreme at point xo yo are f x 0 f y 0 and 2 f x2 2 f y2 2 f x y 2 0 The point xo yo is a relative maximum if 2 f x2 0 or a relative minimum if 2 f x2 0 The value is referred to as the discriminant If 2 f x2 2 f y2 2 f x y 2 0 we have a condition known as a saddle point where the function would at...

Страница 467: ...alculator and edited in the computer illustrates the existence of these two points Using function HESS to analyze extrema Function HESS can be used to analyze extrema of a function of two variables as shown next Function HESS in general takes as input a function of n independent variables φ x1 x2 xn and a vector of the functions x1 x2 xn Function HESS returns the Hessian matrix of the function φ d...

Страница 468: ...tions of two variables For example for the function f X Y X3 3X Y2 5 proceed as follows in RPN mode X 3 3 X Y 2 5 X Y Enter function and variables HESS Apply function HESS SOLVE Find critical points µ Decompose vector s1 K s2 K Store critical points The variables s1 and s2 at this point contain the vectors X 1 Y 0 and X 1 Y 0 respectively The Hessian matrix is at level 1 at this point H K Store He...

Страница 469: ...y plane representing the volume of the solid body contained under the surface f x y above the region R The region R can be described as R a x b f x y g x or as R c y d r y x s y Thus the double integral can be written as d c y s y r b a x g x f R dydx y x dydx y x dA y x φ φ φ Calculating a double integral in the calculator is straightforward A double integral can be built in the Equation Writer s...

Страница 470: ...ng such transformation the expression to use is R R dudv J v u y v u x dydx y x φ φ where R is the region R expressed in u v coordinates Double integral in polar coordinates To transform from polar to Cartesian coordinates we use x r θ r cos θ and y r θ r sin θ Thus the Jacobian of the transformation is r r r y r y x r x J cos sin sin cos θ θ θ θ θ θ With this result integrals in polar coordinates...

Страница 471: ...olar coordinates is R α θ β f θ r g θ Double integrals in polar coordinates can be entered in the calculator making sure that the Jacobian J r is included in the integrand The following is an example of a double integral calculated in polar coordinates shown step by step ...

Страница 472: ...rred to as a vector field The following operator referred to as the del or nabla operator is a vector based operator that can be applied to a scalar or vector function z k y j x i When this operator is applied to a scalar function we can obtain the gradient of the function and when applied to a vector function we can obtain the divergence and the curl of that function A combination of gradient and...

Страница 473: ...st way to obtain the gradient is by using function DERIV available in the CALC menu e g A program to calculate the gradient The following program which you can store into variable GRADIENT uses function DERIV to calculate the gradient of a scalar function of X Y Z Calculations for other base variables will not work If you work frequently in the X Y Z system however this function will facilitate ca...

Страница 474: ... if there exists a function φ x y z such that f φ x g φ y and h φ z then φ x y z is referred to as the potential function for the vector field F It follows that F grad φ φ The calculator provides function POTENTIAL available through the command catalog N to calculate the potential function of a vector field if it exists For example if F x y z xi yj zk applying function POTENTIAL we find Since func...

Страница 475: ...e function i e z h y g x f F divF Function DIV can be used to calculate the divergence of a vector field For example for F X Y Z XY X2 Y2 Z2 YZ the divergence is calculated in ALG mode as follows Laplacian The divergence of the gradient of a scalar function produces an operator called the Laplacian operator Thus the Laplacian of a scalar function φ x y z is given by 2 2 2 2 2 2 2 x x x φ φ φ φ φ T...

Страница 476: ...llows Irrotational fields and potential function In an earlier section in this chapter we introduced function POTENTIAL to calculate the potential function φ x y z for a vector field F x y z f x y z i g x y z j h x y z k such that F grad φ φ We also indicated that the conditions for the existence of φ were f y g x f z h x and g z h y These conditions are equivalent to the vector expression curl F ...

Страница 477: ...x y z j η x y z k such that F curl Φ Φ then function Φ x y z is referred to as the vector potential of F x y z The calculator provides function VPOTENTIAL available through the command catalog N to calculate the vector potential Φ x y z given the vector field F x y z f x y z i g x y z j h x y z k For example given the vector field F x y z yi zj xk function VPOTENTIAL produces i e Φ x y z x2 2j y2 ...

Страница 478: ...z k and those of the vector potential function Φ x y z φ x y z i ψ x y z j η x y z k are related by f η y ψ x g φ z η x and h ψ x φ y A condition for function Φ x y z to exists is that div F F 0 i e f x g y f z 0 Thus if this condition is not satisfied the vector potential function Φ x y z does not exist For example given F X Y X Y Z 2 function VPOTENTIAL returns an error message since function F ...

Страница 479: ...he calculator is typing in the derivatives in the equation The easiest way to enter a differential equation is to type it in the equation writer For example to type the following ODE x 1 dy x dx 2 2 x y x ex sin x use O Ü x 1 x y Ü x Q2 2 x y Ü x x S x The derivative dy dx is represented by x y x or by d1y x For solution or calculation purposes you need to specify y x in the expression i e the dep...

Страница 480: ...unction DERIV in ALG mode as shown next DERIV x f x t g t y h x y t t produces the following expression x d2f x t d1g t y d3h x y t Translated to paper this expression represents the partial differential equation x f t g t h t Because the order of the variable t is different in f x t g t y and h x y t derivatives with respect to t have different indices i e d2f x t d1g t y and d3h x y t All of the...

Страница 481: ...ngential to the solution curves y f x The slope of the segments at any point x y is given by dy dx f x y evaluated at any point x y represents the slope of the tangent line at point x y Example 1 Trace the solution to the differential equation y f x y sin x cos y using a slope field plot To solve this problem follow the instructions in Chapter 12 for slopefield plots If you could reproduce the slo...

Страница 482: ...nts including systems of differential equations with constant coefficients Solution to linear and non linear equations An equation in which the dependent variable and all its pertinent derivatives are of the first degree is referred to as a linear differential equation Otherwise the equation is said to be non linear Examples of linear differential equations are d2 x dt2 β dx dt ωo x A sin ωf t and...

Страница 483: ... the CALC DIFF menu The examples are shown in the RPN mode however translating them to the ALG mode is straightforward Example 1 To solve the homogeneous ODE d3 y dx3 4 d2 y dx2 11 dy dx 30 y 0 enter 0 X 3 4 X 2 11 X 30 LDEC The solution is where cC0 cC1 and cC2 are constants of integration While this result seems very complicated it can be simplified if we take K1 10 cC0 7 cC1 cC2 40 K2 6 cC0 cC1...

Страница 484: ...general solution of the homogeneous equation see Example 1 above If yh represents the solution to the homogeneous equation i e yh K1 e 3x K2 e5x K3 e2x You can prove that the remaining terms in the solution shown above i e yp 450 x2 330 x 241 13500 constitute a particular solution of the ODE Note This result is general for all non homogeneous linear ODEs i e given the solution of the homogeneous e...

Страница 485: ...r allowing the user to see the two components of the vector To see all the details of each component press the EDIT soft menu key Verify that the components are Function DESOLVE The calculator provides function DESOLVE Differential Equation SOLVEr to solve certain types of differential equations The function requires as input the differential equation and the unknown function and returns the solut...

Страница 486: ... holds a string showing the type of ODE used as input for DESOLVE Press ODETY to obtain the string 1st order linear Example 2 Solve the second order ODE d2 y dx2 x dy dx exp x In the calculator use d1d1y x x d1y x EXP x y x DESOLVE The result is an expression having two implicit integrations namely For this particular equation however we realize that the left hand side of the equation represents d...

Страница 487: ... not available in closed form Example 3 Solving an equation with initial conditions Solve d2 y dt2 5y 2 cos t 2 with initial conditions y 0 1 2 y 0 0 5 In the calculator use d1d1y t 5 y t 2 COS t 2 y 0 6 5 d1y 0 1 2 y t DESOLVE Notice that the initial conditions were changed to their Exact expressions y 0 6 5 rather than y 0 1 2 and d1y 0 1 2 rather than d1y 0 0 5 Changing to these Exact expressio...

Страница 488: ...re three 1 Use of the Laplace transform converts the linear ODE involving f t into an algebraic equation 2 The unknown F s is solved for in the image domain through algebraic manipulation 3 An inverse Laplace transform is used to convert the image function found in step 2 into the solution to the differential equation f t Definitions The Laplace transform for function f t is the function F s defin...

Страница 489: ...here VX is the CAS default independent variable which you should set to X Thus the calculator returns the transform or inverse transform as a function of X The functions LAP and ILAP are available under the CALC DIFF menu The examples are worked out in the RPN mode but translating them to ALG mode is straightforward For these examples set the CAS mode to Real and Exact Example 1 You can get the de...

Страница 490: ...uch that f t L 1 sin s Example 4 Determine the inverse Laplace transform of F s 1 s3 Use 1 X 3 ILAP µ The calculator returns the result X 2 2 which is interpreted as L 1 1 s3 t2 2 Example 5 Determine the Laplace transform of the function f t cos a t b Use COS a X b LAP The calculator returns the result Press µ to obtain a sin b X cos b X2 a2 The transform is interpreted as follows L cos a t b s co...

Страница 491: ...the initial velocity is vo v 0 dr dt t 0 then the Laplace transform of the acceleration can be written as A s L a t L d2 r dt2 s2 R s s ro v o Differentiation theorem for the n th derivative Let f k o dk f dxk t 0 and fo f 0 then L dn f dtn sn F s sn 1 fo s f n 2 o f n 1 o Linearity theorem L af t bg t a L f t b L g t Differentiation theorem for the image function Let F s L f t then dn F dsn L t n...

Страница 492: ... L f t and G s L g t use SIN X LAP µ Result 1 X 2 1 i e F s 1 s2 1 Also EXP X LAP Result 1 X 1 i e G s 1 s 1 Thus L f g t F s G s 1 s2 1 1 s 1 1 s 1 s2 1 1 s3 s2 s 1 Shift theorem for a shift to the right Let F s L f t then L f t a e as L f t e as F s Shift theorem for a shift to the left Let F s L f t and a 0 then Similarity theorem Let F s L f t and a 0 then L f a t 1 a F s a Damping theorem Let...

Страница 493: ...tion of a switch Heaviside s step function H t or a sudden instantaneous peak in an input to the system Dirac s delta function δ t These belong to a class of functions known as generalized or symbolic functions e g see Friedman B 1956 Principles and Techniques of Applied Mathematics Dover Publications Inc New York 1990 reprint The formal definition of Dirac s delta function δ x is δ x 0 for x 0 an...

Страница 494: ...of x0 Heaviside s step function H x is defined as 0 0 0 1 x x x H Also for a continuous function f x Dirac s delta function and Heaviside s step function are related by dH dx δ x The two functions are illustrated in the figure below You can prove that L H t 1 s from which it follows that L Uo H t Uo s where Uo is a constant Also L 1 1 s H t and L 1 Uo s Uo H t Also using the shift theorem for a sh...

Страница 495: ...from L 1 1 0 δ t it follows that L δ t 1 0 Also using the shift theorem for a shift to the right L f t a e as L f t e as F s we can write L δ t k e ks L δ t e ks 1 0 e ks Applications of Laplace transform in the solution of linear ODEs At the beginning of the section on Laplace transforms we indicated that you could use these transforms to convert a linear ODE in the time domain into an algebraic ...

Страница 496: ...s 1 Use the calculator to solve for H s by writing X H h0 k H a X 1 H ISOL The result is H X 1 h0 a X 2 k 1 X k To find the solution to the ODE h t we need to use the inverse Laplace transform as follows OBJ ƒ ƒµ Isolates right hand side of last expression ILAP Obtains the inverse Laplace transform The result is Replacing X with t in this expression and simplifying results in h t a k 1 e t k 1 ho ...

Страница 497: ... f 0 f 0 f n 1 0 Example 2 Use Laplace transforms to solve the second order linear equation d2 y dt2 2y sin 3t Using Laplace transforms we can write L d2 y dt2 2y L sin 3t L d2 y dt2 2 L y t L sin 3t Note SIN 3 X LAP µ produces 3 X 2 9 i e L sin 3t 3 s2 9 With Y s L y t and L d2 y dt2 s2 Y s s yo y1 where yo h 0 and y1 h 0 the transformed equation is s2 Y s s yo y1 2 Y s 3 s2 9 Use the calculator ...

Страница 498: ...function LDEC SIN 3 X X 2 2 LDEC µ The result is i e the same as before with cC0 y0 and cC1 y1 Note Using the two examples shown here we can confirm what we indicated earlier i e that function ILAP uses Laplace transforms and inverse transforms to solve linear ODEs given the right hand side of the equation and the characteristic equation of the corresponding homogeneous ODE Example 3 Consider the ...

Страница 499: ...lace transform as follows OBJ ƒ ƒ Isolates right hand side of last expression ILAP µ Obtains the inverse Laplace transform The result is y1 SIN X y0 COS X SIN X 3 Heaviside X 3 Notes 1 An alternative way to obtain the inverse Laplace transform of the expression X y0 y1 EXP 3 X X 2 1 is by separating the expression into partial fractions i e y0 X X 2 1 y1 X 2 1 EXP 3 X X 2 1 and use the linearity t...

Страница 500: ...erse Laplace transform for 1 s2 1 With the calculator try 1 X 2 1 ILAP The result is SIN X Thus L 1 e 3s s2 1 sin t 3 H t 3 Check what the solution to the ODE would be if you use the function LDEC Delta X 3 X 2 1 LDEC µ The result is SIN X 3 Heaviside X 3 cC1 SIN X cC0 COS X Please notice that the variable X in this expression actually represents the variable t in the original ODE Thus the transla...

Страница 501: ...t to X Press L OK to return to normal calculator display Press ò simultaneously to access the PLOT window Change the H VIEW range to 0 to 20 and the V VIEW range to 2 to 2 Press ERASE DRAW to plot the function Use of the function H X with LDEC LAP or ILAP is not allowed in the calculator You have to use the main results provided earlier when dealing with the Heaviside step function i e L H t 1 s L...

Страница 502: ...ribution of the homogeneous solution yh t yo cos t y1 sin t The solution of an equation with a driving signal given by a Heaviside step function is shown below Example 3 Determine the solution to the equation d2 y dt2 y H t 3 where H t is Heaviside s step function Using Laplace transforms we can write L d2 y dt2 y L H t 3 L d2 y dt2 L y t L H t 3 The last term in this expression is L H t 3 1 s e 3...

Страница 503: ...ease notice that the variable X in this expression actually represents the variable t in the original ODE and that the variable ttt in this expression is a dummy variable Thus the translation of the solution in paper may be written as Example 4 Plot the solution to Example 3 using the same values of yo and y1 used in the plot of Example 1 above We now plot the function y t 0 5 cos t 0 25 sin t 1 s...

Страница 504: ...terval a t b f t Uo H t a H t b Triangular pulse with a maximum value Uo increasing from a t b decreasing from b t c f t Uo t a b a H t a H t b 1 t b b c H t b H t c Saw tooth pulse increasing to a maximum value Uo for a t b dropping suddenly down to zero at t b f t Uo t a b a H t a H t b Saw tooth pulse increasing suddenly to a maximum of Uo at t a then decreasing linearly to zero for a t b f t U...

Страница 505: ...cosine functions known as a Fourier series given by 1 0 2 sin 2 cos n n n t T n b t T n a a t f π π where the coefficients an and bn are given by 2 2 2 2 0 2 cos 2 1 T T T T n dt t T n t f T a dt t f T a π 2 2 2 sin T T n dt t T n t f b π The following exercises are in ALG mode with CAS mode set to Exact When you produce a graph the CAS mode will be reset to Approx Make sure to set it back to Exac...

Страница 506: ...ion with the Fourier expansion using these three terms shows that the fitting is acceptable for t 1 or thereabouts But then again we stipulated that T 2 1 Therefore the fitting is valid only between 1 t 1 Function FOURIER An alternative way to define a Fourier series is by using complex numbers as follows n n T t in c t f 2 exp π where ...

Страница 507: ...ine the coefficients c0 c1 and c2 for the function f t t2 t with period T 2 Note Because the integral used by function FOURIER is calculated in the interval 0 T while the one defined earlier was calculated in the interval T 2 T 2 we need to shift the function in the t axis by subtracting T 2 from t i e we will use g t f t 1 t 1 2 t 1 Using the calculator in ALG mode first we define functions f t a...

Страница 508: ...eries with three elements will be written as g t Re 1 3 π i 2 π2 exp i π t π i 1 2π2 exp 2 i π t A plot of the shifted function g t and the Fourier series fitting follows The fitting is somewhat acceptable for 0 t 2 although not as good as in the previous example ...

Страница 509: ...i e i n c 2 3 3 2 2 2 2 2 2 3 2 2 We can simplify this expression even further by using Euler s formula for complex numbers namely e2inπ cos 2nπ i sin 2nπ 1 i 0 1 since cos 2nπ 1 and sin 2nπ 0 for n integer Using the calculator you can simplify the expression in the equation writer O by replacing e2inπ 1 The figure shows the expression after simplification The result is cn i n π 2 n2 π2 Putting to...

Страница 510: ...to write this finite complex Fourier series as 2 exp X T n i n c k X F k k n π However because the function c n is not defined for n 0 we will be better advised to re write the expression as 0 0 c c k X F 2 exp 2 exp 1 X T n i n c X T n i n c k n π π Or in the calculator entry line as DEFINE F X k c0 c0 Σ n 1 k c n EXP 2 i π n X T c n EXP 2 i π n X T where T is the period T 2 The following screen ...

Страница 511: ... in the exponent As expected the coefficients are complex numbers The function F thus defined is fine for obtaining values of the finite Fourier series For example a single value of the series e g F 0 5 2 1 3 can be obtained by using CAS modes set to Exact step by step and Complex Accept change to Approx mode if requested The result is the value 0 40467 The actual value of the function g 0 5 is g ...

Страница 512: ...rm ñ simultaneously if using RPN mode Change the limits of the Plot Window ò as follows Press the soft menu keys ERASE DRAW to produce the plot Notice that the series with 5 terms hugs the graph of the function very closely in the interval 0 to 2 i e through the period T 2 You can also notice a periodicity in the graph of the series This periodicity is easy to visualize by expanding the horizontal...

Страница 513: ...ter finishing example 1 you already have a value of 2 stored in CAS variable PERIOD If you are not sure check the value of this variable and store a 2 in it if needed The coefficient c0 for the Fourier series is calculated as follows The calculator will request a change to Approx mode because of the integration of the function IFTE included in the integrand Accepting the change to Approx produces ...

Страница 514: ...s definition in the calculator i e dX T X n i EXP X 1 0 2 2 1 π 2 1 2 2 2 1 dX T X n i EXP X π where T 2 is the period The value of T can be stored using Typing the first integral above in the Equation Writer selecting the entire expression and using EVAL will produce the following Recall the einπ cos nπ i sin nπ 1 n Performing this substitution in the result above we have ...

