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156

Alphabetical Listing

seqGen()

Catalog >

seqGen(

Expr

Var

depVar

{

Var0

VarMax

}[

ListOfInitTerms

[

VarStep

[

CeilingValue

]]]

)

⇒

list

Generates a list of terms for sequence

depVar

(

Var

Expr

as follows: Increments

independent variable

Var

from

Var0

through

VarMax

VarStep

, evaluates

depVar

(

Var

) for corresponding values of

Var

using the

Expr

formula and

ListOfInitTerms

, and returns the results as

a list.

seqGen(

ListOrSystemOfExpr

Var

ListOfDepVars

{

Var0

VarMax

} [

MatrixOfInitTerms

[

VarStep

[

CeilingValue

]]]

)

⇒

matrix

Generates a matrix of terms for a system
(or list) of sequences

ListOfDepVars

(

Var

ListOrSystemOfExpr

as follows:

Increments independent variable

Var

from

Var0

through

VarMax

VarStep

evaluates

ListOfDepVars

(

Var

) for

corresponding values of

Var

using

ListOrSystemOfExpr

formula and

MatrixOfInitTerms

, and returns the results

as a matrix.

The original contents of

Var

are unchanged

after

seqGen()

is completed.

The default value for

VarStep

Generate the first 5 terms of the sequence

(

) =

(

-1)

/2, with

(1)=

and

VarStep

Example in which Var0=2:

Example in which initial term is symbolic:

System of two sequences:

Note: The Void (_) in the initial term matrix
above is used to indicate that the initial term
for u1(n) is calculated using the explicit
sequence formula u1(n)=1/n.

seqn()

Catalog >

seqn(

Expr

(

[

ListOfInitTerms

[

nMax

[

CeilingValue

]]]

)

⇒

list

Generate the first 6 terms of the sequence

(

) =

(

-1)/2, with

(1)=

Summary of Contents for TI-Nspire CAS

Page 1: ...TI Nspire CAS Reference Guide This guidebook applies to TI Nspire software version 4 3 To obtain the latest version of the documentation go to education ti com guides ...

Page 2: ...ts be liable to anyone for special collateral incidental or consequential damages in connection with or arising out of the purchase or use of these materials and the sole and exclusive liability of Texas Instruments regardless of the form of action shall not exceed the amount set forth in the license for the program Moreover Texas Instruments shall not be liable for any claim of any kind whatsoeve...

Page 3: ... 109 N 117 O 126 P 128 Q 137 R 140 S 153 T 178 U 193 V 194 W 195 X 197 Z 198 Symbols 206 Empty Void Elements 232 Shortcuts for Entering Math Expressions 234 EOS Equation Operating System Hierarchy 236 Error Codes and Messages 238 Warning Codes and Messages 246 Support and Service 249 Texas Instruments Support and Service 249 3 ...

Page 4: ...4 Service and Warranty Information 249 Index 251 ...

Page 5: ...er Position the cursor on each element and type a value or expression for the element Fraction template p keys Note See also divide page 208 Example Exponent template l key Note Type the first value press l and then type the exponent To return the cursor to the baseline press right arrow Note See also power page 209 Example Square root template q keys Note See also square root page 219 Example Nth...

Page 6: ...le Log template s key Calculates log to a specified base For a default of base 10 omit the base Note See also log page 105 Example Piecewise template 2 piece Catalog Lets you create expressions and conditions for a two piece piecewise function To add a piece click in the template and repeat the template Note See also piecewise page 130 Example ...

Page 7: ...te Catalog Creates a system of two equations To add a row to an existing system click in the template and repeat the template Note See also system page 177 Example System of N equations template Catalog Lets you create a system of N equations Prompts for N Note See also system page 177 Example See the example for System of equations template 2 equation Absolute value template Catalog Note See also...

Page 8: ... of decimal degrees mm is the number of minutes and ss ss is the number of seconds Example Matrix template 2 x 2 Catalog Creates a 2 x 2 matrix Example Matrix template 1 x 2 Catalog Example Matrix template 2 x 1 Catalog Example Matrix template m x n Catalog The template appears after you are prompted to specify the number of rows and columns Example ...

Page 9: ... appear Sum template Σ Catalog Note See also Σ sumSeq page 220 Example Product template Π Catalog Note See also Π prodSeq page 219 Example First derivative template Catalog The first derivative template can also be used to calculate first derivative at a point Note See also d derivative page 217 Example Expression Templates 9 ...

Page 10: ...o d derivative page 217 Example Nth derivative template Catalog The nth derivative template can be used to calculate the nth derivative Note See also d derivative page 217 Example Definite integral template Catalog Note See also integral page 217 Example Indefinite integral template Catalog Note See also integral page 217 Example Limit template Catalog Example ...

Page 11: ...Limit template Catalog Use or for left hand limit Use for right hand limit Note See also limit page 10 Expression Templates 11 ...

Page 12: ... argument is a complex number returns the number s modulus Note All undefined variables are treated as real variables amortTbl Catalog amortTbl NPmt N I PV Pmt FV PpY CpY PmtAt roundValue matrix Amortization function that returns a matrix as an amortization table for a set of TVM arguments NPmt is the number of payments to be included in the table The table starts with the first payment N I PV Pmt...

Page 13: ...plified form of the original entry Integer1 andInteger2 integer Compares two real integers bit by bit using an and operation Internally both integers are converted to signed 64 bit binary numbers When corresponding bits are compared the result is 1 if both bits are 1 otherwise the result is 0 The returned value represents the bit results and is displayed according to the Base mode You can enter th...

Page 14: ... element as a complex number that represents a two dimensional rectangular coordinate point ANOVA Catalog ANOVA List1 List2 List3 List20 Flag Performs a one way analysis of variance for comparing the means of two to 20 populations A summary of results is stored in the stat results variable page 172 Flag 0 for Data Flag 1 for Stats Output variable Description stat F Value of the F statistic stat PV...

Page 15: ...tions A summary of results is stored in the stat results variable See page 172 LevRow 0 for Block LevRow 2 3 Len 1 for Two Factor where Len length List1 length List2 length List10 and Len LevRow Î 2 3 Outputs Block Design Output variable Description stat F F statistic of the columnfactor stat PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat df Degrees of freedom of...

Page 16: ...SSCol Sum of squares of the columnfactor stat MSCol Meansquares for columnfactor ROW FACTOR Outputs Output variable Description stat FRow F statistic of the rowfactor stat PValRow Probability value of the rowfactor stat dfRow Degrees of freedom of the rowfactor stat SSRow Sum of squares of the rowfactor stat MSRow Meansquares for rowfactor INTERACTION Outputs Output variable Description stat FInte...

Page 17: ...as an expression containing decimal values when possible regardless of the current Auto or Approximate mode This is equivalent to entering the argument and pressing approx List1 list approx Matrix1 matrix Returns a list or matrix where each element has been evaluated to a decimal value when possible approxFraction Catalog Expr approxFraction Tol expression List approxFraction Tol list Matrix appro...

Page 18: ...alog approxRational Expr Tol expression approxRational List Tol list approxRational Matrix Tol matrix Returns the argument as a fraction using a tolerance of Tol If Tol is omitted a tolerance of 5 E 14 is used arccos See cos page 35 arccosh See cosh page 36 arccot See cot page 37 arccoth See coth page 38 arccsc See csc page 41 arccsch See csch page 41 ...

Page 19: ...uming a function mode definition arcLen List1 Var Start End list Returns a list of the arc lengths of each element of List1 from Start to End with respect to Var arcsec See sec page 154 arcsech See sech page 154 arcsin See sin page 164 arcsinh See sinh page 165 arctan See tan page 179 arctanh See tanh page 180 augment Catalog augment List1 List2 list Alphabetical Listing 19 ...

Page 20: ...1 or Matrix2 avgRC Catalog avgRC Expr1 Var Value Step expression avgRC Expr1 Var Value List1 list avgRC List1 Var Value Step list avgRC Matrix1 Var Value Step matrix Returns the forward difference quotient average rate of change Expr1 can be a user defined function name see Func When Value is specified it overrides any prior variable assignment or any current substitution for the variable Step is ...

Page 21: ...91 If you omit Pmt it defaults to Pmt tvmPmt N I PV FV PpY CpY PmtAt If you omit FV it defaults to FV 0 The defaults for PpY CpY and PmtAt are the same as for the TVM functions roundValue specifies the number of decimal places for rounding Default 2 bal NPmt amortTable calculates the balance after payment number NPmt based on amortization table amortTable The amortTable argument must be a matrix i...

Page 22: ...FFFFFFF in Hex base mode 0b111 111 64 1 s in Binary base mode 263 is displayed as 0h8000000000000000 in Hex base mode 0b100 000 63 zeros in Binary base mode If you enter a decimal integer that is outside the range of a signed 64 bit binary form a symmetric modulo operation is used to bring the value into the appropriate range Consider the following examples of values outside the range 263 becomes ...

Page 23: ...ase16 integer Note You can insert this operator from the computer keyboard by typing Base16 Converts Integer1 to a hexadecimal number Binary or hexadecimal numbers always have a 0b or 0h prefix respectively 0b binaryNumber 0h hexadecimalNumber Zero not the letter O followed by b or h A binary number can have up to 64 digits A hexadecimal number can have up to 16 Without a prefix Integer1 is treate...

Page 24: ...trials and probability p of success on each trial For P X upBound set lowBound 0 binomPdf Catalog binomPdf n p list binomPdf n p XVal number if XVal is a number list if XVal is a list Computes a probability for the discrete binomial distribution with n number of trials and probability p of success on each trial C Catalog ceiling Expr1 integer Returns the nearest integer that is the argument The ar...

Page 25: ...is the step value If Step is omitted it defaults to 0 001 When using List1 or Matrix1 the operation gets mapped across the values in the list or across the matrix elements Note See also avgRC and d cFactor Catalog cFactor Expr1 Var expression cFactor List1 Var list cFactor Matrix1 Var matrix cFactor Expr1 returns Expr1 factored with respect to all of its variables over a common denominator Expr1 i...

Page 26: ...tional expressions in any other variables to increase factorization with respect to Var There might be some incidental factoring with respect to other variables For the Auto setting of the Auto or Approximate mode including Var also permits approximation with floating point coefficients where irrational coefficients cannot be explicitly expressed concisely in terms of the built in functions Even w...

