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Chapter 2: Documenting Polynomial Zero Finding

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Содержание Derive 5

Страница 1: ...Introduction to T I E X P L O R A T I O N S S O F T W A R E by Bernhard Kutzler Vlasta Kokol Voljc ...

Страница 2: ...following Derive 5 documentation is being provided as a courtesy of the authors Bernhard Kutzler Vlasta Kokol and the publisher Texas Instruments We invite you to use the following abbreviated document during your personal evaluation of Derive 5 Use of the document for any other purpose is strictly prohibited ...

Страница 3: ... this book The author and publisher shall not be liable in any event for incidental or consequental damages in connection with or arising out of the furnishing performance or use of this text No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise without prior permission in writ...

Страница 4: ...Chapter 6 Sequences and Families of Curves 95 Chapter 7 Investigations in Space 117 Chapter 8 What Is Simple 135 Chapter 9 Vectors Matrices and Sets 153 Chapter 10 Parametric Plots 171 Chapter 11 Towards a Module for Analytical Geometry 185 Chapter 12 Some Calculus 203 Chapter 13 More on Plotting 221 Chapter 14 What Else Can DERIVE Do 243 Learn More about DERIVE 261 Appendix A DERIVE Startup Optio...

Страница 5: ...Many thanks to Albert Rich and Theresa Shelby the principal authors of DERIVE 5 for their continuous support during the writing of this book Many thanks to Patricia Littlefield and David Stoutemyer who polished the language of this book Bernhard Kutzler Vlasta Kokol Voljc February 2000 ...

Страница 6: ... the mechanical algorithmic parts of solving a problem students can concentrate on the mathematical meaning of concepts Instead of teaching and learning boring technical skills teachers and students can concentrate on the exciting and useful techniques of problem solving It has proven to be highly supportive for the cognitive development of advanced mathematical concepts For an engineer DERIVE is ...

Страница 7: ...h WINDOWS 95 WINDOWS 98 or WINDOWS NT It is assumed that you know how to use computers and the WINDOWS operating system The screen shots in this book were produced from DERIVE running on WINDOWS NT If you are running DERIVE on WINDOWS 95 or 98 some of the screens may appear slightly different This book introduces all features and functions that are required for routine use of DERIVE 5 There is mor...

Страница 8: ... DERIVE it is a factory default version If you use a version of DERIVE that was used by someone else we recommend that you turn it into a factory default version now Appendix B gives instructions on how to do this Start DERIVE by double clicking on the DERIVE icon If there is no DERIVE icon on your computer s desktop you probably will find DERIVE on the Start menu or via Start Programs The followi...

Страница 9: ...ove the mouse pointer onto then click i e press and release the left mouse button Enter the fraction 2 3 End the input with the Enter key DERIVE displays this expression as a fraction with a horizontal line a numerator and a denominator i e in 2 dimensional output format as opposed to the 1 dimensional or linear input format used for entering the number The expression s unique label number 1 is sh...

Страница 10: ...pression is highlighted in reverse video Expression 1 is no longer highlighted If you mistyped the input and want to delete the highlighted expression for a retry use Esc to move the focus into the algebra window use the Delete key Del to delete the highlighted expression then use the Author Expression button to move the focus back into the entry line An alternative technique for replacing an expr...

Страница 11: ...n expression is highlighted the Status Bar s second pane shows the automatically generated expression annotation The third pane shows the computing time in case the expres sion was obtained as a result of a computation For expression 6 this is The automatically generated annotation explains how the expression was obtained Approx 4 means that the expression was obtained by applying the Approximate ...

Страница 12: ...A text object allows simple text editing similar to what you can do in standard text editors Later you will learn how to change the font size alignment color etc As a next example compute 56 1234 Due to the previous activity the focus now is in the algebra window Before you can enter another expression move the focus into the entry line Enter 1234 56 by using the Author Expression button then typi...