Страница 515: ... second integral defining the coefficient cn namely Once again replacing einπ 1 n and using e2inπ 1 we get Press to copy this second result to the screen Now add ANS 1 and ANS 2 to get the full expression for cn Pressing will place this result in the Equation Writer where we can simplify SIMP it to read ...

Страница 516: ... completed example 1 you already have this function stored DEFINE F X k c0 c0 Σ n 1 k c n EXP 2 i π n X T c n EXP 2 i π n X T To compare the original function and the Fourier series we can produce the simultaneous plot of both functions The details are similar to those of example 1 except that here we use a horizontal range of 0 to 2 and a vertical range from 0 to 1 and adjust the equations to plo...

Страница 517: ...he Fourier series approximation Using k 2 or 5 terms in the series shows not so good a fitting The Fourier series can be used to generate a periodic triangular wave or saw tooth wave by changing the horizontal axis range for example from 2 to 4 The graph shown below uses k 5 Fourier series for a square wave A square wave can be generated by using the function 4 3 0 3 1 1 1 0 0 x if x if x if x g ...

Страница 518: ...n the calculator by using DEFINE g X IFTE X 1 AND X 3 1 0 The function plotted as follows horizontal range 0 to 4 vertical range 0 to 1 2 Using a procedure similar to that of the triangular shape in example 2 above you can find that 5 0 1 1 3 1 0 dX T c and We can simplify this expression by using einπ 2 in and e3inπ 2 i n to get ...

Страница 519: ...in the figure to the left above to define function c n The Fourier series is calculated with F X k c0 as in examples 1 and 2 above with c0 0 5 For example for k 5 i e with 11 components the approximation is shown below A better approximation is obtained by using k 10 i e For k 20 the fitting is even better but it takes longer to produce the graph ...

Страница 520: ...tion of X The second input item will be the characteristic equation corresponding to the homogeneous ODE shown above i e X 2 0 25 With these two inputs function LDEC produces the following result decimal format changed to Fix with 3 decimals Pressing allows you to see the entire expression in the Equation writer Exploring the equation in the Equation Writer reveals the existence of two constants o...

Страница 521: ...l part of this function Change the decimal mode to Standard and use the following The solution is shown below Fourier Transforms Before presenting the concept of Fourier transforms we ll discuss the general definition of an integral transform In general an integral transform is a transformation that relates a function f t to a new function F s by an ...

Страница 522: ...f φ ϖ where tan 1 2 2 n n n n n n a b b a A φ for n 1 2 The amplitudes An will be referred to as the spectrum of the function and will be a measure of the magnitude of the component of f x with frequency fn n T The basic or fundamental frequency in the Fourier series is f0 1 T thus all other frequencies are multiples of this basic frequency i e fn n f0 Also we can define an angular frequency ωn 2n...

Страница 523: ...angular frequency ω0 2π T becomes a very small quantity say ω Also the angular frequencies corresponding to ωn n ω0 n ω n 1 2 now take values closer and closer to each other suggesting the need for a continuous spectrum of values The non periodic function can be written therefore as where and dx x x f S sin 2 1 ω π ω The continuous spectrum is given by The functions C ω S ω and A ω are continuous ...

Страница 524: ...inuous spectrum A ω is calculated as Define this expression as a function by using function DEFINE à Then plot the continuous spectrum in the range 0 ω 10 as Definition of Fourier transforms Different types of Fourier transforms can be defined The following are the definitions of the sine cosine and full Fourier transforms and their inverses used in this Chapter Fourier sine transform ...

Страница 525: ... π ω 2 1 F Inverse Fourier transform proper dt e F t f F t iω ω ω 1 F Example 1 Determine the Fourier transform of the function f t exp t for t 0 and f t 0 for t 0 The continuous spectrum F ω is calculated with the integral ε ω ε ω π π 0 1 0 1 2 1 lim 2 1 dt e dt e t i t i 1 1 2 1 1 1 exp 1 2 1 lim ω π ω ε ω π ε i i i This result can be rationalized by multiplying numerator and denominator by the ...

Страница 526: ...mple shown above F ω 1 2π 1 ω2 1 2 The plot of F ω vs ω was shown earlier Some functions such as constant values sin x exp x x2 etc do not have Fourier transform Functions that go to zero sufficiently fast as x goes to infinity do have Fourier transforms Properties of the Fourier transform Linearity If a and b are constants and f and g functions then F a f b g a F f b F g Transformation of partial...

Страница 527: ...defined as 1 0 1 2 1 0 2 exp 1 n j j k n k n kj i x n X π The direct calculation of the sequence Xk involves n2 products which would involve enormous amounts of computer or calculator time particularly for large values of n The Fast Fourier Transform reduces the number of operations to the order of n log2n For example for n 100 the FFT requires about 664 operations while the direct calculation wou...

Страница 528: ... is the uniform random number generator provided by the calculator Generate 128 data points by using values of x in the interval 0 12 8 Store those values in an array and perform a FFT on the array First we define the function f x as a RPN program x 2 SIN 3 x 5 COS 5 x EVAL RAND 5 NUM and store this program in variable f Next type the following program to generate 2m data values between a and b Th...

Страница 529: ...he coefficients Xk represents a frequency spectrum of the original data To obtain the magnitude of the coefficients you could transform the array into a list and then apply function ABS to the list This is accomplished by using OBJ µ ƒ LIST Ê Finally you can convert the list back to a column vector to be stored in ΣDAT as follows OBJ 1 2 LIST ARRY STOΣ To plot the spectrum follow the instructions ...

Страница 530: ...s case is 0 to 64 while the vertical range is 1 to 10 To reproduce the signal whose spectrum is shown use function IFFT Since we left a copy of the spectrum in the stack a row vector all you need to do if find function IFFT in the MTH FFT menu or through the command catalog N As an alternative you could simply type the function name i e type ifft The signal is shown as an array row vector with com...

Страница 531: ... dy dx b y 0 where a and b are real constants is known as the Cauchy or Euler equation A solution to the Cauchy equation can be found by assuming that y x xn Type the equation as x 2 d1d1y x a x d1y x b y x 0 Then type and substitute the suggested solution y x x n SUBST The result is x 2 n x n 1 1 n 1 a x n x n 1 b x n 0 which simplifies to n n 1 x n a n x n b x n 0 Dividing by x n results in an a...

Страница 532: ...lator and can be recalled by using the function LEGENDRE given the order of the polynomial n The function LEGENDRE can be obtained from the command catalog N or through the menu ARITHMETIC POLYNOMIAL menu see Chapter 5 In RPN mode the first six Legendre polynomials are obtained as follows 0 LEGENDRE result 1 i e P0 x 1 0 1 LEGENDRE result X i e P1 x x 2 LEGENDRE result 3 X 2 1 2 i e P2 x 3x2 1 2 3...

Страница 533: ...ssel functions of the first kind for n integer are defined by 0 2 2 2 1 m n m m m n n m n m x x x J Regardless of whether we use ν non integer or n integer in the calculator we can define the Bessel functions of the first kind by using the following finite series Thus we have control over the function s order n and of the number of elements in the series k Once you have typed this function you can...

Страница 534: ...e cannot use them to obtain a general function to the equation Instead we introduce the Bessel functions of the second kind defined as Yν x Jν x cos νπ J ν x sin νπ for non integer ν and for n integer with n 0 by m m n m n m m m n n n x n m m h h x x x J x Y 2 0 2 1 2 1 2 ln 2 π γ π m n m n m n x m m n x 2 1 0 2 2 1 π where γ is the Euler constant defined by 0 5772156649 0 ln 1 3 1 2 1 1 lim r r r...

Страница 535: ...second kind Kν x π 2 I ν x Iν x sin νπ are also solutions of this ODE You can implement functions representing Bessel s functions in the calculator in a similar manner to that used to define Bessel s functions of the first kind but keeping in mind that the infinite series in the calculator need to be translated into a finite series Chebyshev or Tchebycheff polynomials The functions Tn x cos n cos ...

Страница 536: ...ult 1 i e U1 x 1 0 2 TCHEBYCHEFF result 2 X 2 1 i e T2 x 2x2 1 2 TCHEBYCHEFF result 2 X i e U2 x 2x 3 TCHEBYCHEFF result 4 X 3 3 X i e T3 x 4x3 3x 3 TCHEBYCHEFF result 4 X 2 1 i e U3 x 4x2 1 Laguerre s equation Laguerre s equation is the second order linear ODE of the form x d2 y dx2 1 x dy dx n y 0 Laguerre polynomials defined as 2 1 1 0 n dx e x d n e x L x L n x n n x n are solutions to Laguerr...

Страница 537: ...guerre polynomials use L x 0 L x 1 L x 2 L x 3 The results are L0 x L 1 x 1 x L 2 x 1 2x 0 5x2 L 3 x 1 3x 1 5x2 0 16666 x3 Weber s equation and Hermite polynomials Weber s equation is defined as d2 y dx2 n 1 2 x2 4 y 0 for n 0 1 2 A particular solution of this equation is given by the function y x exp x2 4 H x 2 where the function H x is the Hermite polynomial 2 1 1 1 2 2 0 n e dx d e x H H x n n ...

Страница 538: ...inear ordinary differential equations The use of this feature is presented using the following example The method used in the solution is a fourth order Runge Kutta algorithm preprogrammed in the calculator Example 1 Suppose we want to solve the differential equation dv dt 1 5 v1 2 with v 4 at t 0 We are asked to find v for t 2 First create the expression defining the derivative and store it into ...

Страница 539: ...EDIT Solves for v at t 0 25 v 3 285 OK INIT 5 OK SOLVE wait EDIT Changes initial value of t to 0 25 and final value of t to 0 5 solve for v 0 5 2 640 OK INIT 75 OK SOLVE wait EDIT Changes initial value of t to 0 5 and final value of t to 0 75 solve for v 0 75 2 066 OK INIT 1 OK SOLVE wait EDIT Changes initial value of t to 0 75 and final value of t to 1 solve for v 1 1 562 Repeat for t 1 25 1 50 1...

Страница 540: ...culator allows for the plotting of the solution of differential equations of the form Y T F T Y For our case we let Y x and T t therefore F T Y f t x exp t2 Let s plot the solution x t for t 0 to 5 by using the following keystroke sequence ô simultaneously if in RPN mode to enter PLOT environment Highlight the field in front of TYPE using the keys Then press CHOOS and highlight Diff Eq using the k...

Страница 541: ...ore Try the following EDIT L LABEL MENU to see axes labels and range Notice that the labels for the axes are shown as 0 horizontal for t and 1 vertical for x These are the definitions for the axes as given in the PLOT SETUP window ô i e H VAR 0 and V VAR 1 To see the graphical solution in detail use the following LL PICT To recover menu and return to PICT environment X Y To determine coordinates o...

Страница 542: ... want to find x x at t 2 Re write the ODE as w Aw where w x x T and A is the 2 x 2 matrix shown below 962 1 75 18 1 0 x x x x The initial conditions are now written as w 0 6 T for t 0 Note The symbol T means the transpose of the vector or matrix To solve this problem first create and store the matrix A e g in ALG mode Then activate the numerical differential equation solver by using Ï OK To solve ...

Страница 543: ...ows First prepare a table to write down your results Next within the SOLVE environment change the final value of the independent variable to 0 25 use 25 OK SOLVE wait EDIT Solves for w at t 0 25 w 0 968 1 368 OK INIT 5 OK SOLVE wait EDIT Changes initial value of t to 0 25 and final value of t to 0 5 solve again for w 0 5 0 748 2 616 OK INIT 75 OK SOLVE wait EDIT Changes initial value of t to 0 5 a...

Страница 544: ... 75 0 227 0 268 0 75 0 015 2 859 2 00 0 167 0 627 1 00 0 469 0 607 Graphical solution for a second order ODE Start by activating the differential equation numerical solver Ï OK The SOLVE screen should look like this Notice that the initial condition for the solution Soln w Init 0 includes the vector 0 6 Press L OK Next press ô simultaneously if in RPN mode to enter the PLOT environment Highlight t...

Страница 545: ... To plot the second curve we need to use the PLOT SETUP input form once more To reach this form from the graph above use CANCL L OK ô simultaneously if in RPN mode Change the value of the V Var field to 2 and press DRAW do not press ERASE or you would loose the graph produced above Use EDIT L LABEL MENU to see axes labels and range Notice that the x axis label is the number 0 indicating the indepe...

Страница 546: ...d an integration constant by using C Then we divide by FI x by using EXP 100 t The result is t 1 EXP 100 t C EXP 100 t i e y t 1 t C e100t Use of the initial condition y 0 1 results in 1 1 0 C e0 or C 0 the particular solution being y t 1 t Numerical solution If we attempt a direct numerical solution of the original equation dy dt 100y 100t 101 using the calculator s own numerical solver we find t...

Страница 547: ...r the cases C 0 or C 0 e g for C 1 t 0 1 C e100t 22026 The calculator s ODE numerical solver allows for the solution of stiff ODEs by selecting the option _Stiff in the SOLVE Y T F T Y screen With this option selected you need to provide the values of f y and f t For the case under consideration f y 100 and f t 100 Enter those values in the corresponding fields of the SOLVE Y T F T Y screen When d...

Страница 548: ...nction you will prepare your stack as follows 3 x y f x y 2 ε x 1 xfinal The value in the first stack level is the value of the independent variable where you want to find your solution i e you want to find yfinal fs xfinal where fs x represents the solution to the differential equation The second stack level may contain only the value of ε and the step x will be taken as a small default value Aft...

Страница 549: ...he expression Thus the input stack for this function will look as follows 3 x y f x y f x f y 2 ε x 1 xfinal The value in the first stack level is the value of the independent variable where you want to find your solution i e you want to find yfinal fs xfinal where fs x represents the solution to the differential equation The second stack level may contain only the value of ε and the step x will b...

Страница 550: ...imate of the next step in the independent variable The function returns the input list the tolerance and the next step in the independent variable that satisfies that tolerance Thus the input stack looks as follows 3 x y f x y 2 ε 1 x After running this function the stack will show the lines 3 x y f x y 2 ε 1 x next Thus this function is used to determine the appropriate size of a time step to sat...

Страница 551: ...able and the current method CURRENT used to arrive at the next step Thus the input stack looks as follows 4 x y f x y 3 ε 2 x 1 LAST After running this function the stack will show the lines 4 x y f x y 3 ε 2 x next 1 CURRENT Thus this function is used to determine the appropriate size of a time step x next to satisfy the required tolerance and the method used to arrive at that result CURRENT The ...

Страница 552: ... to determine the increment in the solution y as well as the absolute error error The following screen shots show the RPN stack before and after application of function RKFERR These result show that y 0 827 and error 1 89 10 6 Function RSBERR This function performs similarly to RKERR but with the input elements listed for function RRK Thus the input stack for this function will look as follows 2 x...

Страница 553: ...at if Dx is reduced to 0 01 y 0 00307 and error 0 000547 Note As you execute the commands in the DIFF menu values of x and y will be produced and stored as variables in your calculator The results provided by the functions in this section will depend on the current values of x and y Therefore some of the results illustrated above may differ from what you get in your calculator ...

Страница 554: ...d the PROBABILITY option option 7 to show the following functions see right hand side figure below In this section we discuss functions COMB PERM factorial RAND and RDZ Factorials combinations and permutations The factorial of an integer n is defined as n n n 1 n 2 3 2 1 By definition 0 1 Factorials are used in the calculation of the number of permutations and combinations of objects For example t...

Страница 555: ...or provides a random number generator that returns a uniformly distributed random real number between 0 and 1 The generator is able to produce sequences of random numbers However after a certain number of times a very large number indeed the sequence tends to repeat itself For that reason the random number generator is more properly referred to as a pseudo random number generator To generate a ran...

Страница 556: ... with the same number you can reproduce the same sequence more than once For example try the following RDZ 0 25 Use 0 25 as the seed RAND First random number 0 75285 RAND Second random number 0 51109 RAND Third random number 0 085429 Re start the sequence RDZ 0 25 Use 0 25 as the seed RAND First random number 0 75285 RAND Second random number 0 51109 RAND Third random number 0 085429 To generate a...

Страница 557: ... f x X P x F Next we will define a number of functions to calculate discrete probability distributions We suggest that you create a sub directory say HOME STATS DFUN Discrete FUNctions where we will define the probability mass function and cumulative distribution function for the binomial and Poisson distributions Binomial distribution The probability mass function of the binomial distribution is ...

Страница 558: ...currences per unit time length area volume etc The cumulative distribution function for the Poisson distribution is given by 2 1 0 0 x x f x F x k λ λ Next use function DEFINE à to define the following probability mass functions pmf and cumulative distribution functions cdf DEFINE pmfb n p x COMB n x p x 1 p n x DEFINE cdfb n p x Σ k 0 x pmfb n p k DEFINE pmfp λ x EXP λ λ x x DEFINE cdfp λ x Σ k 0...

Страница 559: ...here P X x stands for the probability that the random variable X is less than the value x In this section we describe several continuous probability distributions including the gamma exponential beta and Weibull distributions These distributions are described in any statistics textbook Some of these distributions make use of a the Gamma function defined earlier which is calculated in the calculato...

Страница 560: ...is given by 0 0 1 0 1 1 1 Γ Γ Γ β α β α β α β α x for x x x f As in the case of the gamma distribution the corresponding cdf for the beta distribution is also given by an integral with no closed form solution The Weibull distribution The pdf for the Weibull distribution is given by 0 0 0 exp 1 β α α β α β β x for x x x f While the corresponding cdf is given by 0 0 0 exp 1 β α α β x for x x F Funct...

Страница 561: ... Gamma cdf i e the function gcdf should be modified to read x NUM 0 x gpdf t t and stored back into gcdf Repeat the procedure for βcdf Unlike the discrete functions defined earlier the continuous functions defined in this section do not include their parameters α and or β in their definitions Therefore you don t need to enter them in the display to calculate the functions However those parameters ...

Страница 562: ...ed to statistical inference These distributions are the normal distribution the Student s t distribution the Chi square χ2 distribution and the F distribution The functions provided by the calculator to evaluate probabilities for these distributions are contained in the MTH PROBABILITY menu introduced earlier in this chapter The functions are NDIST UTPN UTPT UTPC and UTPF Their application is desc...

Страница 563: ...value of the upper tail normal distribution UTPN we need to enter the following values the mean µ the variance σ2 and the value x e g UTPN µ σ2 x For example check that for a normal distribution with µ 1 0 σ2 0 5 UTPN 0 75 0 638163 Use UTPN 1 0 0 5 0 75 0 638163 Different probability calculations for normal distributions X is N µ σ2 can be defined using the function UTPN as follows P X a 1 UTPN µ ...