Page 27: ...2way obsMatrix Computes a χ2 test for association on the two way table of counts in the observed matrix obsMatrix A summary of results is stored in the stat results variable page 172 For information on the effect of empty elements in a matrix see Empty Void Elements page 232 Output variable Description stat χ2 Chi square stat sum observed expected 2 expected stat PVal Smallestlevel of significance...

Page 28: ...GOF obsList expList df chi2GOF obsList expList df Performs a test to confirm that sample data is from a population that conforms to a specified distribution obsList is a list of counts and must contain integers A summary of results is stored in the stat results variable See page 172 For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Descripti...

Page 29: ...riables See unLock page 193 ClrErr Catalog ClrErr Clears the error status and sets system variable errCode to zero The Else clause of the Try Else EndTry block should use ClrErr or PassErr If the error is to be processed or ignored use ClrErr If what to do with the error is not known use PassErr to send it to the next error handler If there are no more pending Try Else EndTry error handlers the er...

Page 30: ...rix expression Returns the number of columns contained in Matrix Note See also rowDim colNorm Catalog colNorm Matrix expression Returns the maximum of the sums of the absolute values of the elements in the columns in Matrix Note Undefined matrix elements are not allowed See also rowNorm comDenom Catalog comDenom Expr1 Var expression comDenom List1 Var list comDenom Matrix1 Var matrix comDenom Expr...

Page 31: ...enominator Such results usually save even more time memory and screen space Such partially factored results also make subsequent operations on the result much faster and much less likely to exhaust memory Even when there is no denominator the comden function is often a fast way to achieve partial factorization if factor is too slow or if it exhausts memory Hint Enter this comden function definitio...

Page 32: ...o complete the square with respect to variables Var1 Var2 conj Catalog conj Expr1 expression conj List1 list conj Matrix1 matrix Returns the complex conjugate of the argument Note All undefined variables are treated as real variables constructMat Catalog constructMat Expr Var1 Var2 numRows numCols matrix Returns a matrix based on the arguments Expr is an expression in variables Var1 and Var2 Eleme...

Page 33: ...up creating Var2 if necessary Var1 must be the name of an existing variable group such as the statistics stat nn results or variables created using the LibShortcut function If Var2 already exists this command replaces all members that are common to both groups and adds the members that do not already exist If one or more members of Var2 are locked all members of Var2 are left unchanged corrMat Cat...

Page 34: ... not contain explicit references to degree or gradian angles cos µ key cos Expr1 expression cos List1 list cos Expr1 returns the cosine of the argument as an expression cos List1 returns a list of the cosines of all elements in List1 Note The argument is interpreted as a degree gradian or radian angle according to the current angle mode setting You can use G or r to override the angle mode tempora...

Page 35: ...s A X cos B X where cos B All computations are performed using floating point arithmetic cos µ key cos Expr1 expression cos List1 list cos Expr1 returns the angle whose cosine is Expr1 as an expression cos List1 returns a list of the inverse cosines of each element of List1 Note The result is returned as a degree gradian or radian angle according to the current angle mode setting Note You can inse...

Page 36: ...sh Expr1 expression cosh List1 list cosh Expr1 returns the hyperbolic cosine of the argument as an expression cosh List1 returns a list of the hyperbolic cosines of each element of List1 InDegree angle mode cosh squareMatrix1 squareMatrix Returns the matrix hyperbolic cosine of squareMatrix1 This is not the same as calculating the hyperbolic cosine of each element For information about the calcula...

Page 37: ...e mode andInRectangular Complex Format To see the entire result press andthen use and to move the cursor cot µ key cot Expr1 expression cot List1 list Returns the cotangent of Expr1 or returns a list of the cotangents of all elements in List1 Note The argument is interpreted as a degree gradian or radian angle according to the current angle mode setting You can use G or r to override the angle mod...

Page 38: ...ngent of Expr1 or returns a list containing the inverse hyperbolic cotangents of each element of List1 Note You can insert this function from the keyboard by typing arccoth count Catalog count Value1orList1 Value2orList2 value Returns the accumulated count of all elements in the arguments that evaluate to numeric values Each argument can be an expression value list or matrix You can mix data types...

Page 39: ...he value 3 A Boolean expression containing the symbol as a placeholder for each element For example 5 counts only those elements in List that are less than 5 Within the Lists Spreadsheet application you can use a range of cells in place of List Empty void elements in the list are ignored For more information on empty elements see page 232 Note See also sumIf page 176 and frequency page 77 Counts t...

Page 40: ...t1 List2 list Returns the cross product of List1 and List2 as a list List1 and List2 must have equal dimension and the dimension must be either 2 or 3 crossP Vector1 Vector2 vector Returns a row or column vector depending on the arguments that is the cross product of Vector1 and Vector2 Both Vector1 and Vector2 must be row vectors or both must be column vectors Both vectors must have equal dimensi...

Page 41: ...from the keyboard by typing arccsc InDegree angle mode InGradianangle mode InRadianangle mode csch Catalog csch Expr1 expression csch List1 list Returns the hyperbolic cosecant of Expr1 or returns a list of the hyperbolic cosecants of all elements of List1 csch Catalog csch Expr1 expression csch List1 list Returns the inverse hyperbolic cosecant of Expr1 or returns a list containing the inverse hy...

Page 42: ...mplex during the solution even if the current domain is real In the complex domain fractional powers having odd denominators use the principal rather than the real branch Consequently solutions from solve to equations involving such fractional powers are not necessarily a subset of those from cSolve cSolve starts with exact symbolic methods cSolve also uses iterative approximate complex polynomial...

Page 43: ...or non real number For example x is valid and so is x 3 i If all of the equations are polynomials and if you do NOT specify any initial guesses cSolve uses the lexical Gröbner Buchberger elimination method to attempt to determine all complex solutions Note The following examples use an underscore press _ so thatthe variables will be treatedas complex Complex solutions can include both real and non...

Page 44: ...ble but all equations are linear in all solution variables cSolve uses Gaussian elimination to attempt to determine all solutions If a system is neither polynomial in all of its variables nor linear in its solution variables cSolve determines at most one solution using an approximate iterative method To do so the number of solution variables must equal the number of equations and all other variabl...

Page 45: ...ffect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn Regressionequation a x3 b x2 c x d stat a stat b stat c stat d Regressioncoefficients stat R2 Coefficientof determination stat Resid Residuals from the regression stat XReg Listof data points inthe modifiedX List actually usedinthe regressionbasedon restrictions of Freq Category List andInclu...

Page 46: ...current loop For While or Loop Cycle is not allowed outside the three looping structures For While or Loop Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator section of your product guidebook Functionlisting thatsums the integers from 1 to 100 skipping 50 Cylind Catalog Vector Cylind Note You can insert this operator from ...

Page 47: ...r variables in Expr that might have unreal values Otherwise you may receive unexpected results cZeros Expr1 Expr2 VarOrGuess1 VarOrGuess2 matrix Returns candidate positions where the expressions are zero simultaneously Each VarOrGuess specifies an unknown whose value you seek Optionally you can specify an initial guess for a variable Each VarOrGuess must have the form variable or variable real or ...

Page 48: ...include unknown variables that do not appear in the expressions These zeros show how families of zeros might contain arbitrary constants of the form ck where k is an integer suffix from 1 through 255 For polynomial systems computation time or memory exhaustion may depend strongly on the order in which you list unknowns If your initial choice exhausts memory or your patience try rearranging the var...

Page 49: ...d Catalog dbd date1 date2 value Returns the number of days between date1 and date2 using the actual day count method date1 and date2 can be numbers or lists of numbers within the range of the dates on the standard calendar If both date1 and date2 are lists they must be the same length date1 and date2 must be between the years 1950 through 2049 You can enter the dates in either of two formats The d...

Page 50: ...al expression Note You can insert this operator from the computer keyboard by typing Decimal Displays the argument in decimal form This operator can be used only at the end of the entry line Define Catalog Define Var Expression Define Function Param1 Param2 Expression Defines the variable Var or the user defined function Function Parameters such as Param1 provide placeholders for passing arguments...

Page 51: ... can be either a single statement or a series of statements on separate lines Block also can include expressions and instructions such as If Then Else and For Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator section of your product guidebook Note See also Define LibPriv page 51 and Define LibPub page 52 Define LibPriv Ca...

Page 52: ... Define LibPub Var Expression Define LibPub Function Param1 Param2 Expression Define LibPub Function Param1 Param2 Func Block EndFunc Define LibPub Program Param1 Param2 Prgm Block EndPrgm Operates the same as Define except defines a public library variable function or program Public functions and programs appear in the Catalog after the library has been saved and refreshed Note See also Define pa...

Page 53: ... The dot in this form of the DelVar command limits it to deleting a variable group the simple variable Var is not affected delVoid Catalog delVoid List1 list Returns a list that has the contents of List1 with all empty void elements removed For more information on empty elements see page 232 derivative See d page 217 deSolve Catalog deSolve 1stOr2ndOrderODE Var depVar a general solution Returns an...

Page 54: ...ns When comparing your results with textbook or manual solutions be aware that different methods introduce arbitrary constants at different points in the calculation which may produce different general solutions deSolve 1stOrderODE and initCond Var depVar a particular solution Returns a particular solution that satisfies 1stOrderODE and initCond This is usually easier than determining a general so...

Page 55: ...rderODE and bndCond1 and bndCond2 Var depVar a particular solution Returns a particular solution that satisfies 2ndOrderODE and has specified values at two different points det Catalog det squareMatrix Tolerance expression Returns the determinant of squareMatrix Optionally any matrix element is treated as zero if its absolute value is less than Tolerance This tolerance is used only if the matrix h...

Page 56: ...lumnMatrix matrix Returns a matrix with the values in the argument list or matrix in its main diagonal diag squareMatrix rowMatrix Returns a row matrix containing the elements from the main diagonal of squareMatrix squareMatrix must be square dim Catalog dim List integer Returns the dimension of List dim Matrix list Returns the dimensions of matrix as a two element list rows columns dim String int...