Страница 13: ...same effect as concluding the input with the key Try it Enter the above expression with then simplify as usual using the Command Toolbar s Simplify button Unlike ordinary calculators DERIVE can perform nonnumeric symbolic algebraic computations such as simplifying expression 10 into expression 11 For the next example use the Expression Entry Toolbar s second button To simplify 2 x x immediately ty...

Страница 14: ...ames DERIVE allows this too for example time or x12 Using multicharacter variable names will be explained in Chapter 14 Clearly you cannot omit all parentheses For example you will need to parenthesize the denominator to enter a rational expression such as 2 1 x If the parentheses are omitted in this example the resulting expression has a different meaning Enter 2 x 1 Oops the expression on the sc...

Страница 15: ...nd have no gaps Errors such as omitting a whole pair of parentheses may change the meaning of an expression as was the case in the previous example If only one parenthesis is omitted the input becomes a meaningless character string and DERIVE will issue a warning in the form of an appropriate syntax error message Enter 4x 1 x 5 after moving the focus into the entry line with DERIVE attempts to pos...

Страница 16: ...o type the new input string You could use the backspace key several times to delete the old string but a more elegant way is to use the tab key Highlight the contents of the entry line with the tab key ÿ Enter and simplify x 2 It is up to you to either use the Enter key followed by the Simplify button or to use the entry line s Enter and Simplify button The square root symbol can be obtained from ...

Страница 17: ...reek Symbol Toolbar button Try to expand expression 21 first by simplifying with This did not change anything Now you have an opportunity to apply one of those commands for which there is no equivalent Command Toolbar button Prepare for opening the Simplify menu by moving the mouse pointer above the Menu Bar s Simplify command Open the Simplify menu by clicking the left mouse button This menu offe...

Страница 18: ...the expansion with the suggested parameters by using _Expand_ either press because this is the default button or click on _Expand_ A keyboard alternative for selecting the Expansion command from the Simplify menu is the following standard WINDOWS technique Alt S opens the Simplify menu use S because of the underscore under the letter S in Simplify then press E again the letter with the underscore ...

Страница 19: ...can see in the Command Toolbar The inaccessible buttons and menu commands appear dimmed For example the Approximate button is not available in text editing mode now because a text object is highlighted You need to highlight an expression before you can approximate it Highlight expression 24 then approximate it with Simplify Approximate Other than the Command Toolbar s Approximate button the Simpli...

Страница 20: ... font size requires the Formatting Toolbar to be on Open the View Toolbars submenu with the Window View Toolbars command Turn the Formatting Toolbar on by selecting the Formatting Toolbar command For editing DERIVE text use the same techniques as in standard word processing programs This toolbar indicates that the font size is 12 points Before you can reduce the font size to 10 points you need to ...

Страница 21: ...ct command adds the text object after the highlighted object you need to highlight expression 26 first Select expression 26 Although the frame around the unintentionally inserted empty text object disappeared it is still there It can be deleted like any other object only after it is highlighted Highlight the text object by clicking into it Try to delete it using the Del key This has no effect Reme...

Страница 22: ...ject after the highlighted expression 26 using then start entering the text Next we Note that this text again has font size 12 as you can see in the Formatting Toolbar s Font Size field Earlier you only changed the format of existing text Changing the default format of all new text objects is done via a command from the Options Display menu To change the default setting of future text objects sele...

Страница 23: ... objects only Delete the text object for a retry with the new default text font size Select it by clicking into it then using Esc Delete with Del or Enter a new text object with the following contents The text has font size 10 points now You will not need the Formatting Toolbar any more in this session so switch it off to provide more space for other purposes Switching a toolbar off requires the s...

Страница 24: ... an operator to an expression without simplifying Simplified Expressions are expressions obtained from simplifying or approximating an expression It is helpful to display user input left justified and the answers centered as it is done by the default setting To keep the settings as they are exit the dialog with _Cancel_ or the Esc key Try the next command in the Options Display submenu Try the men...

Страница 25: ...ing the Color selection menu Select a color of your choice by clicking on it then close the dialog with _OK_ Earlier you entered π via the Greek or Math Symbol Toolbar There are several methods for entering special constants such as π the base of the natural logarithm e or the imaginary unit i To enter a sum of three π s first move the focus into the entry line using F2 Enter the first π from one ...