Страница 564: ...er ν and the value of t i e UTPT ν t The definition of this function is therefore t t t T P dt t f dt t f t UTPT 1 1 ν For example UTPT 5 2 5 2 7245 E 2 Other probability calculations for the t distribution can be defined using the function UTPT as follows P T a 1 UTPT ν a P a T b P T b P T a 1 UTPT ν b 1 UTPT ν a UTPT ν a UTPT ν b P T c UTPT ν c Examples Given ν 12 determine P T 0 5 1 UTPT 12 0 5...

Страница 565: ...are variable x i e UTPC ν x For example UTPC 5 2 5 0 776495 Different probability calculations for the Chi squared distribution can be defined using the function UTPC as follows P X a 1 UTPC ν a P a X b P X b P X a 1 UTPC ν b 1 UTPC ν a UTPC ν a UTPC ν b P X c UTPC ν c Examples Given ν 6 determine P X 5 32 1 UTPC 6 5 32 0 4965 P 1 2 X 10 5 UTPC 6 1 2 UTPC 6 10 5 0 8717 P X 20 UTPC 6 20 2 769 E 3 T...

Страница 566: ...fined using the function UTPF as follows P F a 1 UTPF νN νD a P a F b P F b P F a 1 UTPF νN νD b 1 UTPF νN νD a UTPF νN νD a UTPF νN νD b P F c UTPF νN νD a Example Given νN 10 νD 5 find P F 2 1 UTPF 10 5 2 0 7700 P 5 F 10 UTPF 10 5 5 UTPF 10 5 10 3 4693 E 2 P F 5 UTPF 10 5 5 4 4808 E 2 Inverse cumulative distribution functions For a continuous random variable X with cumulative density function cd...

Страница 567: ...the numerical solver will not be feasible because of the integral sign involved in the expression However a graphical solution is possible Details on how to find the root of a graph are presented in Chapter 12 To ensure numerical results change the CAS setting to Approx The function to plot for the Gamma distribution is Y X 0 X z α 1 exp z β β α GAMMA α z p For the Beta distribution the function t...

Страница 568: ...n the plot screen You will only get the message Constant Shown in the screen However if you press at this point the approximate root will be listed in the display Two roots are shown in the right hand figure below Alternatively you can use function TRACE X Y to estimate the roots by tracing the curve near its intercepts with the x axis Two estimates are shown below These estimates suggest solution...

Страница 569: ...tion Also the symbol ν the lower case Greek letter no is not available in the calculator You can use for example γ gamma instead of ν The letter γ is available thought the character set For example to obtain the value of x for a normal distribution with µ 10 σ2 2 with p 0 25 store the equation p 1 UTPN µ σ2 x into variable EQ figure in the left hand side below Then launch the numerical solver to g...

Страница 570: ...r one of the variables Examples of the UTPT UTPC and UPTF are shown below Notice that in all the examples shown above we are working with p P X x In many statistical inference problems we will actually try to find the value of x for which P X x α Furthermore for the normal distribution we most likely will be working with the standard normal distribution in which µ 0 and σ2 1 The standard normal va...

Страница 571: ...Page 17 18 With these four equations whenever you launch the numerical solver you have the following choices Examples of solution of equations EQNA EQTA EQCA and EQFA are shown below ...

Страница 572: ... in this Chapter First however we demonstrate how to enter data for statistical analysis Entering data For the analysis of a single set of data a sample we can use applications number 1 2 and 4 from the list above All of these applications require that the data be available as columns of the matrix ΣDAT This can be accomplished by entering the data in columns using the matrix writer This operation...

Страница 573: ...ress Ù Press OK to select 1 Single var There will be available to you an input form labeled SINGLE VARIABLE STATISTICS with the data currently in your ΣDAT variable listed in the form as a vector Since you only have one column the field Col should have the value 1 in front of it The Type field determines whether you are working with a sample or a population the default setting is Sample Move the c...

Страница 574: ... of measures or statistics There are measures of central tendency such as the mean median and mode and measures of spreading such as the range variance and standard deviation Measures of central tendency The mean or arithmetic mean of the sample x is defined as the average value of the sample elements n i i x n x 1 1 The value labeled Total obtained above represents the summation of the values of ...

Страница 575: ...Store this program under the name MED An example of application of this program is shown next Example 2 To run the program first you need to prepare your ΣDAT matrix Then enter the number of the column in ΣDAT whose median you want to find and press MED For the data currently in ΣDAT entered in an earlier example use program MED to show that Median 2 15 The mode of a sample is better determined fr...

Страница 576: ...riance and standard deviation however will be given by Variance 0 852 Std Dev 0 923 Obtaining frequency distributions The application 2 Frequencies in the STAT menu can be used to obtain frequency distributions for a set of data Again the data must be present in the form of a column vector stored in variable ΣDAT To get started press Ù OK The resulting input form contains the following fields ΣDAT...

Страница 577: ...longs to the i th class if xBi xj xB i 1 The application 2 Frequencies in the STAT menu will perform this frequency count and will keep track of those values that may be below the minimum and above the maximum class boundaries i e the outliers Example 1 In order to better illustrate obtaining frequency distributions we want to generate a relatively large data set say 200 points by using the follow...

Страница 578: ...ack as a column vector in stack level 2 and a row vector of two components in stack level 1 The vector in stack level 1 is the number of outliers outside of the interval where the frequency count was performed For this case I get the values 25 22 indicating that there are in my ΣDAT vector 25 values smaller than 10 and 22 larger than 90 Press ƒ to drop the vector of outliers from the stack The rem...

Страница 579: ...17 49 4 40 50 45 17 66 5 50 60 55 22 88 6 60 70 65 22 110 7 70 80 75 24 134 k 8 80 90 85 19 153 XBk outliers above range 22 Given the column vector of frequencies generated by the calculator you can obtain a cumulative frequency vector by using the following program in RPN mode DUP SIZE 1 GET freq k k 1 0 CON cfreq freq 1 1 EVAL cfreq 1 1 STO 2 k FOR j cfreq j 1 1 freq j 1 EVAL cfreq j 1 STO NEXT ...

Страница 580: ...ow you how to use the first method to generate a histogram Example 1 Using the 200 data points generated in the example above stored as a column vector in ΣDAT generate a histogram plot of the data using X Min 10 Bin Count 16 and Bin Width 5 First press ô simultaneously if in RPN mode to enter the PLOT SETUP screen Within this screen change Type to Histogram and check that the option Col 1 is sele...

Страница 581: ... STAT menu can be used to fit linear logarithmic exponential and power functions to data sets x y stored in columns of the ΣDAT matrix In order for this program to be effective you need to have at least two columns in your ΣDAT variable Example 1 Fit a linear relationship to the data shown in the table below First enter the two rows of data into column in the variable ΣDAT by using the matrix writ...

Страница 582: ...sample of data points x y we define the sample covariance as 1 1 1 y y x x n s i n i i xy The sample correlation coefficient for x y is defined as y x xy xy s s s r Where sx sy are the standard deviations of x and y respectively i e 2 1 2 1 1 x x n s n i i x 2 1 2 1 1 y y n s n i i y The values sxy and rxy are the Covariance and Correlation respectively obtained by using the Fit data feature of th...

Страница 583: ...correlation coefficient rξη is η ξ ξη ξη s s s r The general form of the regression equation is η A Bξ Best data fitting The calculator can determine which one of its linear or linearized relationship offers the best fitting for a set of x y data points We will illustrate the use of this feature with an example Suppose you want to find which one of the data fitting functions provides the best fit ...

Страница 584: ...tatistics To get started press Ù once more move to the fourth option using the down arrow key and press OK The resulting input form contains the following fields ΣDAT the matrix containing the data of interest X Col Y Col these options apply only when you have more than two columns in the matrix ΣDAT By default the x column is column 1 and the y column is column 2 _ΣX _ ΣY summary statistics that ...

Страница 585: ...at divide a data set into 100 parts The basic procedure to calculate the 100 p th Percentile 0 p 1 in a sample of size n is as follows 1 Order the n observations from smallest to largest 2 Determine the product n p A If n p is not an integer round it up to the next integer and find the corresponding ordered value B If n p is an integer say k calculate the mean of the k th and k 1 th ordered observ...

Страница 586: ...soft menu The STAT soft menu can be accessed by using in RPN mode the command 96 MENU You can create your own program say STATm to activate the STAT soft menu directly The contents of this program are simply 96 MENU The STAT soft menu contains the following functions Pressing the key corresponding to any of these menus provides access to different functions as described below The DATA sub menu The...

Страница 587: ...t data fitting Default 0 Slope shows slope of most recent data fitting Default 0 Model shows current data fit model Default LINFIT The functions listed in the soft menu keys operate as follows XCOL entered as n XCOL changes Xcol to n YCOL entered as n YCOL changes Ycol to n ΣPAR shows statistical parameters RESET reset parameters to default values INFO shows statistical parameters The MODL sub men...

Страница 588: ...based on n rather than on n 1 of each column in ΣDATA matrix PVAR shows population variance of each column in ΣDATA matrix MINΣ shows average of each column in ΣDATA matrix The PLOT sub menu The PLOT sub menu contains functions that are used to produce plots with the data in the ΣDATA matrix The functions included are BARPL produces a bar plot with data in Xcol column of the ΣDATA matrix HISTP pro...

Страница 589: ...recent fitting PCOV shows population co variance for the most recent fitting The SUMS sub menu The SUMS sub menu contains functions used to obtain summary statistics of the data in columns Xcol and Ycol of the ΣDATA matrix ΣX provides the sum of values in Xcol column ΣY provides the sum of values in Ycol column ΣX 2 provides the sum of squares of values in Xcol column ΣY 2 provides the sum of squa...

Страница 590: ...produces 11 52 46 08 445084146 33 PSDEV produces 3 142 6 284 19532 04 PVAR produces 9 87 39 49 381500696 85 Data 55066 5 21 0 10 24743 9 19 2 9 2245 1 15 8 6 612 5 12 5 5 25 9 5 2 2 101 9 8 7 3 8 7 7 3 1 1 Generate a scatterplot of the data in columns 1 and 2 and fit a straight line to it STAT PAR RESET resets statistical parameters L STAT PLOT SCATR produces scatterplot STATL draws data fit as a ...

Страница 591: ...0 75 1 PREDY produces 3 50 CORR produces 1 0 COV produces 23 04 L PCOV produces 19 74 Obtain summary statistics for data in columns 1 and 2 STAT SUMS X produces 38 5 Y produces 87 5 X2 produces 280 87 Y2 produces 1370 23 XY produces 619 49 N produces 7 Fit data using columns 1 x and 3 y using a logarithmic fitting L STAT PAR 3 YCOL select Ycol 3 and MODL LOGFI select Model Logfit ...

Страница 592: ...he best fitting by using STAT PAR MODL BESTF shows EXPFIT as the best fit for these data L STAT FIT LINE produces 2 6545 EXP 0 9927 X CORR produces 0 99995 good correlation 2300 PREDX produces 6 8139 5 2 PREDY produces 463 37 L STAT PLOT SCATR produce scattergram of y vs x STATL show line for log fitting To return to STAT menu use L STAT To get your variable menu back use J ...

Страница 593: ... θ a random sample of observations X1 X2 X3 Xn of size n can be used to estimate θ Sampling distribution the joint probability distribution of X1 X2 X3 Xn A statistic any function of the observations that is quantifiable and does not contain any unknown parameters A statistic is a random variable that provides a means of estimation Point estimation when a single value of the parameter θ is provide...

Страница 594: ...val containing an unknown parameter θ Confidence level or confidence coefficient is the quantity 1 α where 0 α 1 such that P Cl θ Cu 1 α where P represents a probability see Chapter 17 The previous expression defines the so called two sided confidence limits A lower one sided confidence interval is defined by Pr Cl θ 1 α An upper one sided confidence interval is defined by Pr θ Cu 1 α The paramete...

Страница 595: ...ation σ The 100 1 α i e 99 95 90 etc central two sided confidence interval for the population mean µ is X tn 1 α 2 S n X tn 1 α 2 S n where tn 1 α 2 is Student s t variate with ν n 1 degrees of freedom and probability α 2 of exceedence The one sided upper and lower 100 1 α confidence limits for the population mean µ are respectively X tn 1 α 2 S n and X tn 1 α 2 S n Small samples and large samples...

Страница 596: ...Let S1 and S2 be independent statistics from two populations based on samples of sizes n1 and n2 respectively Also let the respective means and standard errors of the sampling distributions of those statistics be µS1 and µS2 and σS1 and σS2 respectively The differences between the statistics from the two populations S1 S2 have a sampling distribution with mean µ S1 S2 µS1 µS2 and standard error σ ...

Страница 597: ...n S z X X α α If one of the samples is small i e n1 30 or n2 30 and with unknown but equal population variances σ1 2 σ2 2 we can obtain a pooled estimate of the variance of µ1 µ2 as sp 2 n1 1 s1 2 n2 1 s2 2 n1 n2 2 In this case the centered confidence intervals for the sum and difference of the mean values of the populations i e µ1 µ2 are given by 2 2 2 1 2 2 2 1 p p s t X X s t X X α ν α ν where ...

Страница 598: ...to 1 1 2 2 2 2 1 1 2 1 2 2 2 2 1 2 1 n n S n n S n S n S ν Determining confidence intervals The application 6 Conf Interval can be accessed by using Ù OK The application offers the following options These options are to be interpreted as follows 1 Z INT 1 µ Single sample confidence interval for the population mean µ with known population variance or for large samples with unknown population varian...

Страница 599: ...s Example 1 Determine the centered confidence interval for the mean of a population if a sample of 60 elements indicate that the mean value of the sample is x 23 2 and its standard deviation is s 5 2 Use α 0 05 The confidence level is C 1 α 0 95 Select case 1 from the menu shown above by pressing OK Enter the values required in the input form as shown Press HELP to obtain a screen explaining the m...

Страница 600: ...he mean value 23 2 and the corresponding interval limits 21 88424 and 24 51576 Press TEXT to return to the previous results screen and or press OK to exit the confidence interval environment The results will be listed in the calculator s display Example 2 Data from two samples samples 1 and 2 indicate that x1 57 8 and x2 60 0 The sample sizes are n1 45 and n2 75 If it is known that the populations...

Страница 601: ...ation proportion that would favor increasing taxes Press Ù OK to access the confidence interval feature in the calculator Press OK to select option 3 Z INT µ 1 µ2 Enter the following values When done press OK The results as text and graph are shown below Example 4 Determine a 90 confidence interval for the difference between two proportions if sample 1 shows 20 successes out of 120 trials and samp...

Страница 602: ...wn below Example 5 Determine a 95 confidence interval for the mean of the population if a sample of 50 elements has a mean of 15 5 and a standard deviation of 5 The population s standard deviation is unknown Press Ù OK to access the confidence interval feature in the calculator Press OK to select option 5 T INT µ Enter the following values When done press OK The results as text and graph are shown...

Страница 603: ...13 2 s 2 24 5 Press Ù OK to access the confidence interval feature in the calculator Press OK to select option 6 T INT µ1 µ2 Enter the following values hen done press OK The results as text and graph are shown below These results assume that the values s1 and s2 are the population standard deviations If these values actually represent the samples standard deviations you should enter the same value...

Страница 604: ...istribution with ν n 1 degrees of freedom The 1 α 100 two sided confidence interval is found from Pr χ2 n 1 1 α 2 n 1 S2 σ2 χ2 n 1 α 2 1 α The confidence interval for the population variance σ2 is therefore n 1 S2 χ2 n 1 α 2 n 1 S2 χ2 n 1 1 α 2 where χ2 n 1 α 2 and χ2 n 1 1 α 2 are the values that a χ2 variable with ν n 1 degrees of freedom exceeds with probabilities α 2 and 1 α 2 respectively The...

Страница 605: ...6 7 62116179676 n 1 S2 χ2 n 1 1 α 2 25 1 12 5 12 4011502175 24 1913044144 Thus the 95 confidence interval for this example is 7 62116179676 σ2 24 1913044144 Hypothesis testing A hypothesis is a declaration made about a population for instance with respect to its mean Acceptance of the hypothesis is based on a statistical test on a sample taken from the population The consequent action and decision...

Страница 606: ...termine whether the computed value of the test statistic is within or outside the critical region If the test statistic is within the critical region then we say that the quantity we are testing is significant at the 100α percent level Notes 1 For the example under consideration the alternate hypothesis H1 µ1 µ2 0 produces what is called a two tailed test If the alternate hypothesis is H1 µ1 µ2 0 ...

Страница 607: ...ences of a Type I error are more serious choose smaller values of α say 0 01 or even 0 001 The value of β i e the probability of making an error of Type II depends on α the sample size n and on the true value of the parameter tested Thus the value of β is determined after the hypothesis testing is performed It is customary to draw graphs showing β or the power of the test 1 β as a function of the ...

Страница 608: ...zο or tο and compare it to α to decide whether or not to reject the null hypothesis The P value for a two sided test is defined as either P value P z zo or P value P t to The criteria to use for hypothesis testing is Reject Ho if P value α Do not reject Ho if P value α The P value for a two sided test can be calculated using the probability functions in the calculator as follows If using z P value...

Страница 609: ...d a standard deviation s This test is referred to as a one sided or one tailed test The procedure for performing a one side test starts as in the two tailed test by calculating the appropriate statistic for the test to or zo as indicated above Next we use the P value associated with either zο or tο and compare it to α to decide whether or not to reject the null hypothesis The P value for a two sid...

Страница 610: ...grees of freedom is P value UTPT 24 0 7142 UTPT 24 0 7124 0 2409 since 0 2409 0 05 i e P value α we cannot reject the null hypothesis Ho µ 22 0 Inferences concerning two means The null hypothesis to be tested is Ho µ1 µ2 δ at a level of confidence 1 α 100 or significance level α using two samples of sizes n1 and n2 mean values x1 and x2 and standard deviations s1 and s2 If the populations standard...

Страница 611: ... P value UTPN 0 1 zo If using t P value UTPT ν to The criteria to use for hypothesis testing is Reject Ho if P value α Do not reject Ho if P value α Paired sample tests When we deal with two samples of size n with paired data points instead of testing the null hypothesis Ho µ1 µ2 δ using the mean values and standard deviations of the two samples we need to treat the problem as a single sample of t...

Страница 612: ...istic to test is z0 p p0 sp Instead of using the P value as a criterion to accept or not accept the hypothesis we will use the comparison between the critical value of z0 and the value of z corresponding to α or α 2 Two tailed test If using a two tailed test we will find the value of z α 2 from Pr Z zα 2 1 Φ zα 2 α 2 or Φ z α 2 1 α 2 where Φ z is the cumulative distribution function CDF of the sta...