Page 57: ...tions refer to the Calculator section of your product guidebook DMS Catalog Expr DMS List DMS Matrix DMS Note You can insert this operator from the computer keyboard by typing DMS Interprets the argument as an angle and displays the equivalent DMS DDDDDD MM SS ss number See on page 224 for DMS degree minutes seconds format Note DMS will convert from radians to degrees when used in radian mode If t...

Page 58: ...gs of computer algebra simplification and solver algorithms Certain functions cannot be used as arguments for domain regardless of whether they appear explicitly or within user defined variables and functions In the following example the expression cannot be simplified because is a disallowed function dominantTerm Catalog dominantTerm Expr1 Var Point expression dominantTerm Expr1 Var Point Var Poi...

Page 59: ...ties such as sin 1 z at z 0 e 1 z at z 0 or ez at z or If the series or one of its derivatives has a jump discontinuity at Point the result is likely to contain sub expressions of the form sign or abs for a real expansion variable or 1 floor angle for a complex expansion variable which is one ending with _ If you intend to use the dominant term only for values on one side of Point then append to d...

Page 60: ...e also e exponent template page 6 Note Pressing u to display e is different from pressing the character E on the keyboard You can enter a complex number in reiθ polar form However use this form in Radian angle mode only it causes a Domain error in Degree or Gradian angle mode e List1 list Returns e raised to the power of each element in List1 e squareMatrix1 squareMatrix Returns the matrix exponen...

Page 61: ...omplex squareMatrix where each column in the result corresponds to an eigenvalue Note that an eigenvector is not unique it may be scaled by any constant factor The eigenvectors are normalized meaning that if V x1 x2 xn then x12 x22 xn2 1 squareMatrix is first balanced with similarity transformations until the row and column norms are as close to the same value as possible The squareMatrix is then ...

Page 62: ...orm and the eigenvalues are computed from the upper Hessenberg matrix To see the entire result press andthen use and to move the cursor Else See If page 86 ElseIf Catalog If BooleanExpr1 Then Block1 ElseIf BooleanExpr2 Then Block2 ElseIf BooleanExprN Then BlockN EndIf Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator sect...

Page 63: ...the interval Var0 VarMax Returns a matrix whose first row defines the Var output values and whose second row defines the value of the first solution component at the corresponding Var values and so on Expr is the right hand side that defines the ordinary differential equation ODE SystemOfExpr is the system of right hand sides that define the system of ODEs corresponds to order of dependent variabl...

Page 64: ... all i 0 1 2 such that Var0 i VarStep is in var0 VarMax there may not be a solution value at VarMax eulerStep is a positive integer defaults to 1 that defines the number of euler steps between output values The actual step size used by the euler method is VarStep eulerStep System of equations withy1 0 2 andy2 0 5 eval Hub Menu eval Expr string eval is valid only in the TI Innovator Hub Command arg...

Page 65: ...etAns iostr GetStrAns Note See also Get page 80 GetStr page 83 and Send page 155 exact Catalog exact Expr1 Tolerance expression exact List1 Tolerance list exact Matrix1 Tolerance matrix Uses Exact mode arithmetic to return when possible the rational number equivalent of the argument Tolerance specifies the tolerance for the conversion the default is 0 zero Exit Catalog Exit Exits the current For W...

Page 66: ...ponential e This is a display conversion operator It can be used only at the end of the entry line Note You can insert this operator from the computer keyboard by typing exp exp u key exp Expr1 expression Returns e raised to the Expr1 power Note See also e exponent template page 6 You can enter a complex number in reiθ polar form However use this form in Radian angle mode only it causes a Domain e...

Page 67: ...ded in the results of the solve cSolve fMin and fMax functions Note exp list is not necessary with the zeros and cZeros functions because they return a list of solution values directly You can insert this function from the keyboard by typing exp list expand Catalog expand Expr1 Var expression expand List1 Var list expand Matrix1 Var matrix expand Expr1 returns Expr1 expanded with respect to all it...

Page 68: ...pansion more complete Hint For rational expressions propFrac is a faster but less extreme alternative to expand Note See also comDenom for an expanded numerator over an expanded denominator expand Expr1 Var also distributes logarithms and fractional powers regardless of Var For increased distribution of logarithms and fractional powers inequality constraints might be necessary to guarantee that so...

Page 69: ...ing X and Y data Include is a list of one or more of the category codes Only those data items whose category code is included in this list are included in the calculation For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn Regressionequation a b x stat a stat b Regressioncoefficients stat r2 Coefficientof linear determi...

Page 70: ...n real subexpressions This alternative is appropriate if you want factorization with respect to more than one variable factor Expr1 Var returns Expr1 factored with respect to variable Var Expr1 is factored as much as possible toward real factors that are linear in Var even if it introduces irrational constants or subexpressions that are irrational in other variables The factors and their terms are...

Page 71: ...mber factored into primes For composite numbers the computing time grows exponentially with the number of digits in the second largest factor For example factoring a 30 digit integer could take more than a day and factoring a 100 digit number could take more than a century To stop a calculation manually Handheld Hold down the c key and press repeatedly Windows Hold down the F12 key and press Enter...

Page 72: ...r matrixVar matrix Replaces each element in variable matrixVar with Expr matrixVar must already exist Fill Expr listVar list Replaces each element in variable listVar with Expr listVar must already exist FiveNumSummary Catalog FiveNumSummary X Freq Category Include Provides an abbreviated version of the 1 variable statistics on list X A summary of results is stored in the stat results variable See...

Page 73: ... empty elements see page 232 Output variable Description stat MinX Minimum of x values stat Q1X 1stQuartile of x stat MedianX Medianof x stat Q3X 3rdQuartile of x stat MaxX Maximum of x values floor Catalog floor Expr1 integer Returns the greatest integer that is the argument This function is identical to int The argument can be a real or a complex number floor List1 list floor Matrix1 matrix Retu...

Page 74: ... that contains exactly one local maximum Note See also fMin and max fMin Catalog fMin Expr Var Boolean expression fMin Expr Var lowBound fMin Expr Var lowBound upBound fMin Expr Var lowBound Var upBound Returns a Boolean expression specifying candidate values of Var that minimize Expr or locate its greatest lower bound You can use the constraint operator to restrict the solution interval and or sp...

Page 75: ...r section of your product guidebook format Catalog format Expr formatString string Returns Expr as a character string based on the format template Expr must simplify to a number formatString is a string and must be in the form F n S n E n G n c where indicate optional portions F n Fixed format n is the number of digits to display after the decimal point S n Scientific format n is the number of dig...

Page 76: ... fPart Matrix1 matrix Returns the fractional part of the argument For a list or matrix returns the fractional parts of the elements The argument can be a real or a complex number FPdf Catalog FPdf XVal dfNumer dfDenom number if XVal is a number list if XVal is a list Computes the F distribution probability at XVal for the specified dfNumer degrees of freedom and dfDenom freqTable list Catalog freq...

Page 77: ... 2 b n the specified ranges are b 1 b 1 b 2 b n 1 b n b n The resulting list is one element longer than binsList Each element of the result corresponds to the number of elements from List1 that are in the range of that bin Expressed in terms of the countIf function the result is countIf list b 1 countIf list b 1 b 2 countIf list b n 1 b n countIf list b n Elements of List1 that cannot be placed in...

Page 78: ... on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat F CalculatedF statistic for the data sequence stat PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat dfNumer numerator degrees of freedom n1 1 stat dfDenom denominator degrees of freedom n2 1 stat sx1 stat sx2 Sample standarddeviations of the data sequences i...

Page 79: ...aphing g x G gcd Catalog gcd Number1 Number2 expression Returns the greatest common divisor of the two arguments The gcd of two fractions is the gcd of their numerators divided by the lcm of their denominators In Auto or Approximate mode the gcd of fractional floating point numbers is 1 0 gcd List1 List2 list Returns the greatest common divisors of the corresponding elements in List1 and List2 gcd...

Page 80: ... the first success occurs for the discrete geometric distribution with the specified probability of success p Get Hub Menu Get promptString var statusVar Get promptString func arg1 argn statusVar Programming command Retrieves a value from a connected TI Innovator Hub and assigns the value to variable var The value must be requested In advance through a Send READ command or By embedding a READ requ...

Page 81: ...es as if the program executed the command Define func arg1 argn received string The program can then use the defined function func Note You can use the Get command within a user defined program but not within a function Note See also GetStr page 83 and Send page 155 getDenom Catalog getDenom Expr1 expression Transforms the argument into an expression having a reduced common denominator and then re...

Page 82: ... Var value 0 Var is unlocked or does not exist value 1 Var is locked and cannot be modified or deleted See Lock page 104 and unLock page 193 getMode Catalog getMode ModeNameInteger value getMode 0 list getMode ModeNameInteger returns a value representing the current setting of the ModeNameInteger mode getMode 0 returns a list containing number pairs Each pair consists of a mode integer and a setti...

Page 83: ...Fix5 20 Fix6 21 Fix7 22 Fix8 23 Fix9 24 Fix10 25 Fix11 26 Fix12 Angle 2 1 Radian 2 Degree 3 Gradian Exponential Format 3 1 Normal 2 Scientific 3 Engineering Real or Complex 4 1 Real 2 Rectangular 3 Polar Auto or Approx 5 1 Auto 2 Approximate 3 Exact Vector Format 6 1 Rectangular 2 Cylindrical 3 Spherical Base 7 1 Decimal 2 Hex 3 Binary Unit system 8 1 SI 2 Eng US getNum Catalog getNum Expr1 expres...

Page 84: ... has not been defined returns the string NONE getVarInfo Catalog getVarInfo matrix or string getVarInfo LibNameString matrix or string getVarInfo returns a matrix of information variable name type library accessibility and locked unlocked state for all variables and library objects defined in the current problem If no variables are defined getVarInfo returns the string NONE getVarInfo LibNameStrin...

Page 85: ...e does not support a generalized matrix structure where an element of a matrix can be either a matrix or a list Goto Catalog Goto labelName Transfers control to the label labelName labelName must be defined in the same function using a Lbl instruction Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator section of your produ...

Page 86: ...cution If BooleanExpr evaluates to false continues execution without executing the statement or block of statements Block can be either a single statement or a sequence of statements separated with the character Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator section of your product guidebook If BooleanExpr Then Block1 ...