Страница 26: ...ee e s and the variable e with The postfix equals operator causes an automatic simplification and the generation of an equation whose left hand side is the unsimplified expression and whose right hand side is the simplified expression This method displays both the unsimplified and simplified expression on the same line saving lines on the screen Similarly there are three methods for entering the i...

Страница 27: ... highlighting one expression up down Esc cancel command click left mouse button into row occupied by the expression highlight expression click left mouse button into text object edit contents of text object click onto text object frame or left or right of it or press Esc from within text editing highlight text object without text editing Expression Entry Toolbar or enter expression enter simplifie...

Страница 28: ...isplay Font of New Text Objects command is among the data in this file The Derive Startup dialog gives you the choice to either start DERIVE with the factory default settings or start DERIVE with the settings from the initialization file i e with some of the changes from the first chapter This book is written so that each chapter starts with a factory default DERIVE We recommend that you do the sa...

Страница 29: ...of the old expression with the new one There is no need to delete the old expression when using the Edit Derive Object command Consider looking at a house from several different positions From each position you will see details that you can t see from other positions Based on this idea mathematicians use a variety of different representations for mathematical objects The fourth degree polynomial t...

Страница 30: ...gebra window and a 2D plot window Use the usual WINDOWS techniques to flip between windows or change their sizes and positions Put the two windows side by side using the Window Tile Vertically command Each window is labeled with the window type in its upper left corner 2D plot and Algebra The active window s Title Bar is dark the inactive window s Title Bar is dimmed Since the plot window is activ...

Страница 31: ... window then make the 2D plot window active again by clicking its Title Bar There are several techniques to make a different window active From the algebra window use the Command Toolbar s 2D plot window button and from the 2D plot window use the Command Toolbar s Algebra window button Click on the window you want to make active This method however must be used with care Clicking on an algebra win...

Страница 32: ... by using the 2D plot window s File Embed command This freezes the current status of the plot window into the worksheet The plot window is interactive the embedded plot image is not The embedded plot image can be brought back into an interactive plot window at any time with a double mouse click The graphical representation is useful for exploring the polynomial s zeros However from the current pic...

Страница 33: ...cture Use the arrow keys to move the cross to 0 5 0 5 Try Ctrl Æ Ctrl æ Ctrl and Ctrl to move the cross in bigger steps The Home key moves the cross to the plot window center The trace mode is very useful for inspecting curves This mode can be switched on and off with the Trace Plots button the Options Trace Plots command or the corresponding hot key F3 As is customary in WINDOWS programs a button...

Страница 34: ...Ctrl Æ and Ctrl æ for big steps It can also be done by moving the mouse pointer and clicking with the left mouse button to the new position If there are several graphs displayed use and to select another graph Become familiar with moving the square Use the arrow keys and the mouse to move the square Finally click the left mouse button at the point 2 5 0 What happened to the square It disappeared L...

Страница 35: ...ledge and reasoning about the curve Independent of the follow mode status you may use the Center on origin button Select the Set Plot Range or the Set Plot Region command use the _Reset_ button then leave the dialog If available double click on an embedded version of the original graph This last option is particularly convenient Restore the original graph by double clicking on the embedded graph T...

Страница 36: ...een 1 61 and 1 62 Restore the original scale factors by zooming out with the Zoom out button Find an approximation for the leftmost zero by moving the square to it The leftmost zero seems to be at exactly x 2 Document what you found so far by inserting appropriate text objects Switch to the algebra window Resize the embedded plot Select the image by clicking on it The image is surrounded by 8 blac...

Страница 37: ...ndow active then move the square to the uncertain middle section You will find that there is one zero between 0 62 and 0 6 Another zero seems to be at exactly x 0 5 To obtain a picture with intersections of the graph magnify again Zoom in this time using the Zoom in button twice It becomes obvious that there are two zeros Continue to magnify the graph Zooming in once more with lets the square leav...