Страница 613: ...les will be estimated respectively as s1 2 p1 1 p1 n1 k1 n1 k1 n1 3 and s2 2 p2 1 p2 n2 k2 n2 k2 n2 3 And the variance of the difference of proportions is estimated from sp 2 s1 2 s2 2 Assume that the Z score Z p1 p2 p0 sp follows the standard normal distribution i e Z N 0 1 The particular value of the statistic to test is z0 p1 p2 p0 sp Two tailed test If using a two tailed test we will find the ...

Страница 614: ...wn population variance or for large samples with unknown population variance 2 Z Test µ1 µ2 Hypothesis testing for the difference of the population means µ1 µ2 with either known population variances or for large samples with unknown population variances 3 Z Test 1 p Single sample hypothesis testing for the proportion p for large samples with unknown population variance 4 Z Test p1 p2 Hypothesis te...

Страница 615: ...s OK to select option 1 Z Test 1 µ Enter the following data and press OK You are then asked to select the alternative hypothesis Select µ 150 and press OK The result is Then we reject H0 µ 150 against H1 µ 150 The test z value is z0 5 656854 The P value is 1 54 10 8 The critical values of zα 2 1 959964 corresponding to critical x range of 147 2 152 8 This information can be observed graphically by...

Страница 616: ...ter the following data and press OK Select the alternative hypothesis H1 µ 150 and press OK The result is We reject the null hypothesis H0 µ0 150 against the alternative hypothesis H1 µ 150 The test t value is t0 5 656854 with a P value 0 000000393525 The critical value of t is tα 1 676551 corresponding to a critical x 152 371 Press GRAPH to see the results graphically as follows Example 3 Data fr...

Страница 617: ...t the alternative hypothesis µ1 µ2 and press OK The result is Thus we accept more accurately we do not reject the hypothesis H0 µ1 µ2 0 or H0 µ1 µ2 against the alternative hypothesis H1 µ1 µ2 0 or H1 µ1 µ2 The test t value is t0 1 341776 with a P value 0 09130961 and critical t is tα 1 659782 The graphical results are These three examples should be enough to understand the operation of the hypothe...

Страница 618: ...rly max x y produces the maximum value of x or y UTPC ν x represents the calculator s upper tail probabilities for ν n 1 degrees of freedom The test criteria are the same as in hypothesis testing of means namely Reject Ho if P value α Do not reject Ho if P value α Please notice that this procedure is valid only if the population from which the sample was taken is a Normal population Example 1 Cons...

Страница 619: ...lowing table shows how to select the numerator and denominator for Fo depending on the alternative hypothesis chosen ____________________________________________________________________ Alternative Test Degrees hypothesis statistic of freedom ____________________________________________________________________ H1 σ1 2 σ2 2 one sided Fo s2 2 s1 2 νN n2 1 νD n1 1 H1 σ1 2 σ2 2 one sided Fo s1 2 s2 2 ...

Страница 620: ...88 0 05 i e P value α therefore we cannot reject the null hypothesis that Ho σ1 2 σ2 2 Additional notes on linear regression In this section we elaborate the ideas of linear regression presented earlier in the chapter and present a procedure for hypothesis testing of regression parameters The method of least squares Let x independent non random variable and Y dependent random variable The regressi...

Страница 621: ... b x a y x 1 2 1 1 This is a system of linear equations with a and b as the unknowns which can be solved using the linear equation features of the calculator There is however no need to bother with these calculations because you can use the 3 Fit Data option in the Ù menu as presented earlier ____________________________________________________________________ Notes a b are unbiased estimators of ...

Страница 622: ...variance of x y are given respectively by 1 n S s xx x 1 n S s yy y and 1 n S s yx xy Also the sample correlation coefficient is yy xx xy xy S S S r In terms of x y Sxx Syy and Sxy the solution to the normal equations is x b y a 2 x xy xx xy s s S S b Prediction error The regression curve of Y on x is defined as Y Α Β x ε If we have a set of n data points xi yi then we can write Yi Α Β xi εI i 1 2...

Страница 623: ...ion with ν n 2 degrees of freedom and n represents the number of points in the sample Hypothesis testing on the slope Β Null hypothesis H0 Β Β0 tested against the alternative hypothesis H1 Β Β0 The test statistic is t0 b Β0 se Sxx where t follows the Student s t distribution with ν n 2 degrees of freedom and n represents the number of points in the sample The test is carried out as that of a mean ...

Страница 624: ...α 2 se 1 1 n x0 x 2 Sxx 1 2 Y0 a b x t n 2 α 2 se 1 1 n x0 x 2 Sxx 1 2 Procedure for inference statistics for linear regression using the calculator 1 Enter x y as columns of data in the statistical matrix ΣDAT 2 Produce a scatterplot for the appropriate columns of ΣDAT and use appropriate H and V VIEWS to check linear trend 3 Use Ù OK to fit straight line and get a b sxy Covariance and rxy Correl...

Страница 625: ...a option in the Ù menu to get 3 86 3 24 X 2 Correlation 0 989720229749 1 Covariance 2 025 These results are interpreted as a 0 86 b 3 24 rxy 0 989720229749 and sxy 2 025 The correlation coefficient is close enough to 1 0 to confirm the linear trend observed in the graph From the Single var option of the Ù menu we find x 3 sx 0 790569415042 y 8 86 sy 2 58804945857 Next with n 5 calculate 5 2 42 790...

Страница 626: ...ld be the case we test the null hypothesis H0 Α 0 against the alternative hypothesis H1 Α 0 at the level of significance α 0 05 The test statistic is t0 a 0 1 n x2 Sxx 1 2 0 86 1 5 32 2 5 0 44117 The critical value of t for ν n 2 3 and α 2 0 025 can be calculated using the numerical solver for the equation α UTPT γ t developed in Chapter 17 In this program γ represents the degrees of freedom n 2 a...

Страница 627: ...se t0 tα 2 we must reject the null hypothesis H1 Β 0 at the level of significance α 0 05 for the linear fitting of Example 1 Multiple linear fitting Consider a data set of the form x1 x2 x3 xn y x11 x21 x31 xn1 y1 x12 x22 x32 xn2 y2 x13 x32 x33 xn3 y3 x1 m 1 x 2 m 1 x 3 m 1 x n m 1 ym 1 x1 m x 2 m x 3 m x n m ym Suppose that we search for a data fitting of the form y b0 b1 x1 b2 x2 b3 x3 bn xn You...

Страница 628: ...can proceed as follows First within your HOME directory create a sub directory to be called MPFIT Multiple linear and Polynomial data FITting and enter the MPFIT sub directory Within the sub directory type this program X y X TRAN X INV X TRAN y and store it in a variable called MTREG MulTiple REGression Next enter the matrices X and b into the stack 1 1 2 3 1 2 1 2 5 3 1 2 5 1 3 5 4 5 2 5 1 4 4 5 ...

Страница 629: ...a set In other words we seek a fitting of the form y b0 b1 x b2 x2 b3 x3 bp xp You can obtain the least square approximation to the values of the coefficients b b0 b1 b2 b3 bp by putting together the matrix X _ _ 1 x1 x1 2 x1 3 x1 p 1 y1 p 1 x2 x2 2 x2 3 x2 p 1 y2 p 1 x3 x3 2 x3 3 x3 p 1 y3 p 1 xn x n 2 xn 3 x n p 1 yn p _ _ Then the vector of coefficients is obtained from b XT X 1 XT y where y is...

Страница 630: ...on Thus we can write a program to calculate the polynomial fitting that can take advantage of the program already developed for multiple linear fitting We need to add to this program the steps 1 through 3 listed above The algorithm for the program therefore can be written as follows Enter vectors x and y of the same dimension as lists Note since the function VANDERMONDE uses a list as input it is ...

Страница 631: ... 1 FOR j Start loop j n 1 n 2 p 1 step 1 j COL DROP Remove column and drop it from stack 1 STEP Close FOR STEP loop ELSE IF p n 1 THEN n 1 Calculate n 1 p 1 Calculate p 1 FOR j Start a loop with j n n 1 p 1 x j Calculate xj as a list OBJ ARRY Convert list to array j COL Add column to matrix NEXT Close FOR NEXT loop END Ends second IF clause END Ends first IF clause Its result is X y OBJ ARRY Conve...

Страница 632: ...f the program POLY Thus proceed as follows 2 3 3 2 4 5 1 65 9 32 1 18 6 24 3 45 9 89 1 22 xx K 179 72 562 30 1969 11 65 87 31220 89 32 81 6731 48 737 41 39248 46 33 45 yy K To fit the data to polynomials use the following xx yy 2 POLY Result 4527 73 3958 52 742 23 i e y 4527 73 39 58x 742 23x2 xx yy 3 POLY Result 998 05 1303 21 505 27 79 23 i e y 998 05 1303 21x 505 27x2 79 23x3 xx yy 4 POLY Resul...

Страница 633: ...to program these criteria we present some definitions Given the vectors x and y of data to be fit to the polynomial equation we form the matrix X and use it to calculate a vector of polynomial coefficients b We can calculate a vector of fitted data y by using y X b An error vector is calculated by e y y The sum of square errors is equal to the square of the magnitude of the error vector i e SSE e ...

Страница 634: ...te p 1 FOR j Start loop with j n n 1 p 1 x j Calculate xj as a list OBJ ARRY Convert list to array j COL Add column to matrix NEXT Close FOR NEXT loop END Ends second IF clause END Ends first IF clause Produces X y OBJ ARRY Convert list y to an array X yv Enter matrix and array as X and y Open subprogram 3 X yv MTREG X and y used by program MTREG NUM If needed converts to floating point b Resultin...

Страница 635: ...ient r Using the POLYR program for values of p between 2 and 6 produce the following table of values of the correlation coefficient r and the sum of square errors SSE p r SSE 2 0 9971908 10731140 01 3 0 9999768 88619 36 4 0 9999999 7 48 5 0 9999999 8 92 6 0 9999998 432 61 While the correlation coefficient is very close to 1 0 for all values of p in the table the values of SSE vary widely The small...

Страница 636: ...ystem the relative position of digits determines its value In general a number n in base b can be written as a series of digits n a1a2 an c1c2 cm b The point separates n integer digits from m decimal digits The value of the number converted to our customary decimal system is calculated by using n a1 bn 1 a2 bn 2 anb0 c1 b 1 c2 b 2 cm b m For example 15 234 10 1 101 5 100 2 10 1 3 10 2 4 10 3 and 1...

Страница 637: ... â the 3 key To select which number system current base will be used for numbers preceded by select one of the following functions in the first BASE menu i e HEX adecimal DEC imal OCT al or BIN ary For example if HEX is selected any number written in the calculator that starts with will be a hexadecimal number Thus you can write numbers such as 53 A5B etc in this system As different systems are se...

Страница 638: ...the function B R will convert any hexadecimal number preceded by into a decimal number while the function R B works in the opposite direction Try the following exercises HEX is the current base The following examples show conversions when the base is the octal system We also present transformations using the binary system as the current base Notice that every time you enter a number starting with ...

Страница 639: ...r between 0 and 64 Changing the wordsize will affect the way that binary integer operations are performed For example if a binary integer exceeds the current wordsize the leading bits will be dropped before any operation can be performed on such number Operations with binary integers The operations of addition subtraction change of sign multiplication and division are defined for binary integers S...

Страница 640: ...er true 1 or false 0 Some examples of logical statements are shown below Functions AND OR XOR and NOT can be applied to comparison statements under the following rules 1 AND 1 1 1 AND 0 0 0 AND 1 0 0 AND 0 0 1 OR 1 1 1 OR 0 1 0 OR 1 1 0 OR 0 0 1 XOR 1 0 1 XOR 0 1 0 XOR 1 1 0 XOR 0 0 NOT 1 0 NOT 0 1 These functions can be used to build logical statements for programming purposes In the context of t...

Страница 641: ...ulate bits in a binary integer The definition of these functions are shown below RL Rotate Left one bit e g 1100b 1001b SL Shift Left one bit e g 1101b 11010b ASR Arithmetic Shift Right one bit e g 1100010b 110001b SR Shift Right one bit e g 11011b 1101b RR Rotate Right one bit e g 1101b 1110b The BYTE menu The BYTE menu available through the BASE ã provides the following functions ...

Страница 642: ...11010b SRB Shift Right one byte e g 11011b 1101b RRB Rotate Right one byte e g 1101b 1110b Hexadecimal numbers for pixel references Many plot option specifications use pixel references as input e g 332h A23h Ah 0 360 ARC to draw an arc of a circle We use functions C PX and PX C to convert quickly between user unit coordinates and pixel references These functions can be found through the command ca...

Страница 643: ... CST Thus to create a menu you must put together this variable with the features that you want to display in your menu and the actions required by the soft menu keys To show examples of customizing menus we need to set system flag 117 to SOFT menu Make sure you do this before continuing See Chapter 2 for instructions on setting system flags The PRG MODES MENU menu Commands useful in customizing me...

Страница 644: ...t menu Note The number 96 01 in this example means the first 01 sub menu of menu 96 Custom menus MENU and TMENU functions Suppose that you need to activate four functions for a particular application Say that you need to be able to quickly access the functions EXP LN GAMMA and 2 and you want to place them in a soft menu that you will keep active for a while You could do this by creating a temporar...

Страница 645: ... the prompt e g EXP We need not worry about the closing parenthesis because the calculator will complete the parentheses before executing the function The implementation of function TMENU in ALG mode with the argument list shown above is as follows First we enter the list then we produce the temporary menu see menu key labels by using function TMENU ANS 1 We also show in the left hand side the res...

Страница 646: ...rm of the argument list for commands TMENU or MENU in ALG mode is label1 function1 ls1 rs1 label2 function2 ls2 rs2 While in RPN mode the argument list has this format label1 function1 ls1 rs1 label2 function2 ls2 rs2 In these specifications function1 function 2 etc represent the main operation of the key while ls1 ls2 etc represent the left shift operation of the key Similarly rs1 rs2 etc represe...

Страница 647: ... 61 key simultaneous with Thus the VAR function will be referred to as key 31 0 or 31 1 while the UPDIR function will be key 31 2 the COPY function will be key 31 3 the upper case J is key 31 4 and lower case j is key 31 5 Key 31 6 is not defined In general a key will be described by the arrangement XY Z where X row number Y column number Z shifting We can combine a given key with the USER key lef...

Страница 648: ... introduced with the HP 48G series calculator but currently not directly available from the keyboard The menu number for this menu is 81 01 You can see this menu active by using ALG mode MENU 81 01 RPN mode 81 01 MENU If you want to have a quick way to activate this menu from the keyboard you could assign this menu to the GRAPH key C whose reference number is 13 0 i e first row third column main f...

Страница 649: ...e DELKEYS 13 0 RPN mode 13 0 DELKEYS Assigning multiple user defined keys The simplest way to assign several user defined is to provide a list of commands and key specifications For example suppose that we assign the three trigonometric functions SIN COS TAN and the three hyperbolic functions SINH COSH TANH to keys A through F respectively as user defined keys In RPN mode use SIN 11 0 COS 12 0 TAN...

Страница 650: ...page in this chapter An example of programming Throughout the previous Chapters in this guide we have presented a number of programs that can be used for a variety of applications e g programs CRMC and CRMT used to create a matrix out of a number of lists were presented in Chapter 10 In this section we present a simple program to introduce concepts related to programming the calculator The program...

Страница 651: ...53 1 0017 Global and local variables and subprograms The program g defined above can be displayed as x STO x SINH 1 x SQ ADD x PURGE by using g Notice that the program uses the variable name x to store the value placed in level 1 of stack through the programming steps x STO The variable x while the program is executing is stored in your variable menu as any other variable you had previously stored...

Страница 652: ...lacing 1 in the stack placing x in the stack squaring x adding 1 to x and dividing stack level 2 SINH x by stack level 1 1 x2 The program control is then passed back to the main program but there are no more commands between the first set of closing programming symbols and the second one therefore the program terminates The last value in the stack i e SINH x 1 x2 is returned as the program output ...

Страница 653: ... directory under consideration Consequences of this rule are the following A global variable defined in the HOME directory will be accessible from any directory within HOME unless redefined within a directory or sub directory If you re define the variable within a directory or sub directory this definition takes precedence over any other definition in directories above the current one When running...

Страница 654: ...FT menus With this flag setting sub menus and commands in the PRG menu will be shown as soft menu labels This facilitates entering the programming commands in the line editor when you are putting together a program To access the PRG menu use the keystroke combination Within the PRG menu we identify the following sub menus press L to move to the next collection of sub menus in the PRG menu Here is ...

Страница 655: ...us KEYS To define and activate user defined keys Chapter 20 MENU To define and activate custom menus Chapter 20 MISC Miscellaneous mode changes beep clock etc IN Functions for program input OUT Functions for program output TIME Time related functions ALRM Alarm manipulation ERROR Functions for error handling IFERR IFERR THEN ELSE END construct for error handling RUN Functions for running and debug...

Страница 656: ...LL VARS CASE C R ROLLD TVARS THEN R C PICK ORDER END NUM UNPICK CHR PICK3 MEM ARITH BRCH START DTAG DEPTH STO START AND EQ DUP2 STO NEXT OR TYPE DUPN STOx STEP XOR VTYPE DROP2 STO NOT DROPN INCR BRCH FOR SAME LIST DUPDU DECR FOR TYPE OBJ NIP SINV NEXT SF LIST NDUPN SNEG STEP CF SUB SCONJ FS REPL MEM BRCH DO FC PURGE BRCH DO FS C MEM IFT UNTIL FC C BYTES IFTE END LININ NEWOB ARCHI RESTO ...

Страница 657: ...D STR STOF SIZE HEAD RCLF IN LIST PROC ANIMATE TAIL RESET INFORM DOLIST SREPL NOVAL DOSUB PICT MODES KEYS CHOOSE NSUB PICT MODES FMT ASN INPUT ENDSUB PDIM STD STOKEYS KEY STREAM LINE FIX RECLKEYS WAIT REVLIST TLINE SCI DELKEYS PROMPT SORT BOX ENG SEQ ARC FM MODES MENU OUT PIXON ML MENU PVIEW PIXOF CST TEXT PIX MODES ANGLE TMENU CLLCD PVIEW DEG RCLMENU DISP PX C RAD FREEZE C PX GRAD MSGBOX RECT BEE...

Страница 658: ...gs in the MODES sub menu can be activated by using the input functions provided by the H key Functions from the TIME sub menu can be accessed through the keystroke combination Ó Functions STO and RCL in MEM DIR sub menu are available in the keyboard through the keys K and Functions RCL and PURGE in MEM DIR sub menu are available through the TOOL menu I Within the BRCH sub menu pressing the left sh...

Страница 659: ...ailable after the key word for each construct so you can start typing at the right location Keystroke sequence for commonly used commands The following are keystroke sequences to access commonly used commands for numerical programming within the PRG menu The commands are first listed by menu ...