Page 87: ... Value_If_true If an element of BooleanExpr evaluates to false returns the corresponding element from Value_If_false If you omit Value_If_false returns undef If an element of BooleanExpr is neither true nor false returns the corresponding element Value_If_unknown If you omit Value_If_unknown returns undef If the second third or fourth argument of the ifFn function is a single expression the Boolea...

Page 88: ...known imag Catalog imag Expr1 expression Returns the imaginary part of the argument Note All undefined variables are treated as real variables See also real page 144 imag List1 list Returns a list of the imaginary parts of the elements imag Matrix1 matrix Returns a matrix of the imaginary parts of the elements impDif Catalog impDif Equation Var dependVar Ord expression where the order Ord defaults...

Page 89: ...returns zero int Catalog int Expr integer int List1 list int Matrix1 matrix Returns the greatest integer that is less than or equal to the argument This function is identical to floor The argument can be a real or a complex number For a list or matrix returns the greatest integer of each of the elements intDiv Catalog intDiv Number1 Number2 integer intDiv List1 List2 list intDiv Matrix1 Matrix2 ma...

Page 90: ...ntains xValue If it finds such an interval it returns an interpolated value for f xValue otherwise it returns undef xList yList and yPrimeList must be of equal dimension 2 and contain expressions that simplify to numbers xValue can be an undefined variable a number or a list of numbers Differential equation y 3 y 6 t 5 andy 0 5 To see the entire result press andthen use and to move the cursor Use ...

Page 91: ...nverse cumulative student t probability function specified by degree of freedom df for a given Area under the curve iPart Catalog iPart Number integer iPart List1 list iPart Matrix1 matrix Returns the integer part of the argument For lists and matrices returns the integer part of each element The argument can be a real or a complex number irr Catalog irr CF0 CFList CFFreq value Financial function ...

Page 92: ...y itself and 1 If Number exceeds about 306 digits and has no factors 1021 isPrime Number displays an error message If you merely want to determine if Number is prime use isPrime instead of factor It is much faster particularly if Number is not prime and has a second largest factor that exceeds about five digits Note for entering the example For instructions on entering multi line program and funct...

Page 93: ...nction definitions refer to the Calculator section of your product guidebook lcm Catalog lcm Number1 Number2 expression lcm List1 List2 list lcm Matrix1 Matrix2 matrix Returns the least common multiple of the two arguments The lcm of two fractions is the lcm of their numerators divided by the gcd of their denominators The lcm of fractional floating point numbers is their product For two lists or m...

Page 94: ...ble group in the current problem that contains references to all the objects in the specified library document libNameString Also adds the group members to the Variables menu You can then refer to each object using its ShortcutNameString Set LibPrivFlag 0 to exclude private library objects default Set LibPrivFlag 1 to include private library objects To copy a variable group see CopyVar on page 33 ...

Page 95: ...imit does not exist undef means that the result is either an unknown number with finite or infinite magnitude or it is the entire set of such numbers limit uses methods such as L Hopital s rule so there are unique limits that it cannot determine If Expr1 contains undefined variables other than Var you might have to constrain them to obtain a more concise result Limits can be very sensitive to roun...

Page 96: ...t of one or more of the category codes Only those data items whose category code is included in this list are included in the calculation For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn RegressionEquation a b x stat a stat b Regressioncoefficients stat r2 Coefficientof determination stat r Correlationcoefficient sta...

Page 97: ... or more of the category codes Only those data items whose category code is included in this list are included in the calculation For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn RegressionEquation y m x b stat m stat b Regressioncoefficients stat r2 Coefficientof determination stat r Correlationcoefficient stat Resi...

Page 98: ...is an optional list of frequency values Each element in F specifies the frequency of occurrence for each corresponding X and Y data point The default value is 1 All elements must be integers 0 For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn RegressionEquation a b x stat a stat b Regressioncoefficients stat df Degree...

Page 99: ...tes a linear regression on the X and Y lists and a t test on the value of slope β and the correlation coefficient ρ for the equation y α βx It tests the null hypothesis H0 β 0 equivalently ρ 0 against one of three alternative hypotheses All the lists must have equal dimension X and Y are lists of independent and dependent variables Freq is an optional list of frequency values Each element in Freq ...

Page 100: ...t PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat df Degrees of freedom stat a stat b Regressioncoefficients stat s Standarderror aboutthe line stat SESlope Standarderror of slope stat r2 Coefficientof determination stat r Correlationcoefficient stat Resid Residuals from the regression linSolve Catalog linSolve SystemOfLinearEqns Var1 Var2 list linSolve LinearEqn1...

Page 101: ...ontaining the differences between consecutive elements in List1 Each element of List1 is subtracted from the next element of List1 The resulting list is always one element shorter than the original List1 list mat Catalog list mat List elementsPerRow matrix Returns a matrix filled row by row with the elements from List elementsPerRow if included specifies the number of elements per row Default is t...

Page 102: ...trix Returns the matrix natural logarithm of squareMatrix1 This is not the same as calculating the natural logarithm of each element For information about the calculation method refer to cos on squareMatrix1 must be diagonalizable The result always contains floating point numbers InRadianangle mode andRectangular complex format To see the entire result press andthen use and to move the cursor LnRe...

Page 103: ...Empty Void Elements page 232 Output variable Description stat RegEqn Regressionequation a b ln x stat a stat b Regressioncoefficients stat r2 Coefficientof linear determinationfor transformeddata stat r Correlationcoefficientfor transformeddata ln x y stat Resid Residuals associatedwiththe logarithmic model stat ResidTrans Residuals associatedwithlinear fitof transformeddata stat XReg Listof data ...

Page 104: ...n a multi line function since modifications on global variables are not allowed in a function Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator section of your product guidebook Lock Catalog LockVar1 Var2 Var3 LockVar Locks the specified variables or variable group Locked variables cannot be modified or deleted You cannot...

Page 105: ...rix1 This is not the same as calculating the base Expr logarithm of each element For information about the calculation method refer to cos squareMatrix1 must be diagonalizable The result always contains floating point numbers If the base argument is omitted 10 is used as base InRadianangle mode andRectangular complex format To see the entire result press andthen use and to move the cursor logbase ...

Page 106: ... the corresponding X and Y data Include is a list of one or more of the category codes Only those data items whose category code is included in this list are included in the calculation For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn Regressionequation c 1 a e bx stat a stat b stat c Regressioncoefficients stat Resi...

Page 107: ...he corresponding X and Y data Include is a list of one or more of the category codes Only those data items whose category code is included in this list are included in the calculation For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn Regressionequation c 1 a e bx d stat a stat b stat c stat d Regressioncoefficients st...

Page 108: ...tion of a real or complex matrix The lower triangular matrix is stored in lMatrix the upper triangular matrix in uMatrix and the permutation matrix which describes the row swaps done during the calculation in pMatrix lMatrix uMatrix pMatrix matrix Optionally any matrix element is treated as zero if its absolute value is less than Tol This tolerance is used only if the matrix has floating point ent...

Page 109: ...ter keyboard by typing mat list max Catalog max Expr1 Expr2 expression max List1 List2 list max Matrix1 Matrix2 matrix Returns the maximum of the two arguments If the arguments are two lists or matrices returns a list or matrix containing the maximum value of each pair of corresponding elements max List expression Returns the maximum element in list max Matrix1 matrix Returns a row vector containi...

Page 110: ...x1 Each freqMatrix element counts the number of consecutive occurrences of the corresponding element in Matrix1 Empty void elements are ignored For more information on empty elements see page 232 InRectangular vector format median Catalog median List freqList expression Returns the median of the elements in List Each freqList element counts the number of consecutive occurrences of the correspondin...

Page 111: ...nt and dependent variables Freq is an optional list of frequency values Each element in Freq specifies the frequency of occurrence for each corresponding X and Y data point The default value is 1 All elements must be integers 0 Category is a list of category codes for the corresponding X and Y data Include is a list of one or more of the category codes Only those data items whose category code is ...

Page 112: ...unt characters from character string sourceString beginning with character number Start If Count is omitted or is greater than the dimension of sourceString returns all characters from sourceString beginning with character number Start Count must be 0 If Count 0 returns an empty string mid sourceList Start Count list Returns Count elements from sourceList beginning with element number Start If Cou...

Page 113: ...te reinvestRate CF0 CFList CFFreq Financial function that returns the modified internal rate of return of an investment financeRate is the interest rate that you pay on the cash flow amounts reinvestRate is the interest rate at which the cash flows are reinvested CF0 is the initial cash flow at time 0 it must be a real number CFList is a list of cash flow amounts after the initial cash flow CF0 CF...

Page 114: ...gument The result is either zero or has the same sign as the second argument If the arguments are two lists or two matrices returns a list or matrix containing the modulo of each pair of corresponding elements Note See also remain page 146 mRow Catalog mRow Expr Matrix1 Index matrix Returns a copy of Matrix1 with each element in row Index of Matrix1 multiplied by Expr mRowAdd Catalog mRowAdd Expr ...

Page 115: ... stat yList yList b0 b1 x1 stat Resid Residuals from the regression MultRegIntervals Catalog MultRegIntervals Y X1 X2 X3 X10 XValList CLevel Computes a predicted y value a level C prediction interval for a single observation and a level C confidence interval for the mean response A summary of results is stored in the stat results variable See page 172 All the lists must have equal dimension For in...

Page 116: ...ssion test computes a multiple linear regression on the given data and provides the global F test statistic and t test statistics for the coefficients A summary of results is stored in the stat results variable See page 172 For information on the effect of empty elements in a list see Empty Void Elements page 232 Outputs Output variable Description stat RegEqn RegressionEquation b0 b1 x1 b2 x2 sta...

Page 117: ...the regression stat sResid Standardizedresiduals obtainedby dividing a residual by its standarddeviation stat CookDist Cook s distance measure of the influence of anobservationbasedonthe residual andleverage stat Leverage Measure of howfar the values of the independentvariable are from their mean values N nand keys BooleanExpr1 nand BooleanExpr2 returns Boolean expression BooleanList1 nand Boolean...