Страница 38: ...h allows to choose a crop rectangle graphically Prepare for choosing a crop rectangle by using the Set range with box button The mouse cursor turns into a crosshair Choose a crop rectangle Click and hold the left mouse button at the top left corner of the desired area Drag the mouse down and to the right until the box encloses the desired area Release the mouse button The Set 2D Plot Range dialog ...

Страница 39: ...ul to edit the suggested numerical values to the nearest simple values Start by overwriting the highlighted value of the input field for the Horizontal Minimum Then use the tab key ÿ to make the next input field active Enter the following values 0 7 ÿ 0 4 ÿ 6 ÿ Make the values for the Intervals fields fit to the difference of the values for the Minimum and the Maximum fields For example 6 interval...

Страница 40: ...hen switch to the algebra window and resize the embedded plot appropriately Insert a new text object documenting the method and result of your findings Close the plot window then open the algebra window to full size Close the plot window by clicking the left mouse button on the button that is located in the window s upper right corner Open the algebra window to full size by clicking on the button ...

Страница 41: ...y with 0 then conclude the input with For solving this equation either use the Solve Expression command or the corresponding toolbar button Prepare for solving the equation by applying the Solve Expression button Solve the equation Accept all suggested parameters by selecting _Solve_ Here is the mathematical symbol for the logical operator OR Similar to the Entry Toolbar s Enter and Simplify butto...

Страница 42: ...re documenting author s and date Switch the Formatting Toolbar on using Window View Toolbars Formatting Toolbar All fields and buttons are dimmed as long as there is no text object in editing mode Add a text object at the end of the worksheet using Choose a special format for the signature In the Formatting Toolbar change the font size to 8 points and click on the Right Justify button Enter This d...

Страница 43: ... Look at the print preview using the File Print Preview command Print preview offers various options including a button for zooming in Zoom in with _Zoom_In_ The magnifying glass shaped cursor in the upper right quarter of the page indicates that an alternative to using the _Zoom_In_ button is to click with the left mouse button ...

Страница 44: ...ow with _Close_ then select the command Options Printing Expression Layout Here you can select the expression font size choose between Regular and Bold font and control the printing of Annotations and Computation Times By default neither is printed Change the font size to 11 points then close the dialog with _OK_ Apply again the File Print Preview command this time zooming in twice The worksheet i...

Страница 45: ...e printer is properly connected switched on and set In the Printing dialog box you can change the printer or the printing properties change the print range from All to either a range of pages or the highlighted expressions or change the number of copies from the default 1 to the number you want Send the document to the printer with _OK_ ...

Страница 46: ...from the selection menu that is offered for the Save in field Accept the suggestion and enter the file name chapter02 in the File name input field Close the dialog with _Save_ Notice the Title Bar Previously there was Algebra 1 as the indication of an unnamed algebra worksheet Now there is Algebra 1 chapter02 dfw indicating an algebra worksheet with name chapter02 dfw The suffix dfw is the default...

Страница 47: ...open new 2D plot window double click left mouse button on embedded plot open embedded plot in plot window 2D plot Window or Insert Plot plot highlighted expression or Options Trace Plots or F3 toggle trace mode center plot region on cross center plot region on origin or F9 zoom in or F10 zoom out or F7 zoom in vertically graphically choose a crop rectangle File Embed copy plot window into algebra ...

Страница 48: ... 49 122 F4 49 64 F9 128 Home 28 PgDn 119 PgUp 119 PrtSc 201 ÿ 11 34 Insert expression 49 64 dfw 41 2 1 56 2D plot Window button 25 3D plot window button 117 A abort computation 90 absolute value function 74 139 154 198 accuracy 86 activate window 26 aggressive simplification 151 algebra window 3 25 Algebra Window button 26 algebraic equation 232 algebraic representation 24 Algebraically radio butt...

Страница 49: ... axes titles 107 B back accent operator 164 background color 130 backspace 14 15 base of natural logarithm e 21 Basic File 201 Binary 252 Bold button 37 books about DERIVE 261 Boolean combination 77 border lines curves 76 77 Bottom Maximum 125 Bottom Minimum 125 box 129 brackets 72 137 branch 230 Branch 232 bring embedded plots alive 56 Buchberger 73 built in functions 187 C C File 201 Calculus Di...