Страница 660: ...CH IF IF THEN BRCH IF THEN ELSE BRCH IF ELSE END BRCH IF END BRCH CASE CASE BRCH CASE CASE THEN BRCH CASE THEN END BRCH CASE END BRCH START START BRCH START START NEXT BRCH START NEXT STEP BRCH START STEP BRCH FOR FOR BRCH FOR FOR NEXT BRCH FOR NEXT STEP BRCH FOR STEP BRCH DO DO BRCH DO DO UNTIL BRCH DO UNTIL END BRCH DO END ...

Страница 661: ...ME SF TEST L L SF CF TEST L L CF FS TEST L L FS FC TEST L L FC FS C TEST L L FS C FC C TEST L L FC C TYPE OBJ TYPE OBJ ARRY TYPE ARRY LIST TYPE LIST STR TYPE STR TAG TYPE TAG NUM TYPE L NUM CHR TYPE L CHR TYPE TYPE L TYPE LIST ELEM GET LIST ELEM GET GETI LIST ELEM GETI PUT LIST ELEM PUT PUTI LIST ELEM PUTI SIZE LIST ELEM SIZE HEAD LIST ELEM L HEAD TAIL LIST ELEM L TAIL ...

Страница 662: ...T LL RUN HALT KILL LL RUN KILL Programs for generating lists of numbers Please notice that the functions in the PRG menu are not the only functions that can be used in programming As a matter of fact almost all functions in the calculator can be included in a program Thus you can use for example functions from the MTH menu Specifically you can use functions for list operations such as SORT ΣLIST e...

Страница 663: ...grams is as follows 1 LISC creates a list of n elements all equals to a constant c Operation enter n enter c press LISC Example 5 6 5 LISC creates the list 6 5 6 5 6 5 6 5 6 5 2 CRLST creates a list of numbers from n1 to n2 with increment n i e n1 n1 n n1 2 n n1 N n where N floor n2 n1 n 1 Operation enter n1 enter n2 enter n press CRLST Example 5 3 5 5 CRLST produces 0 5 1 1 5 2 2 5 3 3 5 3 CLIST ...

Страница 664: ... by using function DEFINE à with an argument of the form function_name x1 x2 expression containing variables x1 x2 The program is stored in a variable called function_name When the program is recalled to the stack by using function_name The program shows up as follows x1 x2 expression containing variables x1 x2 To evaluate the function for a set of input variables x1 x2 in RPN mode enter the varia...

Страница 665: ...nding system in use i e m2 s in S I and ft2 s in E S Manning s equation is therefore not dimensionally consistent Suppose that we want to create a function q Cu n y0 S0 to calculate the unit discharge q for this case Use the expression q Cu n y0 S0 Cu n y0 5 3 S0 as the argument of function DEFINE Notice that the exponent 5 3 in the equation represents a ratio of real numbers due to the decimal po...

Страница 666: ...e of programs is with an example Example Velocity head for a rectangular channel Suppose that we want to calculate the velocity head hv in a rectangular channel of width b with a flow depth y that carries a discharge Q The specific energy is calculated as hv Q2 2g by 2 where g is the acceleration of gravity g 9 806 m s2 in S I units or g 32 2 ft s2 in E S units If we were to calculate hv for Q 23 ...

Страница 667: ...ype the following y b º g 2 Q º and keeping only the operations shown below do not type the following 2 º Note When entering the program do not use the keystroke instead use the keystroke sequence STACK SWAP Unlike the interactive use of the calculator performed earlier we need to do some swapping of stack levels 1 and 2 within the program To write the program we use therefore å Opens program symb...

Страница 668: ... now is 2 26618623518E 2 i e hv 2 26618623518 10 2 m Note Since the equation programmed in hv is dimensionally consistent we can use units in the input As mentioned earlier the two types of programs presented in this section are sequential programs in the sense that the program flow follows a single path i e INPUT OPERATION OUTPUT Branching of the program flow is possible by using the commands in ...

Страница 669: ...r s CAS Calculator Algebraic System must be set to symbolic and exact modes This is accomplished by using H CAS and ensuring that the check marks in the options _Numeric and _Approx are removed Press OK OK to return to normal calculator display Press J to display your variables menu We will use this latter approach to check what formula results from using the program hv as follows We know that the...

Страница 670: ...iables From the various methods provided by the User RPL language the simplest is to use an input string and the function INPUT L IN INPUT to load your input data The following program prompts the user for the value of a variable a and places the input in stack level 1 Enter a a 2 0 V INPUT OBJ This program includes the symbol tag and return available through the keystroke combinations ê and ë bot...

Страница 671: ...e FUNCa FUNCtion of a Run the program by pressing FUNCa When prompted to enter the value of a enter for example 2 and press The result is simply the algebraic 2a2 3 which is an incorrect result The calculator provides functions for debugging programs to identify logical errors during program execution as shown below Debugging the program To figure out why it did not work we use the DBUG function i...

Страница 672: ... RUN DBG Starts debugger SST Step by step debugging result Enter a SST Result a 2 0 V SST Result user is prompted to enter value of a 2 Enter a value of 2 for a Result a 2 SST Result a 2 SST Result empty stack executing a SST Result empty stack entering subprogram At this point we are within the subprogram 2 a 2 3 which uses the local variable a To see the value of a use aµ This indeed shows that ...

Страница 673: ...ME to hold examples of input strings for one two and three input data values These will be generic input strings that can be incorporated in any future program taking care of changing the variable names according to the needs of each program Let s get started by creating a sub directory called PTRICKS Programming TRICKS to hold programming tidbits that we can later borrow from to use in more compl...

Страница 674: ...e ideal gas law pV nRT where p gas pressure Pa V gas volume m3 n number of moles gmol R universal gas constant 8 31451_J gmol K and T absolute temperature K We can define the pressure p as a function of two variables V and T as p V T nRT V for a given mass of gas since n will also remain constant Assume that n 0 2 gmol then the function to program is V T K J V T T V p _ 662902 1 2 0 31451 8 We can...

Страница 675: ...ediately above The resulting program can then be stored in a variable called INPT3 With this program we complete the collection of input string programs that will allow us to enter one two or three data values Keep these programs as a reference and copy and modify them to fulfill the requirements of new programs you write Application evaluating a function of three variables Suppose that we want to...

Страница 676: ...ications of the form label helpInfo type0 type1 typen The label is a field label The helpInfo is a character string describing the field label in detail and the type specifications is a list of types of variables allowed for the field see Chapter 24 for object types 3 Field format information a single number col or a list col tabs In this specification col is the number of columns in the input box...

Страница 677: ...hezy s formula Q C R S 1 2 where C is the Chezy coefficient a function of the channel surface s roughness typical values 80 150 R is the hydraulic radius of the channel a length and S is the channel bed s slope a dimensionless numbers typically 0 01 to 0 000001 The following program defines an input form through function INFORM CHEZY S EQN C Chezy s coefficient 0 R Hydraulic radius 0 S Channel bed...

Страница 678: ...the effect of resetting these values use L RESET select Reset all to reset field values Now enter different values for the three fields say C 95 R 2 5 and S 0 003 pressing OK after entering each of these new values After these substitutions the input form will look like this Now to enter these values into the program press OK once more This activates the function INFORM producing the following res...

Страница 679: ...e of Q and put a tag or label to it On the other hand if the value in stack level 1 is 0 which happens when a CANCEL is entered while using the input box the program control is passed to the commands Operation cancelled MSGBOX These commands will produce a message box indicating that the operation was cancelled Note Function MSGBOX belongs to the collection of output functions under the PRG OUT su...

Страница 680: ...ws the user to create a choose box in a program This function requires three arguments 1 A prompt a character string describing the choose box 2 A list of choice definitions c1 c2 cn A choice definition ci can have any of two formats a An object e g a number algebraic etc that will be displayed in the choose box and will also be the result of the choice b A list object_displayed object_result so t...

Страница 681: ...s 1 E S units 1 486 1 CHOOSE Running this program press CHP1 shows the following choose box Depending on whether you select S I units or E S units function CHOOSE places either a value of 1 or a value of 1 486 in stack level 2 and a 1 in level 1 If you cancel the choose box CHOICE returns a zero 0 The values returned by function CHOOSE can be operated upon by other program commands as shown in the...

Страница 682: ...vel 2 then use the TAG function TYPE TAG For example to produce the tagged result B 5 use 5 Õ b TYPE TAG Decomposing a tagged numerical result into a number and a tag To decompose a tagged result into its numerical value and its tag simply use function OBJ TYPE OBJ The result of decomposing a tagged number with OBJ is to place the numerical value in stack level 2 and the tag in stack level 1 If yo...

Страница 683: ...l function program reads Enter a a 2 0 V INPUT OBJ a 2 a 2 3 NUM Modify it to read Enter a a 2 0 V INPUT OBJ a 2 a 2 3 NUM F TAG Store the program back into FUNCa by using FUNCa Next run the program by pressing FUNCa Enter a value of 2 when prompted and press The result is now the tagged result F 11 Example 2 tagging input and output from function FUNCa In this example we modify the program FUNCa ...

Страница 684: ...n the stack It should be pointed out that in performing the calculation of the function the tag of the tagged input a is dropped automatically and only its numerical value is used in the calculation To see the operation of the function FUNCa step by step you could use the DBUG function as follows FUNCa Copies program name to stack level 1 LL RUN DBG Starts debugger SST Step by step debugging resul...

Страница 685: ...nce of instructions contained within the inner set of program symbols This is necessary because without the program symbol separating the two listings of input variables V T N V T n the program will assume that the input command V T N V T n requires six input values while only three are available The result would have been the generation of an error message and the interruption of the program exec...

Страница 686: ...e examples shown here is the use of tags to identify input and output variables If we use an input string to get our input values those values are already pre tagged and can be easily recall into the stack for output Use of the TAG command allows us to identify the output from a program Using a message box A message box is a fancier way to present output from a program The message box command in t...

Страница 687: ... STR available at TYPE STR Using a message box for program output The function p from the last example can be modified to read Enter V T and n V T n 2 0 V INPUT OBJ V T n V T n 8 31451_J K mol n T V EVAL p TAG STR MSGBOX Store the program back into variable p by using p Run the program by pressing p Enter values of V 0 01_m 3 T 300_K and n 0 8_mol when prompted As in the earlier version of p befor...

Страница 688: ...n 2 0 V INPUT OBJ V T n V STR T STR n STR 8 31451_J K mol n T V EVAL p TAG STR MSGBOX Notice that you need to add the following piece of code after each of the variable names V T and n within the sub program STR To get this piece of code typed in the first time use TYPE STR Õ ë Because the functions for the TYPE menu remain available in the soft menu keys for the second and third occurrences of th...

Страница 689: ...individual character strings To see the program operating Store the program back into variable p by using p Run the program by pressing p Enter values of V 0 01_m 3 T 300_K and n 0 8_mol when prompted As in the earlier version of p before pressing ENTER for input the stack will look like this The first program output is a message box containing the string Press OK to cancel message box output Inco...

Страница 690: ...oes not show in the stack instead it produces a new line Enter V T n S I V T n 2 0 V INPUT OBJ V T n V 1_m 3 T 1_K n 1_mol V T n V V TAG STR T T TAG STR n n TAG STR 8 31451_J K mol n T V EVAL p TAG STR MSGBOX This new version of the program includes an additional level of sub programming i e a third level of program symbols and some steps using lists i e V 1_m 3 T 1_K n 1_mol EVAL V T n The interp...

Страница 691: ...3 2 and 1 are passed on to the next level of sub programming To see this version of the program in action do the following Store the program back into variable p by using p Run the program by pressing p Enter values of V 0 01 T 300 and n 0 8 when prompted no units required now Before pressing for input the stack will look like this Press to run the program The output is a message box containing th...

Страница 692: ...ar we have worked mainly with sequential programs The User RPL language provides statements that allow branching and looping of the program flow Many of these make decisions based on whether a logical statement is true or not In this section we present some of the elements used to construct such logical statements namely relational and logical operators Relational operators Relational operators ar...

Страница 693: ...also available in TEST Two numbers variables or algebraics connected by a relational operator form a logical expression that can take value of true 1 false 0 or could simply not be evaluated To determine whether a logical statement is true or not place the statement in stack level 1 and press EVAL µ Examples 2 10 µ result 1 true 2 10 µ result 0 false In the next example it is assumed that the vari...

Страница 694: ...nts All of the others apply to two logical statements Tabulating all possible combinations of one or two statements together with the resulting value of applying a certain logical operator produces what is called the truth table of the operator The following are truth tables of each of the standard logical operators available in the calculator p NOT p 1 0 0 1 p q p AND q 1 1 1 1 0 0 0 1 0 0 0 0 p ...

Страница 695: ...ogram makes a decision among two or more possible flow paths The User RPL language provides a number of commands that can be used for program branching The menus containing these commands are accessed through the keystroke sequence BRCH This menu shows sub menus for the program constructs The program constructs IF THEN ELSE END and CASE THEN END will be referred to as program branching constructs ...

Страница 696: ... false skip program_statements and continue program flow after the END statement To type in the particles IF THEN ELSE and END use BRCH IF The functions IF THEN ELSE END are available in that menu to be typed selectively by the user Alternatively to produce an IF THEN END construct directly on the stack use BRCH IF This will create the following input in the stack With the cursor in front of the I...

Страница 697: ...ve program flow paths based on the truth value of the logical_statement The general format of this construct is IF logical_statement THEN program_statements_if_true ELSE program_statements_if_false END The operation of this construct is as follows 1 Evaluate logical_statement 2 If logical_statement is true perform program statements_if_true and continue program flow after the END statement 3 If lo...

Страница 698: ...te For this particular case a valid alternative would have been to use an IFTE function of the form f2 x IFTE x 3 x 2 1 x Nested IF THEN ELSE END constructs In most computer programming languages where the IF THEN ELSE END construct is available the general format used for program presentation is the following IF logical_statement THEN program_statements_if_true ELSE program_statements_if_false EN...

Страница 699: ... x x if x x if x x if x x f 2 15 3 exp 3 5 sin 5 3 1 3 2 3 π π Here is a possible way to evaluate this function using IF THEN ELSE END constructs IF x 3 THEN x2 ELSE IF x 5 THEN 1 x ELSE IF x 3π THEN sin x ELSE IF x 15 THEN exp x ELSE 2 END END END END A complex IF construct like this is called a set of nested IF THEN ELSE END constructs A possible way to evaluate f3 x based on the nested IF const...

Страница 700: ...ssible program flux paths as in the case of the nested IF constructs presented earlier The general format of this construct is as follows CASE Logical_statement1 THEN program_statements1 END Logical_statement2 THEN program_statements2 END Logical_statement THEN program_statements END Default_program_statements optional END When evaluating this construct the program tests each of the logical_statem...

Страница 701: ...n is defined by the following 5 expressions elsewhere x if x x if x x if x x if x x f 2 15 3 exp 3 5 sin 5 3 1 3 2 3 π π Using the CASE statement in User RPL language we can code this function as x CASE x 3 THEN x 2 END x 5 THEN 1 x END x 3 π THEN SIN x END x 15 THEN EXP x END 2 END EVAL Store the program into a variable called f3c Then try the following exercises 1 5 f3c Result 2 25 i e x2 2 5 f3...

Страница 702: ...in Chapters 2 and 13 However in order to illustrate the use of programming loops we will calculate this summation with our own User RPL codes There are four different commands that can be used to code a program loop in User RPL these are START FOR DO and WHILE The commands START and FOR use an index or counter to determine how many times the loop is executed The commands DO and WHILE rely on a log...

Страница 703: ... is called an infinite never ending loop Example calculating of the summation S defined above The START NEXT construct contains an index whose value is inaccessible to the user Since for the calculation of the sum the index itself k in this case is needed we must create our own index k that we will increment within the loop each time the loop is executed A possible implementation for the calculati...

Страница 704: ...t the updated values of S and k are available in stack levels 2 and 1 respectively The piece of code k STO stores the value from stack level 1 into local variable k The updated value of S now occupies stack level 1 9 The piece of code S STO stores the value from stack level 1 into local variable k The stack is now empty 10 The particle NEXT increases the index by one and sends the control to the b...

Страница 705: ... k2 SST SL1 k SL2 1 SL3 0 S k2 SST SL1 0 S k2 Stores value of SL2 1 into SL1 k SST SL1 S SL2 0 S k2 SST Empty stack Stores value of SL2 0 into SL1 S SST Empty stack NEXT end of loop loop execution number 2 for k 1 SST SL1 1 k SST SL1 1 SQ k k2 SST SL1 0 S SL2 1 k2 SST SL1 1 S k2 SST SL1 1 SL2 1 S k2 SST SL1 1 k SL2 1 SL3 1 S k2 SST SL1 2 k 1 SL2 1 S k2 SST SL1 k SL2 2 SL3 1 S k2 SST SL1 1 S k2 Sto...

Страница 706: ... Empty stack NEXT end of loop for n 2 the loop index is exhausted and control is passed to the statement following NEXT SST SL1 5 S is recalled to the stack SST SL1 S SL2 5 S is placed in the stack SST SL1 S 5 tagging output value SST SL1 S 5 leaving sub program SST SL1 S 5 leaving main program The step by step listing is finished The result of running program S1 with n 2 is S 5 Check also the fol...

Страница 707: ... 5 You can write the following program xs xe dx xs DUP xe START DUP dx dx STEP DROP xe xs dx ABS 1 LIST and store it in variable GLIST In this program xs starting value of the loop xe ending value of the loop dx increment value for loop The program places values of xs xs dx xs 2 dx xs 3 dx in the stack Then it calculates the number of elements generated using the piece of code xe xs dx ABS 1 Final...

Страница 708: ...one in the examples using START The value corresponding to the index is available for calculations Commands involved in the FOR construct are available through BRCH FOR Within the BRCH menu BRCH the following keystrokes are available to generate FOR constructs the symbol indicates cursor position FOR Starts the FOR NEXT construct FOR NEXT FOR Starts the FOR STEP construct FOR STEP The FOR NEXT con...

Страница 709: ...roducing such increments The FOR STEP construct The general form of this statement is start_value end_value FOR loop_index program_statements increment STEP The start_value end_value and increment of the loop index can be positive or negative quantities For increment 0 execution occurs as long as the index is less than or equal to end_value For increment 0 execution occurs as long as the index is ...

Страница 710: ... command starts an indefinite loop executing the program_statements until the logical_statement returns FALSE 0 The logical_statement must contain the value of an index whose value is changed in the program_statements Example 1 This program produces a counter in the upper left corner of the screen that adds 1 in an indefinite loop until a keystroke press any key stops the counter 0 DO DUP 1 DISP 1...