Page 118: ... the 0b or 0h prefix respectively Without a prefix integers are treated as decimal base 10 nCr Catalog nCr Expr1 Expr2 expression For integer Expr1 and Expr2 with Expr1 Expr2 0 nCr is the number of combinations of Expr1 things taken Expr2 at a time This is also known as a binomial coefficient Both arguments can be integers or symbolic expressions nCr Expr 0 1 nCr Expr negInteger 0 nCr Expr posInte...

Page 119: ...Value is specified it overrides any prior variable assignment or any current substitution for the variable Order of the derivative must be 1 or 2 newList Catalog newList numElements list Returns a list with a dimension of numElements Each element is zero newMat Catalog newMat numRows numColumns matrix Returns a matrix of zeros with the dimension numRows by numColumns nfMax Catalog nfMax Expr Var v...

Page 120: ...ound Var upBound value Returns a candidate numerical value of variable Var where the local minimum of Expr occurs If you supply lowBound and upBound the function looks in the closed interval lowBound upBound for the local minimum Note See also fMin and d nInt Catalog nInt Expr1 Var Lower Upper expression If the integrand Expr1 contains no variable other than Var and if Lower and Upper are constant...

Page 121: ...g nom effectiveRate CpY value Financial function that converts the annual effective interest rate effectiveRate to a nominal rate given CpY as the number of compounding periods per year effectiveRate must be a real number and CpY must be a real number 0 Note See also eff page 61 nor keys BooleanExpr1 nor BooleanExpr2 returns Boolean expression BooleanList1 nor BooleanList2 returns Boolean list Boo...

Page 122: ...e You can enter the integers in any number base For a binary or hexadecimal entry you must use the 0b or 0h prefix respectively Without a prefix integers are treated as decimal base 10 norm Catalog norm Matrix expression norm Vector expression Returns the Frobenius norm normalLine Catalog normalLine Expr1 Var Point expression normalLine Expr1 Var Point expression Returns the normal line to the cur...

Page 123: ...r Internally Integer1 is converted to a signed 64 bit binary number The value of each bit is flipped 0 becomes 1 and vice versa for the one s complement Results are displayed according to the Base mode You can enter the integer in any number base For a binary or hexadecimal entry you must use the 0b or 0h prefix respectively Without a prefix the integer is treated as decimal base 10 If you enter a...

Page 124: ...ent pairs in the two lists The arguments must be the same size list nPr Matrix1 Matrix2 matrix Returns a matrix of permutations based on the corresponding element pairs in the two matrices The arguments must be the same size matrix npv Catalog npv InterestRate CFO CFList CFFreq Financial function that calculates net present value the sum of the present values for the cash inflows and outflows A po...

Page 125: ...approximate real numeric solution to Equation for its one variable Specify the variable as variable or variable real number For example x is valid and so is x 3 Note If there are multiple solutions youcan use a guess to helpfinda particular solution nSolve is often much faster than solve or zeros particularly if the operator is used to constrain the search to a small interval containing exactly on...

Page 126: ... The default value is 1 All elements must be integers 0 Category is a list of numeric category codes for the corresponding X values Include is a list of one or more of the category codes Only those data items whose category code is included in this list are included in the calculation An empty void element in any of the lists X Freq or Category results in a void for the corresponding element of al...

Page 127: ...pression BooleanList1 or BooleanList2 returns Boolean list BooleanMatrix1 or BooleanMatrix2 returns Boolean matrix Returns true or false or a simplified form of the original entry Returns true if either or both expressions simplify to true Returns false only if both expressions evaluate to false Note See xor Note for entering the example For instructions on entering multi line program and function...

Page 128: ...fix integers are treated as decimal base 10 If you enter a decimal integer that is too large for a signed 64 bit binary form a symmetric modulo operation is used to bring the value into the appropriate range For more information see Base2 page 21 Note See xor Note A binary entry canhave upto 64 digits notcounting the 0bprefix A hexadecimal entry canhave upto 16 digits ord Catalog ord String intege...

Page 129: ...angle mode If the argument is an expression you can use G or r to override the angle mode setting temporarily Note You can insert this function from the computer keyboard by typing P Ry InRadianangle mode PassErr Catalog PassErr Passes an error to the next level If system variable errCode is zero PassErr does not do anything The Else clause of the Try Else EndTry block should use ClrErr or PassErr...

Page 130: ...ewise definitions by using a template Note See also Piecewise template page 7 poissCdf Catalog poissCdf λ lowBound upBound number if lowBound and upBound are numbers list if lowBound and upBound are lists poissCdf λ upBound for P 0 X upBound number if upBound is a number list if upBound is a list Computes a cumulative probability for the discrete Poisson distribution with specified mean λ For P X ...

Page 131: ...lar form Degree angle mode returns r θ Radian angle mode returns reiθ complexValue can have any complex form However an reiθ entry causes an error in Degree angle mode Note You must use the parentheses for an r θ polar entry InRadianangle mode InGradianangle mode InDegree angle mode polyCoeffs Catalog polyCoeffs Poly Var list Returns a list of the coefficients of polynomial Poly with respect to va...

Page 132: ...al expression in Var We recommend that you do not omit Var unless Poly is an expression in a single variable Constantpolynomials The degree canbe extractedeventhough the coefficients cannot This is because the degree canbe extractedwithoutexpanding the polynomial polyEval Catalog polyEval List1 Expr1 expression polyEval List1 List2 expression Interprets the first argument as the coefficient of a d...

Page 133: ...the specified variable Var Poly1 and Poly2 must be polynomial expressions in Var We recommend that you do not omit Var unless Poly1 and Poly2 are expressions in the same single variable polyRemainder Catalog polyRemainder Poly1 Poly2 Var expression Returns the remainder of polynomial Poly1 divided by polynomial Poly2 with respect to the specified variable Var Poly1 and Poly2 must be polynomial exp...

Page 134: ...s for the coefficients in ListOfCoeffs Note See also cPolyRoots page 40 PowerReg Catalog PowerReg X Y Freq Category Include Computes the power regressiony a x b on lists X and Y with frequency Freq A summary of results is stored in the stat results variable See page 172 All the lists must have equal dimension except for Include X and Y are lists of independent and dependent variables Freq is an op...

Page 135: ...ththe power model stat ResidTrans Residuals associatedwithlinear fitof transformeddata stat XReg Listof data points inthe modifiedX List actually usedinthe regressionbasedon restrictions of Freq Category List andInclude Categories stat YReg Listof data points inthe modifiedY List actually usedinthe regressionbasedon restrictions of Freq Category List andInclude Categories stat FreqReg Listof frequ...

Page 136: ...in List Start and End are optional They specify a range of elements product Matrix1 Start End matrix Returns a row vector containing the products of the elements in the columns of Matrix1 Start and end are optional They specify a range of rows Empty void elements are ignored For more information on empty elements see page 232 propFrac Catalog propFrac Expr1 Var expression propFrac rational_number ...

Page 137: ...tive to expand You can use the propFrac function to represent mixed fractions and demonstrate addition and subtraction of mixed fractions Q QR Catalog QR Matrix qMatrix rMatrix Tol Calculates the Householder QR factorization of a real or complex matrix The resulting Q and R matrices are stored to the specified Matrix The Q matrix is unitary The R matrix is upper triangular Optionally any matrix el...

Page 138: ...y matrix QuadReg Catalog QuadReg X Y Freq Category Include Computes the quadratic polynomial regression y a x2 b x c on lists X and Y with frequency Freq A summary of results is stored in the stat results variable See page 172 All the lists must have equal dimension except for Include X and Y are lists of independent and dependent variables Freq is an optional list of frequency values Each element...

Page 139: ...ories stat YReg Listof data points inthe modifiedY List actually usedinthe regressionbasedon restrictions of Freq Category List andInclude Categories stat FreqReg Listof frequencies corresponding to stat XReg andstat YReg QuartReg Catalog QuartReg X Y Freq Category Include Computes the quartic polynomial regression y a x4 b x3 c x2 d x e on lists X and Y with frequency Freq A summary of results is...

Page 140: ...ession stat XReg Listof data points inthe modifiedX List actually usedinthe regressionbasedon restrictions of Freq Category List andInclude Categories stat YReg Listof data points inthe modifiedY List actually usedinthe regressionbasedon restrictions of Freq Category List andInclude Categories stat FreqReg Listof frequencies corresponding to stat XReg andstat YReg R R Pθ Catalog R Pθ xExpr yExpr e...

Page 141: ...ng R Pr InRadianangle mode Rad Catalog Expr1 Rad expression Converts the argument to radian angle measure Note You can insert this operator from the computer keyboard by typing Rad InDegree angle mode InGradianangle mode rand Catalog rand expression rand Trials list rand returns a random value between 0 and 1 rand Trials returns a list containing Trials random values between 0 and 1 Setthe random ...

Page 142: ...Bound upBound expression randInt lowBound upBound Trials list randInt lowBound upBound returns a random integer within the range specified by lowBound and upBound integer bounds randInt lowBound upBound Trials returns a list containing Trials random integers within the specified range randMat Catalog randMat numRows numColumns matrix Returns a matrix of integers between 9 and 9 of the specified di...

Page 143: ...the specified Order The coefficients are random integers in the range 9 through 9 The leading coefficient will not be zero Order must be 0 99 randSamp Catalog randSamp List Trials noRepl list Returns a list containing a random sample of Trials trials from List with an option for sample replacement noRepl 0 or no sample replacement noRepl 1 The default is with sample replacement RandSeed Catalog Ra...

Page 144: ...ector in rectangular form x y z The vector must be of dimension 2 or 3 and can be a row or a column Note Rect is a display format instruction not a conversion function You can use it only at the end of an entry line and it does not update ans Note See also Polar page 131 complexValue Rect Displays complexValue in rectangular form a bi The complexValue can have any complex form However an reiθ entr...

Page 145: ... are done using floating point arithmetic If Tol is omitted or not used the default tolerance is calculated as 5E 14 max dim Matrix1 rowNorm Matrix1 Avoid undefined elements in Matrix1 They can lead to unexpected results For example if a is undefined in the following expression a warning message appears and the result is shown as The warning appears because the generalized element 1 a would not be...