Страница 50: ...etic 90 data matrix 159 decide 87 decimal fractions 83 decimal numbers 249 Declare Function Definition 193 Declare Input Settings 66 245 Declare Output Settings 47 246 249 Declare Simplification Set 84 144 232 Declare Variable Domain 136 229 default 68 141 default text font size 17 definite integral 211 DEG 147 degrees 148 155 Delete 122 delete all plots 98 delete expression 60 Delete Object butto...

Страница 51: ...ables 47 explicit form 70 explicit solution 217 exponentiation operator 7 Expression display 247 Expression Entry Toolbar 3 expression font size 39 extent of expression 44 extract element from list 72 F factor polynomial 59 factorial 256 factory default 1 23 245 264 265 factory default box 117 factory default plot range 102 factory default precision mode 86 FALSE 65 family member 95 family of curv...

Страница 52: ...ight text object 16 highlighting subexpression 43 holes 139 horizontal drag line 51 horizontal length 101 horizontal line 97 105 horizontal scroll bar 86 HOWOFTEN 254 hyperbola 70 I i 21 IF 200 256 imaginary component 139 231 imaginary unit 21 implicit plot 70 93 235 implicit solution 217 import numerical data 256 inaccessible buttons 14 inactive window 25 increment button 84 incrementally rotated...

Страница 53: ... 84 models 63 move cross 28 move highlighting 6 move objects 51 move the annotation 196 multi character variable names 9 multi line display 201 multiple angles 144 Multiplication Operator 248 N named constants 109 naming convention 187 New button 54 Newsletter 261 noncontiguous objects 53 nonnegative 136 nonreal valued function 228 nonsquare matrices 165 notation 249 NSOLVE 91 Number of Panels 123...

Страница 54: ...tor 185 perpendicular vector form 185 persistent parentheses 43 47 phase angle 232 photograph 243 physical screen length 69 pi 13 20 83 pixels 69 plot 24 74 plot a vector 97 Plot button 117 126 Plot Color tab 124 Plot Expression button 26 Plot Parameters tab 123 Plot Properties dialog 123 plot unsimplified expressions 102 PLOTINT 212 point 98 point of intersection 189 polar 235 236 polynomial 56 1...

Страница 55: ...ed plot 122 semi colons 162 190 Sensitive radio button 66 Set 2D Aspect Ratio dialog 69 Set Eye Position button 120 Set range with box button 33 Set Aspect Ratio 69 80 172 198 240 Set Coordinate System 182 236 237 Set Eye Position 120 Set Plot Range 30 33 99 130 Set Plot Region 30 101 130 sets 168 shaded area 75 shallowest operator 44 shift the function 175 show the equation 199 shrink 119 Shrink ...

Страница 56: ...7 14 15 text object 7 THROWS 253 TIFF File 201 tile windows 25 Titlebar 8 29 41 189 too many windows 68 toolbars 244 Top Maximum 125 Top Minimum 125 touch 240 trace curve 121 trace mode 28 120 trace mode does not work 70 Trace Plots button 28 120 122 Tracing Toolbar 121 Transformation Direction 144 transformation rule 136 translation 175 transpose of matrix 164 triangle TRIANGLE 193 195 221 Trig P...

Страница 57: ... View 3 11 visualize 93 117 155 W window type 25 Window Display Tabs 55 Window New 2D plot Window 25 Window New 3D plot Window 132 179 Window Tile Vertically 25 Window View Toolbars 15 Window View Toolbars Formatting T 37 word mode 66 246 worksheet 4 Wrap button 86 Z Zoom in button 31 127 Zoom out button 31 133 213 Zoom vertical in button 31 Zoom vertical out button 31 213 ...

Страница 58: ...Texas Instruments U S A 7800 Banner Drive Dallas Texas 75251 2000 Texas Instrumentso www ti com calc DERIVE OM 1L3 A ti cares ti com ...

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