Страница 711: ... program call 0 5 2 5 0 5 GLIS3 produces the list 0 5 1 1 5 2 2 5 To see step by step operation use the program DBUG for a short list for example J1 1 5 0 5 Enter parameters 1 1 5 0 5 GLIS3 Enter the program name in level 1 LL RUN DBG Start the debugger Use SST to step into the program and see the detailed operation of each command The WHILE construct The general structure of this command is WHILE...

Страница 712: ...S STO n 1 n STO END S S TAG Store this program in a variable S4 Verify the following exercises J 3 S4 Result S 14 4 S4 Result S 30 5 S4 Result S 55 8 S4 Result S 204 10 S4 Result S 385 20 S4 Result S 2870 30 S4 Result S 9455 100 S4 Result S 338350 Example 2 generate a list using a WHILE REPEAT END construct Type in the following program xs xe dx xe xs dx ABS 1 xs n x xs WHILE x xe REPEAT x dx EVAL...

Страница 713: ...or thus causing the calculator to behave as if that particular error has occurred The function can take as argument either an integer number a binary integer number an error message or the number zero 0 For example in RPN mode entering 5 DOERR produces the following error message Error Memory Clear If you enter 11h DOERR produces the following message Error Undefined FPTR Name If you enter TRY AGA...

Страница 714: ...uted most recently For example in RPN mode if you use 3 2 and then use function LASTARG LASTA you will get the values 3 and 2 listed in the stack Another example in RPN mode is the following 5U Using LASTARG after these entries produces a 5 Sub menu IFERR The IFERR sub menu provides the following functions These are the components of the IFERR THEN END construct or of the IFERR THEN ELSE END const...

Страница 715: ...d b 5 6 A simple division of these two arguments produces an error Error Invalid Dimension However with the error trapping construct of the program ERR1 with the same arguments produces 0 262295 0 442622 User RPL programming in algebraic mode While all the programs presented earlier are produced and run in RPN mode you can always type a program in User RPL when in algebraic mode by using function ...

Страница 716: ...store the program use the STO command as follows îK p2 An evaluation of program P2 for the argument X 5 is shown in the next screen While you can write programs in algebraic mode without using the function RPL some of the RPL constructs will produce an error message when you press for example Whereas using RPL there is no problem when loading this program in algebraic mode ...

Страница 717: ...th graphics To accomplish such tasks we first introduce function in the PLOT menu The PLOT menu Commands for setting up and producing plots are available through the PLOT menu You can access the PLOT menu by using 81 01 L MODES MENU MENU The menu thus produced provides the user access to a variety of graphics functions For application in subsequent examples let s user define the C GRAPH key to pro...

Страница 718: ...f graphs available in the calculator To activate a user defined key you need to press Ì same as the key before pressing the key or keystroke combination of interest To activate the PLOT menu with the key definition used above press Ì C You will get the following menu press L to move to second menu Description of the PLOT menu The following diagram shows the menus in PLOT The number accompanying th...

Страница 719: ...y axis scale equal to the x axis scale POLAR based on the values of the independent variable typically θ it samples the function in EQ and determines minimum and maximum values of both x and y PARAMETRIC produces a similar result as POLAR based on the values of the parameter defining the equations for x and y TRUTH produces no action BAR the x axis range is set from 0 to n 1 where n is the number ...

Страница 720: ...unction DRAX draws the axes in the current plot if any is visible DRAW 6 The function DRAW draws the plot defined in PPAR The PTYPE menu under PLOT 1 The PTYPE menu lists the name of all two dimensional plot types pre programmed in the calculator The menu contains the following menu keys These keys correspond to the plot types Function Conic Polar Parametric Truth and Diff Eq presented earlier Pre...

Страница 721: ...PPAR while in this menu you will get a listing of the current PPAR settings for example This information indicates that X is the independent variable Indep Y is the dependent variable Depnd the x axis range goes from 6 5 to 6 5 Xrng the y axis range goes from 3 1 to 3 2 Yrng The last piece of information in the screen the value of Res resolution determines the interval of the independent variable ...

Страница 722: ...ons for the DEPND variable are the same as those for the INDEP variable XRNG c and YRNG d The command XRNG specifies the plotting range for the x axis while the command YRNG specifies the plotting range for the y axis The input for any of these commands is two numbers representing the minimum and maximum values of x or y The values of the x and y axis ranges are stored as the ordered pairs xmin ym...

Страница 723: ...n a factor xfactor the command SCALEW multiplies the horizontal scale by that factor The W in SCALEW stands for width The execution of SCALEW changes the values of xmin and xmax in PPAR SCALEH j Given a factor yfactor the command SCALEH multiplies the vertical scale by that factor The H in SCALEH stands for height The execution of SCALEW changes the values of ymin and ymax in PPAR Note Changes int...

Страница 724: ...only an ordered pair is given as input to AXES only the axes origin is altered The argument to the command AXES whether an ordered pair or a list of values is stored as the fifth parameter in PPAR To return to the PLOT menu press PLOT Press L to reach the second menu of the PLOT menu set RESET f This button will reset the plot parameters to default values The 3D menu within PLOT 7 The 3D menu cont...

Страница 725: ... VPAR in the 3D menu you will get the following functions Press L to move to the next menu Next we describe the meaning of these functions INFO S and VPAR W When you press INFO S you get the information shown in the left hand side screen shot above The ranges in Xvol Yvol and Zvol describe the extent of the parallelepiped in space where the graph will be generated Xrng and Yrng describe the range ...

Страница 726: ...on EYEPT takes as input real values x y and z representing the location of the viewpoint for a three dimensional graph The viewpoint is a point in space from which the three dimensional graph is observed Changing the viewpoint will produce different views of the graph The figure below illustrates the idea of the viewpoint with respect to the actual graphic space and its projection in the plane of ...

Страница 727: ...lots related to statistical analysis Within this menu we find the following menus The diagram below shows the branching of the STAT menu within PLOT The numbers and letters accompanying each function or menu are used for reference in the descriptions that follow the figure ...

Страница 728: ...AT statistical matrix The functions Σ D and Σ E add or remove data rows from the matrix ΣDAT CLΣ F clears the ΣDAT G matrix and the soft menu key labeled ΣDAT is just used as a reference for interactive applications More details on the use of these functions are presented in a later chapter on statistical applications Press STAT to return to the STAT menu The ΣPAR menu within STAT III The ΣPAR men...

Страница 729: ... in ΣDAT if a data fitting is implemented To see which options are available press MODL You will get the following menu These functions correspond to Linear Fit Logarithmic Fit Exponential Fit Power Fit or Best Fit Data fitting is described in more detail in a later chapter Press PAR to return to the ΣPAR menu ΣPAR K ΣPAR is just a reference to the variable ΣPAR for interactive use RESET L This fu...

Страница 730: ...e commands shown in the previous section help you in setting up such variables Following we describe the general format for the variables necessary to produce the different types of plots available in the calculator Two dimensional graphics The two dimensional graphics generated by functions namely Function Conic Parametric Polar Truth and Differential Equation use PPAR with the format xmin ymin x...

Страница 731: ...eters shown above The variable EQ All plots except those based on ΣDAT also require that you define the function or functions to be plotted by storing the expressions or references to those functions in the variable EQ In summary to produce a plot in a program you need to load EQ if required Then load PPAR PPAR and ΣPAR or PPAR and VPAR Finally use the name of the proper plot type FUNCTION CONIC P...

Страница 732: ...arlier in this Chapter Example 2 A parametric plot Use RAD as angles ÌC Get PLOT menu PTYPE PARAM Select PARAMETRIC as the plot type SIN t i SIN 2 t Define complex function X iY EQ Store complex function into EQ PPAR Show plot parameters t 0 6 29 INDEP Define t as the indep variable y DEPND Define Y as the dependent variable 2 2 2 2 XRNG Define 2 2 2 2 as the x range 1 1 1 1 YRNG L Define 1 1 1 1 ...

Страница 733: ...OT menu 1 Select PTYPE 2 Store function to plot in variable EQ using the proper format e g X t iY t for PARAMETRIC 3 Enter name and range if necessary of independent and dependent variables 4 Enter axes specifications as a list center atick x label y label 5 Use ERASE DRAX LABEL DRAW to produce a fully labeled graph with axes This same approach can be used to produce plots with a program except th...

Страница 734: ... To run it press J if needed then press PLOT1 Example 2 A parametric plot Enter the following program Start program RAD PPAR EQ PURGE Change to radians purge vars SIN t i SIN 2 t STEQ Store X t iY t into EQ t 0 6 29 INDEP Set indep variable to r with range Y DEPND Set dependent variable to Y PARAMETRIC Select PARAMETRIC as the plot type 0 0 5 5 X t Y t AXES Set axes information 2 2 2 2 XRNG Set x ...

Страница 735: ...mmands in programs They just scratch the surface of programming applications of plots I invite the reader to try their own exercises on programming plots Drawing commands for use in programming You can draw figures in the graphics window directly from a program by using commands such as those contained in the PICT menu accessible by L PICT The functions available in this menu are the following Pre...

Страница 736: ...a for the current graph can be thought of as a two dimensional graph with a minimum size of 131 pixels wide by 64 pixels high The maximum width of PICT is 2048 pixels with no restriction on the maximum height A pixel is each one of the dots in the calculator s screen that can be turned on dark or off clear to produce text or graphs The calculator screen has 131 pixels by 64 pixels i e the minimum ...

Страница 737: ...s of pixel coordinates n1 m1 n2 m2 It draws the box whose diagonals are represented by the two pairs of coordinates in the input ARC This command is used to draw an arc ARC takes as input the following objects Coordinates of the center of the arc as x y in user coordinates or n m in pixels Radius of arc as r user coordinates or k pixels Initial angle θ1 and final angle θ2 ...

Страница 738: ...icture is centered in the screen PVIEW does not activate the graphics cursor or the picture menu To activate any of those features use PICTURE PX C The function PX C converts pixel coordinates n m to user unit coordinates x y C PX The function C PX converts user unit coordinates x y to pixel coordinates n m Programming examples using drawing functions In this section we use the commands described ...

Страница 739: ...e useful for determining area and wetted perimeters of natural river cross sections Typically a natural river cross section is surveyed and a series of points representing coordinates x and y with respect to an arbitrary set of coordinates axes These points can be plotted and a sketch of the cross section produced for a given water surface elevation The figure below illustrate the terms presented ...

Страница 740: ...names such as XYD1 X Y Data set 1 and XYD2 X Y Data set 2 To run the program place one of the data sets in the stack e g J XYD1 then type in a water surface elevation say 4 0 and press XSECT The calculator will show an sketch of the cross section with the corresponding water surface To exit the graph display press Try the following examples XYD1 2 XSECT XYD1 3 XSECT XYD1 4 XSECT XYD1 6 XSECT Pleas...

Страница 741: ... AXL 20 DUP NEG SWAP 2 COL ROW DROP SWAP yR xR 131 DUP R B SWAP yR OBJ DROP xR OBJ DROP FLOOR R B PDIM yR OBJ DROP YRNG xR OBJ DROP XRNG ERASE This program keeps the width of the PICT variable at 131 pixels the minimum pixel size for the horizontal axis and adjusts the number of pixels in the vertical axes so that a 1 1 scale is maintained between the vertical and horizontal axes Pixel coordinates...

Страница 742: ...Slice plot type Suppose that you want to animate the traveling wave f X Y 2 5 sin X Y We can treat the X as time in the animation producing plots of f X Y vs Y for different values of X To produce this graph use the following ô simultaneously Select Y Slice for TYPE 2 5 SIN X Y for EQ X for INDEP Press L OK ò simultaneously in RPN mode Use the following values Press ERASE DRAW Allow some time for ...

Страница 743: ...constant value in each subsequent graph Begin program RAD Set angle units to radians 131 R B 64 R B PDIM Set PICT to 131 64 pixels 0 100 XRNG 0 100 YRNG Set x and y ranges to 0 100 1 11 FOR j Start loop with j 1 11 ERASE Erase current PICT 50 50 5 j 1 NUM Centers of circles 50 50 0 2 π NUM ARC Draw circle center r 5 j 1 PICT RCL Place current PICT on stack NEXT End FOR NEXT loop 11 ANIMATE Animate...

Страница 744: ...your list of variables The variable WLIST should now be listed in your soft menu keys To re animate this list of variables you could use the following program Start program WLIST Place list WLIST in stack OBJ Decompose list stack level 1 11 ANIMATE Start animation End program Save this program in a variable called RANIM Re ANIMate To run it press RANIM The following program will animate the graphi...

Страница 745: ...ction ANimation To run the program press J if needed PWAN You will see the calculator drawing each individual power function before starting the animation in which the five functions will be plotted quickly one after the other To stop the animation press More information on the ANIMATE function The ANIMATE function as used in the two previous examples utilized as input the graphics to be animated ...

Страница 746: ... stack level 1 the line Graphic 131 64 if using the standard screen size followed by a sketch of the top part of the graph For example If you press then the graph contained in level 1 is shown in the calculator s graphics display Press CANCL to return to normal calculator display The graph in level 1 is still not in GROB format although it is by definition a graphics object To convert a graph in t...

Страница 747: ...uations or text in the graphics display The GROB menu The GROB menu accessible through L GROB GROB contains the following functions Press L to move to the next menu GROB Of these functions we have already used SUB REPL from the graphics EDIT menu ANIMATE ANIMA and GROB PRG is simply a way to return to the programming menu While using GROB in the two previous examples you may have noticed that I us...

Страница 748: ...b1 and grob2 GXOR The function GXOR Graphics XOR performs the same operation as GOR but using XOR to determine the final status of pixels in the overlapping area between graphic objects grob1 and grob2 Note In both GOR and GXOR when grob2 is replaced by PICT these functions produce no output To see the output you need to recall PICT to the stack by using either PICT RCL or PICTURE LCD Takes a spec...

Страница 749: ...ph PICT RCL Place contents of PICT on stack SINE FUNCTION Place graph label string in stack 1 GROB Convert string into a small GROB 6 1 5 SWAP Coordinates to place label GROB GOR Combine PICT with the label GROB PICT STO Save combined GROB into PICT PVIEW Bring PICT to the stack End program Save the program under the name GRPR GROB PRogram Press GRPR to run the program The output will look like th...

Страница 750: ...o the normal stresses σ and the y axis corresponding to the shear stresses τ Locate the points A σxx τxy and B σyy τxy and draw the segment AB The point C where the segment AB crosses the σn axis will be the center of the circle Notice that the coordinates of point C are σyy σxy 0 When constructing the circle by hand you can use a compass to trace the circle since you know the location of the cent...

Страница 751: ... yy at point E To obtain the principal stresses you need to rotate the coordinate system x y by an angle φn counterclockwise with respect to the system x y In Mohr s circle the angle between segments AC and D C measures 2φn The stress condition for which the shear stress τ xy is a maximum is given by segment F G Under such conditions both normal stresses σ xx σ yy are equal The angle corresponding...

Страница 752: ...m user produces a list σL σx σy τxy as output CC r Uses σL as input produces σc σx σy r radius of Mohr s circle φn angle for principal stresses as output DAXES Uses σc and r as input determines axes ranges and draws axes for the Mohr s circle construct PCIRC Uses σc r and φn as input draw s Mohr s circle by producing a PARAMETRIC plot DDIAM Uses σL as input draws the segment AB see Mohr s circle f...

Страница 753: ...ides the figure itself To obtain additional information out of the Mohr s circle end the program by pressing Then press š to recover the contents of PICT in the graphics environment The Mohr s circle now looks like the picture to the right see above Press the soft menu keys TRACE and x y At the bottom of the screen you will find the value of φ corresponding to the point A σx τxy i e φ 0 2 50E1 5 0...

Страница 754: ...s τ xy becomes zero To find the actual value of φn press Then type the list corresponding to the values σx σy τxy for this case it will be 25 75 50 ENTER Then press CC r The last result in the output 58 2825255885o is the actual value of φn A program to calculate principal stresses The procedure followed above to calculate φn can be programmed as follows Program PRNST Start program PRNST PRiNcipal...

Страница 755: ...g the list by using MEM DIR ORDER After this call to the function ORDER is performed press J You will now see that we have the programs MOHRCIRCL and PRNST being the first two variables in the menu as we expected A second example of Mohr s circle calculations Determine the principal stresses for the stress state defined by σxx 12 5 kPa σyy 6 25 kPa and τxy 5 0 kPa Draw Mohr s circle and determine ...

Страница 756: ...over the circle showing φ and x y Next press until you read φ 35 The corresponding coordinates are 1 63E0 1 05E1 i e at φ 35o σ xx 1 63 kPa and σ yy 10 5kPa An input form for the Mohr s circle program For a fancier way to input data we can replace sub program INDAT with the following program that activates an input form MOHR S CIRCLE σx Normal stress in x 0 σy Normal stress in y 0 τxy Shear stress...

Страница 757: ...gram execution The result is the following figure Since program INDAT is used also for program PRNST PRiNcipal STresses running that particular program will now use an input form for example The result after pressing OK is the following ...

Страница 758: ...ogram output are also strings String related functions in the TYPE sub menu The TYPE sub menu is accessible through the PRG programming menu i e The functions provided in the TYPE sub menu are also shown below Among the functions in the TYPE menu that are useful for manipulating strings we have OBJ Converts string to the object it represents STR Converts an object to its string representation TAG ...

Страница 759: ...ign for example Concatenating strings is a practical way to create output in programs For example concatenating YOU ARE AGE YEAR OLD creates the string YOU ARE 25 YEAR OLD where 25 is stored in the variable called AGE The CHARS menu The CHARS sub menu is accessible through the PRG programming menu i e The functions provided by the CHARS sub menu are the following ...

Страница 760: ...ter in a string TAIL removes first character in a string SUB extract sub string given starting and ending position REPL replace characters in a string with a sub string starting at given position SREPL replaces a sub string by another sub string in a string To see those effects on action try the following exercises Store the string MY NAME IS CYRILLE into variable S1 We ll use this string to show ...

Страница 761: ... you highlight M you will see αM displayed at the lower left side of the screen indicating the use of m On the other hand m shows the keystroke combination α M or m Greek characters such as σ will show the code α S or s Some characters like ρ do not have a keystroke sequence associated with them Therefore the only way to obtain such characters is through the character list by highlighting the desi...

Страница 762: ...______________________________________________________________ Number Type Example _________________________________________________________________ 0 Real Number 1 23E 5 1 Complex Number 1 2 2 3 2 String Hello world 3 Real Array 1 2 3 4 4 Complex Array 1 2 3 4 5 6 7 8 5 List 3 1 PI 6 Global Name X 7 Local Name y 8 Program a a 2 9 Algebraic object a 2 b 2 10 Binary Integer A2F1E h 11 Graphic Objec...

Страница 763: ...ternal Object External 30 External Object External ____________________________________________________________________ Function TYPE This function available in the PRG TYPE sub menu or through the command catalog is used to determine the type of an object The function argument is the object of interest The function returns the object type as indicated by the numbers specified above Function VTYPE...