Page 146: ...splays a dialog box containing the message promptString and an input box for the user s response When the user types a response and clicks OK the contents of the input box are assigned to variable var If the user clicks Cancel the program proceeds without accepting any input The program uses the previous value of var if var was already defined The optional DispFlag argument can be any expression I...

Page 147: ... user s response that completes the function definition Note You can use the Request command within a user defined program but not within a function To stop a program that contains a Request command inside an infinite loop Handheld Hold down the c key and press repeatedly Windows Hold down the F12 key and press Enter repeatedly Macintosh Hold down the F5 key and press Enter repeatedly iPad The app...

Page 148: ... the F5 key and press Enter repeatedly iPad The app displays a prompt You can continue waiting or cancel Note See also Request page 146 Define requestStr_demo Prgm RequestStr Your name name 0 Disp Response has dim name characters EndPrgm Runthe program andtype a response requestStr_demo Resultafter selecting OK Note thatthe DispFlag argumentof 0 omits the prompt andresponse from the history reques...

Page 149: ...ix Uses the Runge Kutta method to solve the system with depVar Var0 depVar0 on the interval Var0 VarMax Returns a matrix whose first row defines the Var output values as defined by VarStep The second row defines the value of the first solution component at the corresponding Var values and so on Expr is the right hand side that defines the ordinary differential equation ODE SystemOfExpr is a system...

Page 150: ... Var0 and solutions are returned at Var0 i VarStep for all i 0 1 2 such that Var0 i VarStep is in var0 VarMax may not get a solution value at VarMax if VarStep evaluates to zero solutions are returned at the Runge Kutta Var values diftol is the error tolerance defaults to 0 001 with y1 0 2 and y2 0 5 root Catalog root Expr root root Expr1 Expr2 root root Expr returns the square root of Expr root E...

Page 151: ...ightmost bit rotates to leftmost produces 0b10000000000000111101011000011010 The result is displayed according to the Base mode Important To enter a binary or hexadecimal number always use the 0bor 0hprefix zero notthe letter O rotate List1 ofRotations list Returns a copy of List1 rotated right or left by of Rotations elements Does not alter List1 If ofRotations is positive the rotation is to the ...

Page 152: ...ed to the specified number of digits round Matrix1 digits matrix Returns a matrix of the elements rounded to the specified number of digits rowAdd Catalog rowAdd Matrix1 rIndex1 rIndex2 matrix Returns a copy of Matrix1 with row rIndex2 replaced by the sum of rows rIndex1 and rIndex2 rowDim Catalog rowDim Matrix expression Returns the number of rows in Matrix Note See also colDim page 30 rowNorm Ca...

Page 153: ...ntries and does not contain any symbolic variables that have not been assigned a value Otherwise Tol is ignored If you use or set the Auto or Approximate mode to Approximate computations are done using floating point arithmetic If Tol is omitted or not used the default tolerance is calculated as 5E 14 max dim Matrix1 rowNorm Matrix1 Note See also ref page 145 S sec µ key sec Expr1 expression sec L...

Page 154: ... according to the current angle mode setting Note You can insert this function from the keyboard by typing arcsec InDegree angle mode InGradianangle mode InRadianangle mode sech Catalog sech Expr1 expression sech List1 list Returns the hyperbolic secant of Expr1 or returns a list containing the hyperbolic secants of the List1 elements sech Catalog sech Expr1 expression sech List1 list Returns the ...

Page 155: ... built inRGB LED for 0 5 seconds Example Requestthe currentvalue of the hub s built inlight level sensor A Get commandretrieves the value andassigns it to variable lightval Example Senda calculatedfrequency to the hub s built inspeaker Use special variable iostr SendAnsto showthe hubcommand withthe expressionevaluated seq Catalog seq Expr Var Low High Step list Increments Var from Low through High...

Page 156: ...r as follows Increments independent variable Var from Var0 through VarMax by VarStep evaluates ListOfDepVars Var for corresponding values of Var using ListOrSystemOfExpr formula and MatrixOfInitTerms and returns the results as a matrix The original contents of Var are unchanged after seqGen is completed The default value for VarStep 1 Generate the first5 terms of the sequence u n u n 1 2 2 withu 1...

Page 157: ...t If nMax is missing nMax is set to 2500 If nMax 0 nMax is set to 2500 Note seqn calls seqGen with n0 1 and nstep 1 series Catalog series Expr1 Var Order Point expression series Expr1 Var Order Point Var Point expression series Expr1 Var Order Point Var Point expression Returns a generalized truncated power series representation of Expr1 expanded about Point through degree Order Order can be any r...

Page 158: ...e one of Var Point Var Point Var Point or Var Point to obtain a simpler result series can provide symbolic approximations to indefinite integrals and definite integrals for which symbolic solutions otherwise can t be obtained series distributes over 1st argument lists and matrices series is a generalized version of taylor As illustrated by the last example to the right the display routines downstr...

Page 159: ...hose integer pairs represent the original modes and settings If you have saved all mode settings with getMode 0 var you can use setMode var to restore those settings until the function or program exits See getMode page 82 Note The current mode settings are passed to called subroutines If any subroutine changes a mode setting the mode change will be lost when control returns to the calling routine ...

Page 160: ...orm If the magnitude of Integer1 is too large for this form a symmetric modulo operation brings it within the range For more information see Base2 page 21 If ofShifts is positive the shift is to the left If ofShifts is negative the shift is to the right The default is 1 shift right one bit In a right shift the rightmost bit is dropped and 0 or 1 is inserted to match the leftmost bit In a left shif...

Page 161: ... ofShifts string Returns a copy of String1 shifted right or left by ofShifts characters Does not alter String1 If ofShifts is positive the shift is to the left If ofShifts is negative the shift is to the right The default is 1 shift right one character Characters introduced at the beginning or end of string by the shift are set to a space sign Catalog sign Expr1 expression sign List1 list sign Mat...

Page 162: ... ignored If you set the Auto or Approximate mode to Approximate computations are done using floating point arithmetic If Tol is omitted or not used the default tolerance is calculated as 5E 14 max dim coeffMatrix rowNorm coeffMatrix Solve for x andy x 2y 1 3x 4y 1 The solutionis x 3 andy 2 Solve ax by 1 cx dy 2 simult coeffMatrix constMatrix Tol matrix Solves multiple systems of linear equations w...

Page 163: ...le modes Before using it make sure that the Angle mode is set to Radians and that Expr does not contain explicit references to degree or gradian angles sin µ key sin Expr1 expression sin List1 list sin Expr1 returns the sine of the argument as an expression sin List1 returns a list of the sines of all elements in List1 Note The argument is interpreted as a degree gradian or radian angle according ...

Page 164: ...eturned as a degree gradian or radian angle according to the current angle mode setting Note You can insert this function from the keyboard by typing arcsin InDegree angle mode InGradianangle mode InRadianangle mode sin squareMatrix1 squareMatrix Returns the matrix inverse sine of squareMatrix1 This is not the same as calculating the inverse sine of each element For information about the calculati...

Page 165: ...ent as an expression sinh List1 returns a list of the inverse hyperbolic sines of each element of List1 Note You can insert this function from the keyboard by typing arcsinh sinh squareMatrix1 squareMatrix Returns the matrix inverse hyperbolic sine of squareMatrix1 This is not the same as calculating the inverse hyperbolic sine of each element For information about the calculation method refer to ...

Page 166: ...clude is a list of one or more of the category codes Only those data items whose category code is included in this list are included in the calculation The output of SinReg is always in radians regardless of the angle mode setting For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat RegEqn RegressionEquation a sin bx c d stat a...

Page 167: ...upplemented by iterative searches with approximate arithmetic when exact solutions are impractical Due to default cancellation of the greatest common divisor from the numerator and denominator of ratios solutions might be solutions only in the limit from one or both sides For inequalities of types or explicit solutions are unlikely unless the inequality is linear and contains only Var For the Exac...

Page 168: ...InRadianangle mode In Real mode fractional powers having odd denominators denote only the real branch Otherwise multiple branched expressions such as fractional powers logarithms and inverse trigonometric functions denote only the principal branch Consequently solve produces only solutions corresponding to that one real or principal branch Note See also cSolve cZeros nSolve and zeros solve Eqn1 an...

Page 169: ...adius r at the origin and another circle of radius r centered where the first circle crosses the positive x axis Use solve to find the intersections As illustrated by r in the example to the right simultaneous polynomial equations can have extra variables that have no values but represent given numeric values that could be substituted later You can also or instead include solution variables that d...

Page 170: ... solution using an approximate iterative method To do so the number of solution variables must equal the number of equations and all other variables in the equations must simplify to numbers To see the entire result press andthen use and to move the cursor Each solution variable starts at its guessed value if there is one otherwise it starts at 0 0 Use guesses to seek additional solutions one by o...

Page 171: ...ormation on empty elements see page 232 Sphere Catalog Vector Sphere Note You can insert this operator from the computer keyboard by typing Sphere Displays the row or column vector in spherical form ρ θ φ Vector must be of dimension 3 and can be either a row or a column vector Note Sphere is a display format instruction not a conversion function You can use it only at the end of an entry line Note...

Page 172: ...splays results from a statistics calculation The results are displayed as a set of name value pairs The specific names shown are dependent on the most recently evaluated statistics function or command You can copy a name or value and paste it into other locations Note Avoid defining variables that use the same names as those used for statistical analysis In some cases an error condition could occu...

Page 173: ...at r stat RegEqn stat Resid stat ResidTrans stat σx stat σy stat σx1 stat σx2 stat Σx stat Σx stat Σxy stat Σy stat Σy stat s stat SE stat SEList stat SEPred stat sResid stat SEslope stat sp stat SS stat SSBlock stat SSCol stat SSX stat SSY stat SSError stat SSInteract stat SSReg stat SSRow stat tList stat UpperPred stat UpperVal stat v stat v1 stat v2 stat vDiff stat vList stat XReg stat XVal sta...

Page 174: ... vector of the population standard deviations of the columns in Matrix1 Each freqMatrix element counts the number of consecutive occurrences of the corresponding element in Matrix1 Note Matrix1must have at least two rows Empty void elements are ignored For more information on empty elements see page 232 stDevSamp Catalog stDevSamp List freqList expression Returns the sample standard deviation of t...

Page 175: ... Empty void elements are ignored For more information on empty elements see page 232 Stop Catalog Stop Programming command Terminates the program Stop is not allowed in functions Note for entering the example For instructions on entering multi line program and function definitions refer to the Calculator section of your product guidebook Store See store page 229 string Catalog string Expr string S...

Page 176: ...ty void elements in List are ignored For more information on empty elements see page 232 sum Matrix1 Start End matrix Returns a row vector containing the sums of all elements in the columns in Matrix1 Start and End are optional They specify a range of rows Any void argument produces a void result Empty void elements in Matrix1 are ignored For more information on empty elements see page 232 sumIf C...

Page 177: ...nt meets the Criteria the element is added to the accumulating sum If you include sumList the corresponding element from sumList is added to the sum instead Within the Lists Spreadsheet application you can use a range of cells in place of List and sumList Empty void elements are ignored For more information on empty elements see page 232 Note See also countIf page 39 sumSeq See Σ page 220 system C...

Page 178: ...t1 returns a list of the tangents of all elements in List1 Note The argument is interpreted as a degree gradian or radian angle according to the current angle mode You can use g or r to override the angle mode setting temporarily InDegree angle mode InGradianangle mode InRadianangle mode tan squareMatrix1 squareMatrix Returns the matrix tangent of squareMatrix1 This is not the same as calculating ...

Page 179: ...rom the keyboard by typing arctan InDegree angle mode InGradianangle mode InRadianangle mode tan squareMatrix1 squareMatrix Returns the matrix inverse tangent of squareMatrix1 This is not the same as calculating the inverse tangent of each element For information about the calculation method refer to cos squareMatrix1 must be diagonalizable The result always contains floating point numbers InRadia...

Page 180: ...uareMatrix1 This is not the same as calculating the hyperbolic tangent of each element For information about the calculation method refer to cos squareMatrix1 must be diagonalizable The result always contains floating point numbers InRadianangle mode tanh Catalog tanh Expr1 expression tanh List1 list tanh Expr1 returns the inverse hyperbolic tangent of the argument as an expression tanh List1 retu...

Page 181: ...ial includes non zero terms of integer degrees from zero through Order in Var minus Point taylor returns itself if there is no truncated power series of this order or if it would require negative or fractional exponents Use substitution and or temporary multiplication by a power of Var minus Point to determine more general power series Point defaults to zero and is the expansion point tCdf Catalog...

Page 182: ...eparate steps simplifies an expression tExpand Catalog tExpand Expr1 expression Returns an expression in which sines and cosines of integer multiple angles angle sums and angle differences are expanded Because of the identity sin x 2 cos x 2 1 there are many possible equivalent results Consequently a result might differ from a result shown in other publications Sometimes tExpand will accomplish yo...

Page 183: ...page 147 Note You can use this command within a user defined program but not within a function Define a program thatpauses to display eachof five random numbers ina dialog box Withinthe Prgm EndPrgm template complete eachline by pressing instead of Onthe computer keyboard hold downAlt andpress Enter Define text_demo Prgm For i 1 5 strinfo Random number string rand i Text strinfo EndFor EndPrgm Run...

Page 184: ... tInterval_2Samp List1 List2 Freq1 Freq2 CLevel Pooled Data list input tInterval_2Samp v1 sx1 n1 v2 sx2 n2 CLevel Pooled Summary stats input Computes a two sample t confidence interval A summary of results is stored in the stat results variable See page 172 Pooled 1 pools variances Pooled 0 does not pool variances For information on the effect of empty elements in a list see Empty Void Elements pa...

Page 185: ...lsius _ F Fahrenheit _ K Kelvin _ R Rankine To type select it from the Catalog symbols to type _ press _ For example 100_ C converts to 212_ F To convert a temperature range use ΔtmpCnv instead Note Youcanuse the Catalog to select temperature units ΔtmpCnv Catalog ΔtmpCnv Expr_ tempUnit _ tempUnit2 expression _ tempUnit2 Note You can insert this function from the keyboard by typing deltaTmpCnv Con...

Page 186: ... 100_ C is equivalent to a 180_ F range To convert a particular temperature value instead of a range use tmpCnv tPdf Catalog tPdf XVal df number if XVal is a number list if XVal is a list Computes the probability density function pdf for the Student t distribution at a specified x value with specified degrees of freedom df trace Catalog trace squareMatrix expression Returns the trace sum of all th...

Page 187: ... example For instructions on entering multi line program and function definitions refer to the Calculator section of your product guidebook To see the commands Try ClrErr and PassErr in operation enter the eigenvals program shown at the right Run the program by executing each of the following expressions Note See also ClrErr page 29 and PassErr page 129 Define eigenvals a b Prgm Program eigenvals ...

Page 188: ...r Ha μ μ0 default set Hypoth 0 For Ha μ μ0 set Hypoth 0 For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat t v μ0 stdev sqrt n stat PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat df Degrees of freedom stat v Sample meanof the data sequence inList stat sx Sample standarddeviationof the data s...

Page 189: ...ans stat PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat df Degrees of freedom for the t statistic stat v1 stat v2 Sample means of the data sequences inList 1 andList 2 stat sx1 stat sx2 Sample standarddeviations of the data sequences inList 1 andList 2 stat n1 stat n2 Size of the samples stat sp The pooledstandarddeviation CalculatedwhenPooled 1 tvmFV Catalog tvm...

Page 190: ...in the TVM functions are described in the table of TVM arguments page 191 See also amortTbl page 12 tvmPmt Catalog tvmPmt N I PV FV PpY CpY PmtAt value Financial function that calculates the amount of each payment Note Arguments used in the TVM functions are described in the table of TVM arguments page 191 See also amortTbl page 12 tvmPV Catalog tvmPV N I Pmt FV PpY CpY PmtAt value Financial funct...

Page 191: ...re their argument values or results to the TVM variables TwoVar Catalog TwoVar X Y Freq Category Include Calculates the TwoVar statistics A summary of results is stored in the stat results variable See page 172 All the lists must have equal dimension except for Include X and Y are lists of independent and dependent variables Freq is an optional list of frequency values Each element in Freq specifi...

Page 192: ...Σx Sum of x values stat Σx2 Sum of x2 values stat sx Sample standarddeviationof x stat σx Populationstandarddeviationof x stat n Number of data points stat w Meanof y values stat Σy Sum of y values stat Σy2 Sum of y2 values stat sy Sample standarddeviationof y stat σy Populationstandarddeviationof y stat Σxy Sum of x y values stat r Correlationcoefficient stat MinX Minimum of x values stat Q1X 1st...

Page 193: ...urns either a row or column unit vector depending on the form of Vector1 Vector1 must be either a single row matrix or a single column matrix To see the entire result press andthen use and to move the cursor unLock Catalog unLock Var1 Var2 Var3 unLock Var Unlocks the specified variables or variable group Locked variables cannot be modified or deleted See Lock page 104 and getLockInfo page 82 Alpha...

Page 194: ... 232 varSamp Catalog varSamp List freqList expression Returns the sample variance of List Each freqList element counts the number of consecutive occurrences of the corresponding element in List Note List must contain at least two elements If an element in either list is empty void that element is ignored and the corresponding element in the other list is also ignored For more information on empty ...

Page 195: ...l value in the range 0 through 100 The command rounds this value up to the nearest 0 1 seconds To cancel a Wait that is in progress Handheld Hold down the c key and press repeatedly Windows Hold down the F12 key and press Enter repeatedly Macintosh Hold down the F5 key and press Enter repeatedly iPad The app displays a prompt You can continue waiting or cancel Note You can use the Wait command wit...

Page 196: ...or a list of warning codes and associated messages see page 246 To see the entire result press andthen use and to move the cursor when Catalog when Condition trueResult falseResult unknownResult expression Returns trueResult falseResult or unknownResult depending on whether Condition is true false or unknown Returns the input if there are too few arguments to specify the appropriate result Omit bo...

Page 197: ... Returns true if BooleanExpr1 is true and BooleanExpr2 is false or vice versa Returns false if both arguments are true or if both are false Returns a simplified Boolean expression if either of the arguments cannot be resolved to true or false Note See or page 127 Integer1 xor Integer2 integer Compares two real integers bit by bit using an xor operation Internally both integers are converted to sig...

Page 198: ...e upto 16 digits Z zeros Catalog zeros Expr Var list zeros Expr Var Guess list Returns a list of candidate real values of Var that make Expr 0 zeros does this by computing exp list solve Expr 0 Var Var For some purposes the result form for zeros is more convenient than that of solve However the result form of zeros cannot express implicit solutions solutions that require inequalities or solutions ...

Page 199: ...es that could be substituted later Each row of the resulting matrix represents an alternate zero with the components ordered the same as the varOrGuess list To extract a row index the matrix by row Extractrow2 You can also or instead include unknowns that do not appear in the expressions For example you can include z as an unknown to extend the previous example to two parallel intersecting cylinde...

Page 200: ...numbers Each unknown starts at its guessed value if there is one otherwise it starts at 0 0 Use guesses to seek additional zeros one by one For convergence a guess may have to be rather close to a zero zInterval Catalog zInterval σ List Freq CLevel Data list input zInterval σ v n CLevel Summary stats input Computes a z confidence interval A summary of results is stored in the stat results variable...

Page 201: ... containing confidence level probability of distribution stat Ç The calculatedproportionof successes stat ME Marginof error stat n Number of samples indata sequence zInterval_2Prop Catalog zInterval_2Prop x1 n1 x2 n2 CLevel Computes a two proportion z confidence interval A summary of results is stored in the stat results variable See page 172 x1 and x2 are non negative integers For information on ...

Page 202: ...le See page 172 For information on the effect of empty elements in a list see Empty Void Elements page 232 Output variable Description stat CLower stat CUpper Confidence interval containing confidence level probability of distribution stat x1 x2 Sample means of the data sequences from the normal random distribution stat ME Marginof error stat x1 stat x2 Sample means of the data sequences from the ...

Page 203: ... Description stat z x μ0 σ sqrt n stat P Value Leastprobability atwhichthe null hypothesis canbe rejected stat x Sample meanof the data sequence inList stat sx Sample standarddeviationof the data sequence Only returnedfor Data input stat n Size of the sample zTest_1Prop Catalog Output variable Description stat p0 Hypothesizedpopulationproportion stat z Standardnormal value computedfor the proporti...