Страница 764: ...between CHOOSE boxes and SOFT menus which are selected by setting or un setting system flag 117 Another example of system flag setting is that of system flags 60 and 61 that relate to the constant library CONLIB see Chapter 3 These flags operate in the following manner user flag 60 clear default SI units set ENGL units user flag 61 clear default use units set value only Functions for setting and c...

Страница 765: ...tion of these functions is as follows SF Set a flag CF Clear a flag FS Returns 1 if flag is set 0 if not set FC Returns 1 if flag is clear not set 0 if flag is set FS C Tests flag as FS does then clears it FC C Tests flag as FC does then clears it STOF Stores new system flag settings RCLF Recalls existing flag settings RESET Resets current field values could be used to reset a flag User flags For ...

Страница 766: ...r set an alarm The input form looks like in the following figure The Message input field allows you to enter a character string identifying the alarm The Time field lets you enter the time for activating the alarm The Date field is used to set the date for the alarm or for the first time of activation if repetition is required For example you could set the following alarm The left hand side figure...

Страница 767: ...ed alarm providing an alarm set input form NEW For programming a new alarm PURG For deleting an alarm OK Returns to normal display Setting time and date Option 3 Set time date provides the following input form that let s the user set the current time and date Details were provided in Chapter 1 TIME Tools Option 4 Tools provides a number of functions useful for clock operation and calculations with...

Страница 768: ...te DDAYS x y Returns number of days between dates x and y HMS Converts time from decimal to HH MMSS HMS Converts time from HH MMSS to decimal HMS Add two times in HH MMSS format HMS Subtract two times in HH MMSS format TSTR time date converts time date to string format CLKADJ x Adds x ticks to system time 1 tick 1 8192 sec Functions DATE TIME CLKADJ are used to adjust date and time There are no ex...

Страница 769: ...larm x into system alarm list RCLALARM x Recalls specified alarm x from system alarm list DELALARM x Deletes alarm x from system alarm list FINDALARM x Returns first alarm due after specified time The argument x in function STOALARM is a list containing a date reference mm ddyyy time of day in 24 hr format hh mm a string containing the text of the alarm and the number of repetitions of the alarm F...

Страница 770: ...le result is shown below This screen indicates the existence of one memory port port 0 which includes the HOME directory See Chapter 2 in the User s Guide Port 0 and the HOME directory share the same area of memory so that the more data stored in the HOME directory for example the less memory is available for Port 0 storage As mentioned above the total size of memory for the Port 0 HOME directory ...

Страница 771: ... described below Checking objects in memory To see the objects stored in memory you can use the FILES function The screen shows the HOME directory with at least four directories namely GRPHS MPFIT MATRX and TRIANG Additional directories can be viewed by moving the cursor downwards in the directory tree Or you can move the cursor upwards to select a memory port When a given directory sub directory ...

Страница 772: ...r checksum value based on the binary data contained in the object This value is stored with the backup object and is used by the calculator to monitor the integrity of the backup object When you restore a backup object into the HOME directory the calculator obtains again the CRC value and compares it to the original value If a discrepancy is noticed the calculator warns the user that the restored ...

Страница 773: ...ered by using the keystroke sequence ê For example to back up HOME into HOME1 in Port 0 use To back up the HOME directory in RPN mode use the command Port_Number Backup_Name ARCHIVE Restoring the HOME directory To restore the home directory in algebraic mode use the command RESTORE Port_Number Backup_Name For example to restore the HOME directory out of backup object HOME1 use RESTORE 0 HOME1 In R...

Страница 774: ...e the PURGE command as follows In algebraic mode use PURGE Port_Number Backup_Name In RPN mode use Port_Number Backup_Name PURGE To restore a backup object Use the File Manager to copy the backup object from Port memory to the HOME directory When a backup object is restored the calculator performs an integrity check on the restored object by calculating its CRC value Any discrepancy between the ca...

Страница 775: ... sub directory of the HOME directory Libraries can be downloaded into the calculator as a regular variable and then installed and attached to the HOME directory Installing and attaching a library To install a library list the library contents in the stack use variable soft menu key or function RCL and store it into port 0 For example to install a library variable into a port use In algebraic mode ...

Страница 776: ...Assembler language in System RPL language or by using a matrix creating library such as LBMKR The latter program is available online see for example http www hpcalc org The details of programming the calculator in Assembler language or in System RPL language are beyond the scope of this document The user is invited to find additional information on the subject online Backup battery A CR2032 back u...

Страница 777: ...Page 26 8 ...

Страница 778: ...rm In such case use the CHK soft menu key to toggle the check mark on and off Press the CANCL soft menu key to close an input form and return to the stack display You can also press the key or the key to close the input form Example Using input forms in the NUM SLV menu Before discussing these items in detail we will present some of the characteristics of the input forms by using input forms from ...

Страница 779: ... be presented later Try the following 10 OK Enter n 10 8 5 OK Enter I YR 8 5 10000 OK Enter PV 10000 1000 OK Enter FV 1000 š SOLVE Select and solve for PMT The resulting screen is In this input form you will notice the following soft menu key labels EDIT Press to edit highlighted field AMOR Amortization menu option specific to this application SOLVE Press to solve for highlighted field Pressing L ...

Страница 780: ...ault value If instead you select Rest all all the fields will be reset to their default values typically 0 At this point you can accept your choice press OK or cancel the operation press CANCL Press CANCL in this instance Press CALC to access the stack The resulting screen is the following At this point you have access to the stack and the value last highlighted in the input form is provided for y...

Страница 781: ...owing specification This indicates that the value in the PMT field must be a real number Press OK to return to the input form and press L to recover the first menu Next press the key or the key to return to the stack In this instance the following values will be shown The top result is the value that was solved for PMT in the first part of the exercise The second value is the calculation we made t...

Страница 782: ...f the calculator s keyboard with the numbering of its rows and columns The figure shows 10 rows of keys combined with 3 5 or 6 columns Row 1 has 6 keys rows 2 and 3 have 3 keys each and rows 4 through 10 have 5 keys each There are 4 arrow keys located on the right hand side of the ...

Страница 783: ...ions The main key functions are shown in the figure below To operate this main key functions simply press the corresponding key We will refer to the keys by the row and column where they are located in the sketch above thus key 10 1 is the ON key Main key functions in the calculator s keyboard ...

Страница 784: ...rectory The HIST function allows you access to the algebraic mode history i e the collection of recent command entries in that mode The EVAL key is used to evaluate algebraic and numeric expressions the apostrophe key is used to enter a set of apostrophes for algebraic expressions The SYMB activates the symbolic operations menu The delete key ƒ is used to delete characters in a line The yx key cal...

Страница 785: ...h some of the other keys to activate the alternative functions shown in the keyboard For example the P key key 4 4 has the following six functions associated with it P Main function to activate the SYMBolic menu Left shift function to activate the MTH Math menu N Right shift function to activate the CATalog function p ALPHA function to enter the upper case letter P p ALPHA Left Shift function to e...

Страница 786: ...ons in the calculator s Algebraic mode of operation press the left shift key first and then any of the keys in Row 1 When using these functions in the calculator s RPN mode you need to press the left shift key simultaneously with the key in Row 1 of your choice Function Y is used to enter functions of the form y f x for plotting function WIN is used to set parameters of the plot window function GR...

Страница 787: ...on activates the Matrix Writer Left shift functions of the calculator s keyboard The CMD function shows the most recent commands The PRG function activates the programming menus The MTRW function activates the Matrix Writer The MTH function activates a menu of mathematical function The DEL key is used to delete variables ...

Страница 788: ...ions The CALC function activates a menu of calculus functions The MATRICES function activates a menu for creating and manipulation of matrices The CONVERT function activates a menu for conversion of units and other expressions The ARITH function activates a menu of arithmetic functions The DEF key is used to define a simple function as a variable in the calculator menu The CONT key is used to cont...

Страница 789: ... menus associated with the different calculator keys when the right shift key is activated The functions BEGIN END COPY CUT and PASTE are used for editing purposes The UNDO key is used to undo the last calculator operation The CHARS function activates the special characters menu The EQW function is used to start the Equation Writer ...

Страница 790: ...s menu The UNITS function activates the menu for units of measurement The CMPLX function activates the complex number functions menu The LIB function activates the library functions The BASE function activates the numeric base conversion menu The OFF key turns the calculator off the NUM key produces a numeric or floating point value of an expression The key enters a set of double quotes used for e...

Страница 791: ...ain functions of these keys The function produces an asterisk when combined with the times key i e Alpha functions of the calculator s keyboard Alpha left shift characters The following sketch shows the characters associated with the different calculator keys when the ALPHA is combined with the left shift key ...

Страница 792: ... of the English alphabet A through Z The numbers mathematical symbols decimal point and the space SPC are the same as the main functions of these keys The ENTER and CONT keys also work as their main function even when the combination is used Alpha functions of the calculator s keyboard ...

Страница 793: ...t calculator keys when the ALPHA is combined with the right shift key Alpha functions of the calculator s keyboard Notice that the combination is used mainly to enter a number of special characters from into the calculator stack The CLEAR OFF comma key enters and OFF keys also work as their main function even ...

Страница 794: ...Page B 13 when the combination is used The special characters generated by the combination include Greek letters α β δ ε ρ µ λ σ θ τ ω and Π other characters generated by the combination are __ and ...

Страница 795: ... FLAGS Provides menus for manipulating calculator flags CHOOS Lets the user chose options in the different fields in the form CAS Provides an input form to change CAS settings DISP Provides an input form to change display settings CANCL Closes this input form and returns to normal display OK Use this key to accept settings Flags are variables in the calculator referred to by numbers which can be s...

Страница 796: ...options will show no check mark in the underline preceding the option of interest e g the _Numeric _Approx _Complex _Verbose _Step Step _Incr Pow options above After having selected and unselected all the options that you want in the CAS MODES input form press the OK soft menu key This will take you back to the CALCULATOR MODES input form To return to normal calculator display at this point press ...

Страница 797: ... variable VX in your programs or equations so as to not get it confused with the CAS VX If you need to refer to the x component of velocity for example you can use vx or Vx Selecting the modulus The Modulo option of the CAS MODES input box represents a number default value 13 used in modular arithmetic More details about modular arithmetic are presented elsewhere Numeric vs symbolic CAS mode When ...

Страница 798: ...ic expressions whenever possible The following screen shows a couple of symbolic expressions entered with an active exact mode in Algebraic operating mode In Algebraic mode the object entered by the user is shown in the left hand side of the screen followed immediately by a result in the right hand side of the screen The results shown above show the symbolic expressions for ln 2 i e the natural lo...

Страница 799: ...t shift key and pressing the ENTER key simultaneously i e hold Real numbers vs integer numbers CAS operations utilize integer numbers in order to keep full precision in the calculations Real numbers are stored in the form of a mantissa and an exponent and have limited precision In APPROX mode however whenever you enter an integer number it is automatically transformed into a real number as illustr...

Страница 800: ...if an operation results in a complex number then the result will be shown in the form a bi or in the form of an ordered pair a b On the other hand if the _Complex CAS option is unset i e the Real CAS option is active and an operation results in a complex number you will be asked to switch to Complex mode If you decline the calculator will report an error Please notice that in COMPLEX mode the CAS ...

Страница 801: ...Verbose CAS option is selected certain calculus applications are provided with comment lines in the main display If the _Verbose CAS option is not selected then those calculus applications will show no comment lines The comment lines will appear momentarily in the top lines of the display while the operation is being calculated Step by step CAS mode When the _Step step CAS option is selected certa...

Страница 802: ...n A X3 5X2 3X 2 and B X 2 These polynomials are represented in the screen by lists of their coefficients For example the expression A 1 5 3 2 represents the polynomial A X3 5X2 3X 2 B 1 2 represents the polynomial B X 2 Q 1 represents the polynomial Q X and R 3 3 2 represents the polynomial R 3X2 3X 2 At this point press for example the key Continue pressing the key to produce additional steps Thu...

Страница 803: ...powers of the independent variable An example is shown next in Algebraic mode In the first case the polynomial X 3 5 is expanded in increasing order of the powers of X while in the second case the polynomial shows decreasing order of the powers of X The keystrokes in both cases are the following Üx 3 Q5 In the first case the _Incr pow option was selected while in the second it was not selected The...

Страница 804: ... non rational expressions will not be automatically simplified Using the CAS HELP facility Turn on the calculator and press the I key to activate the TOOL menu Next press the Bsoft menu key followed by the key the key in the lowest right corner of the keyboard to activate the HELP facility The display will look as follows At this point you will be provided with a list of all CAS commands in alphab...

Страница 805: ...about the command ATAN2S The help facility indicates that the command or function ATAN2S replaces the value of atan x the arc tangent of a value x by its equivalent in terms of the function asin arcsine i e The fourth and fifth lines in the display provide an example of application of the function ATAN2S Line four namely ATAN2S ATAN X is the statement of the operation to be performed while line fi...

Страница 806: ...ith output The first two lines from the top correspond to the first exercise with the HELP facility in which we cancel the request for help The third line from the top shows the most recent call to the HELP facility while the last line shows the ECHO of the example command To activate the command press the key The result is Notice that as new lines of output are produced the display or stack pushe...

Страница 807: ... CAS Computer Algebraic System There is a large number of other functions and commands that were originally developed for the HP 48G series calculators that are not included in the help facility Good references for those commands are the HP 48G Series user s guide HP Part No 00048 90126 and the HP 48G Series Advanced User s Reference Manual HP Part No 00048 90136 both published by Hewlett Packard ...

Страница 808: ... limited to loss of data or data being rendered inaccurate or losses sustained by you or third parties or a failure of the CAS Software to operate with any other programs even if such holder or other party has been advised of the possibility of such damages If required by applicable law the maximum amount payable for damages by the copyright holder shall not exceed the royalty amount paid by Hewle...

Страница 809: ...eystroke combination associated with the EVAL key The result is the following screen By using the arrow keys š we can navigate through the collection of characters For example moving downwards in the screen produces more characters in the display Moving farther down we see these characters There will be one character highlighted at all times The lower line in the display will show the short cut fo...

Страница 810: ...cter to the command line or equation writer EQW but the cursor remains in the character set screen to allow the user to select additional characters i e echoes a string of characters to the stack To exit the character set screen press For example suppose you have to type the expression λ2 2µ 5 Here is a suggested approach using the stack in either Algebraic or RPN mode Use the keystrokes to get to...

Страница 811: ... tau u ω omega v upper case delta c Π upper case pi p Other characters tilde 1 factorial 2 question mark 3 backward slash 5 angle symbol 6 at Some characters commonly used that do not have simple keystroke shortcuts are x x bar γ gamma η eta Ω upper case omega These characters can be echoed from the CHARS screen ...

Страница 812: ... Execution from left to right means that if two operations of the same hierarchy say two multiplications exist in an expression the first multiplication to the left will be executed before the second and so on Consider for example the expression shown below in the equation writer The insertion cursor at this point is located to the right of the 2 in the argument of the SIN function in the denomina...

Страница 813: ...on It is important to point out that the multiplication in Step A5 includes the first term y 3 x 5 with a second term x2 4 which is already calculated To see the steps in calculating these second term press the down arrow key continuously until the clear editing cursor is triggered around the y once more Then press the right arrow key until these cursor is over the x in the second term in the nume...

Страница 814: ...function in the denominator Press the down arrow key continuously until the clear editing cursor is triggered around the y once more Then press the right arrow key until these cursor is over the 4 in the denominator Then press the upper arrow key to select this 4 The steps in the evaluation of the expression starting from this point are shown below Step C1 Step C2 ...

Страница 815: ... Step A6 The expression tree for the expression presented above is shown next The steps in the evaluation of the three terms A1 through A6 B1 through B5 and C1 through C5 are shown next to the circle containing numbers variables or operators ...

Страница 816: ...ns The different applications are described next Plot functions Selecting option 1 Plot functions in the APPS will produce the following menu list of graph related options The six options shown are equivalent to the keystroke sequences listed below Equation entry ñ Plot window ò Graph display ó Plot setup ô Table setup õ Table display ö These applications are presented in detail in Chapter 12 ...

Страница 817: ...eive data from another calculator Print display Send screen to printer Print Print selected object from calculator Transfer Transfer data to other device Start Server Calculator set as a server for communication with computers Constants lib Selecting option 3 Constants lib in the APPS menu opens the Constant Library application that provides values of standard physical constants The Constants Libr...

Страница 818: ...The numerical solver menu is presented in detail in Chapters 6 and 7 Time date Selecting option 5 Time date in the APPS menu produces the time and date menu This operation is equivalent to the keystroke sequence Ó The time and date menu is presented in detail in Chapter 26 Equation writer Selecting option 6 Equation writer in the APPS menu opens the equation writer ...

Страница 819: ...File manager Selecting option 7 File manager in the APPS menu launches the file manager application This operation is equivalent to the keystroke sequence The file manager is introduced in Chapter 2 Matrix Writer Selecting option 8 Matrix Writer in the APPS menu launches the matrix writer This operation is equivalent to the keystroke sequence The Matrix Writer is presented in detail in Chapter 10 ...

Страница 820: ...itor is introduced in Chapter 2 and presented in detail in Appendix L Math menu Selecting option 10 Math menu in the APPS menu produces the MTH mathematics menu This operation is equivalent to the keystroke sequence The MTH menu is introduced in Chapter 3 real numbers Other functions from the MTH menu are presented in Chapters 4 complex numbers 8 lists 9 vectors 10 matrix creation 11 matrix operat...

Страница 821: ...so available by pressing the Pkey The CAS or SYMBOLIC menu is introduced in Chapter 5 algebraic and arithmetic operations Other functions from the CAS menu are presented in Chapters 4 complex numbers 6 equations solutions 10 matrix creation 11 matrix operation 13 calculus 14 multivariate calculus and 15 vector analysis ...

Страница 822: ...ing mode H FLAGS In ALG mode CF 95 selects RPN mode In RPN mode 95 SF selects ALG mode A keyboard short cut to toggle between APPROX and EXACT mode is by holding the right shift key and pressing the ENTER key simultaneously i e hold Set clear system flag 105 EXACT vs APPROX CAS mode H FLAGS In ALG mode SF 105 selects APPROX CAS mode CF 105 selects EXACT CAS mode In RPN mode 105 SF selects APPROX C...

Страница 823: ...sure o To degrees deg o To radian rad Special characters o Angle symbol 6 o Factorial symbol 2 o Degree symbol o hold 6 Lock unlock alpha keyboard o Lock alpha keyboard upper case o Unlock alpha keyboard upper case o Lock alpha keyboard lower case o Unlock alpha keyboard lower case Greek letters Alpha α a Beta β b DELTA c Delta d d Epsilon ε e Rho ρ f Mu µ m Lambda λ n PI Π p Sigma σ s Theta θ t T...

Страница 824: ...ng alarm Menus not accessible through keyboard In RPN enter menu_number type MENU In ALG mode type MENU menu_number Menu_number is one of the following o STAT soft menu 96 o PLOT soft menu 81 o SOLVE soft menu 74 or use hold 7 o UTILITY soft menu 113 Other menus o MATHS menu maths o MAIN menu main Other keyboard short cuts o hold 7 SOLVE menu menu 74 o hold H PRG MODES menu Chapter 21 o hold Start...

Страница 825: ...ikely the command of interest will not be selected at this point you may overshoot or undershoot it However you can use the vertical keys one stroke at a time to locate the command you want and then press OK If while holding down the down arrow key you overshoot the command of interest you can hold down the up arrow key to move back towards that command Refine the selection with the vertical keys ...

Страница 826: ...a keyboard and use the vertical arrow keys to locate the command if needed Press OK to locate the to activate the command For example to locate the command PROPFRAC you can use one of the following keystroke sequences I L HELP pr OK I L HELP pro OK I L HELP prop OK See Appendix C for more information on the CAS Computer Algebraic System Appendix C includes other examples of application of the CAS ...

Страница 827: ... given command if the soft menu key HELP shows up when you highlight that particular command Press this soft menu key to get the CAS help facility entry for the command The first few screens of the catalog are shown below User installed library commands would also appear on the command catalog list using italic font If the library includes a help item then the soft menu key HELP shows up when you ...

Страница 828: ... contains the following sub menus The CMPLX sub menu The CMPLX sub menu contains functions pertinent to operations with complex numbers These functions are described in Chapter 4 The CONSTANTS sub menu The CONSTANTS sub menu provides access to the calculator mathematical constants These are described in Chapter 3 ...

Страница 829: ...scribed in Chapter 3 The INTEGER sub menu The INTEGER sub menu provides functions for manipulating integer numbers and some polynomials These functions are presented in Chapter 5 The MODULAR sub menu The MODULAR sub menu provides functions for modular arithmetic with numbers and polynomials These functions are presented in Chapter 5 ...

Страница 830: ...es relational operators e g etc logical operators e g AND OR etc the IFTE function and the ASSUME and UNASSUME commands Relational and logical operators are presented in Chapter 21 in the context of programming the calculator in User RPL language The IFTE function is introduced in Chapter 3 Functions ASSUME and UNASSUME are presented next using their CAS help facility entries see Appendix C ASSUME...

Страница 831: ...enu This command configures the CAS For CAS configuration information see Appendix C The ALGB sub menu The ALGB sub menu includes the following commands These functions except for 0 MAIN MENU and 11 UNASSIGN are available in the ALG keyboard menu Detailed explanation of these functions can be found in Chapter 5 Function UNASSIGN is described in the following entry from the CAS menu ...

Страница 832: ...s are described in Chapters 13 14 and 15 except for function TRUNC which is described next using its CAS help facility entry The MATHS sub menu The MATHS menu is described in detail in Appendix J The TRIGO sub menu The TRIGO menu contains the following functions These functions are also available in the TRIG menu Ñ Description of these functions is included in Chapter 5 ...

Страница 833: ...e CMPLX sub menu The CMPLX menu includes the following functions The CMPLX menu is also available in the keyboard ß Some of the functions in CMPLX are also available in the MTH COMPLEX menu start with Complex number functions are presented in Chapter 4 The ARIT sub menu The ARIT menu includes the following sub menus The sub menus INTEGER MODULAR and POLYNOMIAL are presented in detail in Appendix J...

Страница 834: ...by using Ð The functions in this menu are presented in Chapter 5 The MATR sub menu The MATR menu contains the following functions These functions are also available through the MATRICES menu in the keyboard Ø The functions are described in Chapters 10 and 11 The REWRITE sub menu The REWRITE menu contains the following functions ...

Страница 835: ...e available through the CONVERT REWRITE menu start with Ú The functions are presented in Chapter 5 except for functions XNUM and XQ which are described next using the corresponding entries in the CAS help facility IL HELP XNUM XQ ...

Страница 836: ...s characters to beginning of word SKIP Skips characters to end of word DEL Delete characters to beginning of word DEL Delete characters to end of word DEL L Delete characters in line INS When selected inserts characters at cursor location If not selected the cursor replaces characters overwrites instead of inserting characters EDIT Edits selection BEG Move to beginning of word END Mark end of sele...

Страница 837: ...y Clip Size is the number of characters in the clipboard Sel Size is the number of characters in the current selection EXEC Execute command selected HALT Stop command execution The line editor also provide the following sub menus SEARCH Search characters or words in the command line It includes the following functions GOTO Move to a desired location in the command line It includes the following fu...

Страница 838: ...h pattern as defined in Find Replace Selection Replace selection with replacement pattern defined with Replace command Replace Find Next Replace a pattern and search for another occurrence The pattern is defined in Replace Replace All Replace all occurrence of a certain pattern This command asks for confirmation from the user before replacing pattern Fast Replace All Replace all occurrences of a c...

Страница 839: ...mand line The input form provided for this command is Labels move to a specified label in the command line The Style sub menu The Style sub menu includes the following styles BOL Bold ITALI Italics UNDE Underline INV Inverse The command FONT allow the user to select the font for the command editor Examples of the different styles are shown below ...

Страница 840: ...Page L 5 ...

Страница 841: ...0 ALRM menu 25 3 AMORT 6 32 AMORTIZATION 6 11 AND 19 5 Angle between vectors 9 16 Angle measure 1 22 Angle symbol G 2 Angle units 22 27 22 29 22 33 Angular measure G 2 ANIMATE 22 27 Animating graphics 22 26 Animation 22 26 Anti derivatives 13 14 Approximate CAS mode C 4 Approximate vs Exact CAS mode C 4 APPS menu F 1 ARC 22 21 AREA in plots 12 7 Area units 3 19 ARG 4 6 ARITHMETIC menu 5 9 ASIN 3 6...

Страница 842: ... input form C 1 Calculator restart G 3 Calculator system tests G 3 Calculus 13 1 Cancel next repeating alarm G 3 Cartesian representation 4 1 CAS help facility C 10 CAS help facility listing H 1 CAS independent variable C 2 CAS menu F 6 CAS modulus C 3 CAS settings 1 24 C 1 CASDIR 2 35 16 36 CASE construct 21 51 CASINFO 2 35 Cauchy equation 16 53 CEIL 3 14 CENTR 22 7 Chain rule 13 6 Change sign 4 ...

Страница 843: ...ervals for the variance 18 33 Confidence intervals in linear regression 18 52 Confidence intervals in the calculator 18 27 Conic curve graphs 12 21 Conic curves 12 21 CONJ 4 6 CONLIB 3 29 Constants lib F 2 Continuous self test G 3 CONVERT 3 27 CONVERT menu 5 27 Convolution 16 49 Coordinate system 1 25 Coordinate transformation 14 7 COPY 2 26 COPY 2 33 Correlation coefficient 18 11 COS 3 7 COSH 3 9...

Страница 844: ...artial 14 1 Derivatives of equations 13 6 Derivatives with 13 4 DERVX 13 3 DESOLVE 16 7 DET 11 11 De tagging 21 33 Determinants 11 12 11 45 DIAG 10 13 Diagonal matrix 10 12 DIFF menu 16 4 DIFFE sub menu 6 31 Differential equation graph 12 26 Differential equations 16 1 Differential equations Fourier series 16 42 Differential equations graphical solutions 16 60 Differential equations Laplace transf...

Страница 845: ...ctric units 3 20 END 2 26 ENDSUB 8 11 Energy units 3 19 Engineering format 1 20 ENGL 3 29 Entering vectors 9 2 EPS 2 35 EPSX0 5 23 EQ 6 28 Equation Writer EQW 2 10 Equation writer properties 1 28 Equation Writer Selection Tree E 1 Equations linear systems 11 16 EQW BIG 2 11 EQW CMDS 2 11 EQW CURS 2 11 EQW Derivatives 2 30 EQW EDIT 2 11 EQW EVAL 2 11 EQW FACTOR 2 11 EQW HELP 2 11 EQW Integrals 2 31...

Страница 846: ...Finite arithmetic ring 5 14 Finite population 18 3 Fitting data 18 10 Fixed format 1 18 Flags 24 1 FLOOR 3 14 FOR construct 21 59 Force units 3 19 FOURIER 16 27 Fourier series 16 27 Fourier series and ODEs 16 42 Fourier series for square wave 16 40 Fourier series for triangular wave 16 35 Fourier series complex 16 27 Fourier transforms 16 43 Fourier transforms convolution 16 49 Fourier transforms ...

Страница 847: ...26 Graphs truth plots 12 29 Graphs bar plots 12 30 Graphs histograms 12 30 Graphs scatterplots 12 30 Graphs slope fields 12 34 Graphs Fast 3D plots 12 35 Graphs wireframe plots 12 34 Graphs Ps Contour plots 12 39 Graphs Y Slice plots 12 41 Graphs Gridmap plots 12 42 Graphs Pr Surface plots 12 43 Graphs Zooming 12 49 Graphs SYMBOLIC menu 12 51 Graphs saving 12 7 GRD 3 1 Greek letters D 3 G 2 Gridma...

Страница 848: ... 21 48 IF THEN END 21 47 IFERR sub menu 21 65 IFTE 3 35 ILAP 16 11 Illumination units 3 20 IM 4 6 IMAGE 11 54 Imaginary part 4 1 Implicit derivatives 13 7 Improper integrals 13 21 Increasing power CAS mode C 9 INDEP 22 6 Independent variable in CAS C 2 Infinite series 13 21 13 23 INFO 22 4 INPUT 21 22 Input forms programming 21 21 Input forms use of A 1 Input string prompt programming 21 21 Input ...

Страница 849: ...Keyboard alternate key functions B 4 Keyboard left shift functions B 5 Keyboard right shift functions B 8 Keyboard ALPHA characters B 9 Keyboard ALPHA left shift characters B 10 Keyboard ALPHA right shift characters B 12 Keyboard ALPHA function 1 12 Keyboard left shift function 1 12 Keyboard main function 1 12 Keyboard right shift function 1 12 Kronecker s delta 10 1 L LABEL 12 47 Labels L 4 LAGRA...

Страница 850: ...y H 1 List of command catalog I 1 Lists 8 1 LN 3 5 Ln X graph 12 9 LNCOLLECT 5 5 LNP1 3 9 Local variables 21 2 LOG 3 5 LOGIC menu 19 5 Logical operators 21 43 Lower triangular matrix 11 49 LQ 11 51 LQ decomposition 11 51 LSQ 11 23 LU 11 49 LU decomposition 11 49 LVARI 7 12 M Maclaurin series 13 21 MAD 11 47 Main diagonal 10 1 MAIN menu G 3 K 1 MAIN ALGB menu K 1 MAIN ARIT menu K 3 MAIN CASCFG comm...

Страница 851: ... 14 5 MAXR 3 16 Mean 18 3 Measures of central tendency 18 3 Measures of spreading 18 3 Median 18 3 MENU 12 47 Menu numbers 20 2 Menus 1 3 Menus not accessible through keyboard G 3 MES 7 10 Message box programming 21 37 Method of least squares 18 50 MIN 3 13 Minimum 13 12 Minimum 14 5 MINIT 7 13 MINR 3 16 MITM 7 12 MKSISOM 11 55 MOD 3 13 Mode 18 4 MODL 22 13 MODSTO 5 12 Modular arithmetic 5 12 Modu...

Страница 852: ... 68 Numerical solver 6 5 NUMX 22 10 NUMY 22 10 O OBJ 9 19 Objects 2 1 objects 24 1 OCT 19 2 Octal numbers 3 2 ODEs Laplace transform applications 16 17 ODEs Fourier series 16 46 ODEs Graphical solution 16 60 ODEs Numerical solution 16 60 ODEs ordinary differential equations 16 1 ODETYPE 16 8 OFF 1 2 ON 1 2 OPER menu 11 14 Operating mode 1 13 Operations with units 3 25 Operators 3 7 OR 19 5 ORDER 2...

Страница 853: ...ts program generated 22 17 Poisson distribution 17 5 Polar coordinate plot 12 19 Polar coordinates double integrals 14 9 Polar plot 12 19 Polar representation 4 1 POLY sub menu 6 30 Polynomial equations 6 6 Polynomial fitting 18 56 Polynomials 5 18 Population 18 3 POS 8 11 POTENTIAL 15 3 Potential function 15 3 15 6 Potential of a gradient 15 3 Power units 3 20 POWEREXPAND 5 29 POWMOD 5 12 PPAR 12...

Страница 854: ...22 36 Programming drawing commands 22 19 Programming input forms 21 27 Programming plots 22 14 Programming with GROBs 22 33 Programs with drawing functions 22 24 PROOT 5 22 PROPFRAC 5 10 5 25 Pr Surface plots 12 43 Ps Contour plots 12 39 PSI 3 15 Psi 3 15 PTAYL 5 11 5 22 PTYPE 22 4 PUT 8 10 PUTI 10 6 PVIEW 22 22 PX C 19 7 Q QR 11 51 QR decomposition 11 51 QUADF 11 52 Quadratic form diagonal repres...

Страница 855: ...RISCH 13 14 RKF 16 71 RKFERR 16 74 RKFSTEP 16 72 RL 19 6 RLB 19 7 RND 3 14 RNRM 11 8 ROOT 6 27 ROOT in plots 12 6 ROOT sub menu 6 27 ROW 10 24 Row norm 11 8 Row vectors 9 19 ROW 10 24 ROW 10 23 RPN mode 1 13 RR 19 6 RRB 19 7 REF RREF rref 11 47 RRK 16 71 RSBERR 16 74 RSD 11 43 RSWP 10 25 R Z 3 1 S Saddle point 14 5 Sample correlation coefficient 18 11 Sample covariance 18 11 Sample vs population 1...

Страница 856: ... SNRM 11 7 SOFT menus 1 3 SOLVE 5 5 SOLVE 6 2 7 1 SOLVE menu 6 27 SOLVE menu menu 74 G 3 SOLVE DIFF menu 16 69 SOLVEVX 6 4 SOLVR menu 6 28 SORT 2 34 Special characters G 2 Speed units 3 19 SPHERE 9 15 SQ 3 5 Square root 3 5 Square wave Fourier series 16 39 SR 19 6 SRAD 11 9 SRB 19 7 SREPL 23 3 SST 21 35 Stack properties 1 27 Standard deviation 18 4 Standard format 1 17 Standard normal distribution...

Страница 857: ...c division 5 26 SYST2MAT 11 42 System flag EXACT APPROX G 1 System flag 117 CHOOSE SOFT 1 4 G 2 System flag 95 ALG RPN G 1 system flags 24 3 System level operation G 3 System of equations 11 16 System tests G 3 T Table 12 17 12 26 TABVAL 12 52 13 9 TABVAR 12 52 13 11 Tagged output programming 21 34 TAIL 8 11 TAN 3 7 TANH 3 9 Taylor polynomial 13 23 Taylor series 13 24 TAYLR 13 25 TAYLR0 13 24 TCHE...

Страница 858: ...N 10 8 TRNC 3 14 Truth plots 12 29 TSTR 25 3 TVM menu 6 32 TVMROOT 6 32 Two dimensional plot programs 22 14 Two dimensional vector 9 12 TYPE 24 2 U UBASE 3 21 UFACT 3 27 UNASSIGN K 1 UNASUMME J 3 UNDE L 4 UNDO 2 61 UNIT 3 29 Unit prefixes 3 24 Units 3 17 Units in programming 21 37 Upper triangular matrix 11 41 User RPL language 21 1 User defined keys 20 1 Using input forms A 1 UTILITY menu menu 11...

Страница 859: ...ZIN 12 50 W Warm calculator restart G 3 Weber s equation 16 59 Weibull distribution 17 7 Weighted average 8 17 WHILE construct 21 62 Wireframe plots 12 37 Wordsize 19 4 X XCOL 22 13 XNUM K 5 XOR 19 5 XPON 3 14 XQ K 5 XRNG 22 6 XROOT 3 5 XSEND 2 34 XVOL 22 10 XXRNG 22 10 XYZ 3 1 Y YCOL 22 13 YRNG 22 6 Y Slice plots 12 41 YVOL 22 10 YYRNG 22 10 Z ZAUTO 12 50 ZDECI 12 50 ZDFLT 12 50 ZEROS 6 4 ZFACT 1...

Страница 860: ...5 LIST 8 9 ΣLIST 8 9 ΠLIST 8 9 ΣPAR 22 13 ARRY 9 21 ARRY 9 6 BEG L 1 COL 10 18 DATE 25 3 DEL L 1 DIAG 10 13 END L 1 GROB 22 31 HMS 25 3 LCD 22 32 LIST 9 23 ROW 10 22 SKIP L 1 STK 3 30 STR 23 1 TAG 21 33 TAG 23 1 TIME 25 3 UNIT 3 27 V2 9 12 V3 9 13 ...

Страница 861: ... or replace any product to a condition as warranted you will be entitled to a refund of the purchase price upon prompt return of the product with proof of purchase 4 HP products may contain remanufactured parts equivalent to new in performance or may have been subject to incidental use 5 Warranty does not apply to defects resulting from a improper or inadequate maintenance or calibration b softwar...

Страница 862: ...anty statements accompanying such products and services HP shall not be liable for technical or editorial errors or omissions contained herein FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND NEW ZEALAND THE WARRANTY TERMS CONTAINED IN THIS STATEMENT EXCEPT TO THE EXTENT LAWFULLY PERMITTED DO NOT EXCLUDE RESTRICT OR MODIFY AND ARE IN ADDITION TO THE MANDATORY STATUTORY RIGHTS APPLICABLE TO THE SALE OF T...

Страница 863: ...ephone numbers Argentina 0 810 555 5520 Brazil Sao Paulo 3747 7799 ROTC 0 800 157751 Mexico Mx City 5258 9922 ROTC 01 800 472 6684 Venezuela 0800 4746 8368 Chile 800 360999 Columbia 9 800 114726 Peru 0 800 10111 Central America Caribbean 1 800 711 2884 Guatemala 1 800 999 5105 Puerto Rico 1 877 232 0589 Costa Rica 0 800 011 0524 N America Country Telephone numbers U S 1800 HP INVENT Canada 905 206...

Страница 864: ... to the receiver Connections to Peripheral Devices To maintain compliance with FCC rules and regulations use only the cable accessories provided Canada This Class B digital apparatus complies with Canadian ICES 003 Cet appareil numerique de la classe B est conforme a la norme NMB 003 du Canada Japan この装置は情報処理装置等電波障害自主規制協議会 VCCI の基準に基づく第二情報技術装置ですこの装置は家庭環境で使用することを目的としていますがこの装置がラジオやテレビジョン受信機に近接...

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