Page 204: ...omputedfor the difference of proportions stat PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat Ç1 Firstsample proportionestimate stat Ç2 Secondsample proportionestimate stat Ç Pooledsample proportionestimate stat n1 stat n2 Number of samples takenintrials 1 and2 zTest_2Samp Catalog zTest_2Samp σ1 σ2 List1 List2 Freq1 Freq2 Hypoth Data list input zTest_2Samp σ1 σ2 v...

Page 205: ...tandardnormal value computedfor the difference of means stat PVal Smallestlevel of significance atwhichthe null hypothesis canbe rejected stat x1 stat x2 Sample means of the data sequences inList1 andList2 stat sx1 stat sx2 Sample standarddeviations of the data sequences inList1 andList2 stat n1 stat n2 Size of the samples Alphabetical Listing 205 ...

Page 206: ...2 Dimensions of the arguments must be equal Expr List1 list List1 Expr list Returns a list containing the sums of Expr and each element in List1 Expr Matrix1 matrix Matrix1 Expr matrix Returns a matrix with Expr added to each element on the diagonal of Matrix1 Matrix1 must be square Note Use dot plus to add an expression to each element subtract key Expr1 Expr2 expression Returns Expr1 minus Expr2...

Page 207: ...e identity matrix minus Matrix1 Matrix1 must be square Matrix1 Expr returns a matrix of Expr times the identity matrix subtracted from Matrix1 Matrix1 must be square Note Use dot minus to subtract an expression from each element multiply r key Expr1 Expr2 expression Returns the product of the two arguments List1 List2 list Returns a list containing the products of the corresponding elements in Lis...

Page 208: ...xpression by each element divide p key Expr1 Expr2 expression Returns the quotient of Expr1 divided by Expr2 Note See also Fraction template page 5 List1 List2 list Returns a list containing the quotients of List1 divided by List2 Dimensions of the lists must be equal Expr List1 list List1 Expr list Returns a list containing the quotients of Expr divided by List1 orList1 divided by Expr Matrix1 Ex...

Page 209: ...the real domain fractional powers that have reduced exponents with odd denominators use the real branch versus the principal branch for complex mode Expr List1 list Returns Expr raised to the power of the elements in List1 List1 Expr list Returns the elements in List1 raised to the power of Expr squareMatrix1 integer matrix Returns squareMatrix1 raised to the integer power squareMatrix1 must be a ...

Page 210: ...lement dot add keys Matrix1 Matrix2 matrix Expr Matrix1 matrix Matrix1 Matrix2 returns a matrix that is the sum of each pair of corresponding elements in Matrix1 and Matrix2 Expr Matrix1 returns a matrix that is the sum of Expr and each element in Matrix1 dot subt keys Matrix1 Matrix2 matrix Expr Matrix1 matrix Matrix1 Matrix2 returns a matrix that is the difference between each pair of correspond...

Page 211: ...ix1 Matrix2 returns a matrix that is the quotient of each pair of corresponding elements in Matrix1 and Matrix2 Expr Matrix1 returns a matrix that is the quotient of Expr and each element in Matrix1 dot power l keys Matrix1 Matrix2 matrix Expr Matrix1 matrix Matrix1 Matrix2 returns a matrix where each element in Matrix2 is the exponent for the corresponding element in Matrix1 Expr Matrix1 returns ...

Page 212: ... list or matrix returns a list or matrix with each element divided by 100 Note To force anapproximate result Handheld Press Windows Press Ctrl Enter Macintosh Press Enter iPad Holdenter andselect equal key Expr1 Expr2 Boolean expression List1 List2 Boolean list Matrix1 Matrix2 Boolean matrix Returns true if Expr1 is determined to be equal to Expr2 Returns false if Expr1 is determined to not be equ...

Page 213: ...to Expr2 Returns false if Expr1 is determined to be equal to Expr2 Anything else returns a simplified form of the equation For lists and matrices returns comparisons element by element Note You can insert this operator from the keyboard by typing See equal example less than keys Expr1 Expr2 Boolean expression List1 List2 Boolean list Matrix1 Matrix2 Boolean matrix Returns true if Expr1 is determin...

Page 214: ... Expr2 Anything else returns a simplified form of the equation For lists and matrices returns comparisons element by element Note You can insert this operator from the keyboard by typing See equal example greater than keys Expr1 Expr2 Boolean expression List1 List2 Boolean list Matrix1 Matrix2 Boolean matrix Returns true if Expr1 is determined to be greater than Expr2 Returns false if Expr1 is det...

Page 215: ...by element Note You can insert this operator from the keyboard by typing See equal example logical implication keys BooleanExpr1 BooleanExpr2 returns Boolean expression BooleanList1 BooleanList2 returns Boolean list BooleanMatrix1 BooleanMatrix2 returns Boolean matrix Integer1 Integer2 returns Integer Evaluates the expression not argument1 or argument2 and returns true false or a simplified form o...

Page 216: ...peration on the two arguments Returns true false or a simplified form of the equation For lists and matrices returns comparisons element by element Note You can insert this operator from the keyboard by typing factorial º key Expr1 expression List1 list Matrix1 matrix Returns the factorial of the argument For a list or matrix returns a list or matrix of factorials of the elements append k keys Str...

Page 217: ... simplified arguments Instead d performs the following steps 1 Simplify the second argument only to the extent that it does not lead to a non variable 2 Simplify the first argument only to the extent that it does recall any stored value for the variable determined by step 1 3 Determine the symbolic derivative of the result of step 2 with respect to the variable from step 1 If the variable from ste...

Page 218: ... make an anti derivative valid over a larger interval than the usual formula returns itself for pieces of Expr1 that it cannot determine as an explicit finite combination of its built in functions and operators When you provide Lower and Upper an attempt is made to locate any discontinuities or discontinuous derivatives in the interval Lower Var Upper and to subdivide the interval at those places ...

Page 219: ... For a list returns the square roots of all the elements in List1 Note You can insert this function from the keyboard by typing sqrt Note See also Square root template page 5 Π prodSeq Catalog Π Expr1 Var Low High expression Note You can insert this function from the keyboard by typing prodSeq Evaluates Expr1 for each value of Var from Low to High and returns the product of the results Note See al...

Page 220: ...eq Catalog Σ Expr1 Var Low High expression Note You can insert this function from the keyboard by typing sumSeq Evaluates Expr1 for each value of Var from Low to High and returns the sum of the results Note See also Sum template page 9 Σ Expr1 Var Low Low 1 0 Σ Expr1 Var Low High μ Σ Expr1 Var High 1 Low 1 if High Low 1 The summation formulas used are derived from the following reference Ronald L ...

Page 221: ...t If you omit FV it defaults to FV 0 The defaults for PpY CpY and PmtAt are the same as for the TVM functions roundValue specifies the number of decimal places for rounding Default 2 ΣInt NPmt1 NPmt2 amortTable calculates the sum of the interest based on amortization table amortTable The amortTable argument must be a matrix in the form described under amortTbl page 12 Note See also ΣPrn below and ...

Page 222: ...lue specifies the number of decimal places for rounding Default 2 ΣPrn NPmt1 NPmt2 amortTable calculates the sum of the principal paid based on amortization table amortTable The amortTable argument must be a matrix in the form described under amortTbl page 12 Note See also ΣInt above and Bal page 21 indirection k keys varNameString Refers to the variable whose name is varNameString This lets you u...

Page 223: ...e 2 3 E4 to enter 2 3E4 g gradian key Expr1g expression List1g list Matrix1g matrix This function gives you a way to specify a gradian angle while in the Degree or Radian mode In Radian angle mode multiplies Expr1 by π 200 In Degree angle mode multiplies Expr1 by g 100 In Gradian mode returns Expr1 unchanged Note You can insert this symbol from the computer keyboard by typing g InDegree Gradianor ...

Page 224: ...er keyboard by typing r degree key Expr1 expression List1 list Matrix1 matrix This function gives you a way to specify a degree angle while in Gradian or Radian mode In Radian angle mode multiplies the argument by π 180 In Degree angle mode returns the argument unchanged In Gradian angle mode multiplies the argument by 10 9 Note You can insert this symbol from the computer keyboard by typing d InD...

Page 225: ...drical input Radius θ_Angle θ_Angle vector spherical input Returns coordinates as a vector depending on the Vector Format mode setting rectangular cylindrical or spherical Note You can insert this symbol from the computer keyboard by typing InRadianmode andvector formatsetto rectangular cylindrical spherical Magnitude Angle complexValue polar input Enters a complex value in r θ polar form The Angl...

Page 226: ...its For a list of pre defined units open the Catalog and display the Unit Conversions tab You can select unit names from the Catalog or type the unit names directly Note Youcanfindthe conversionsymbol inthe Catalog Click andthenclick Math Operators Variable_ When Variable has no value it is treated as though it represents a complex number By default without the _ the variable is treated as real If...

Page 227: ...type _ when typing unit names on the handheld press _ Note To convert temperature units use tmpCnv and ΔtmpCnv The conversion operator does not handle temperature units 10 Catalog 10 Expr1 expression 10 List1 list Returns 10 raised to the power of the argument For a list returns 10 raised to the power of the elements in List1 10 squareMatrix1 squareMatrix Returns 10 raised to the power of squareMa...

Page 228: ...rves as a binary operator The operand to the left of is an expression The operand to the right of specifies one or more relations that are intended to affect the simplification of the expression Multiple relations after must be joined by logical and or or operators The constraint operator provides three basic types of functionality Substitutions Interval constraints Exclusions Substitutions are in...

Page 229: ...or to exclude a specific value from consideration They are used primarily to exclude an exact solution when using cSolve cZeros fMax fMin solve zeros and so on store h key Expr Var List Var Matrix Var Expr Function Param1 List Function Param1 Matrix Function Param1 If the variable Var does not exist creates it and initializes it to Expr List or Matrix If the variable Var already exists and is not ...

Page 230: ...le type pi 4 myvar assign t keys Var Expr Var List Var Matrix Function Param1 Expr Function Param1 List Function Param1 Matrix If variable Var does not exist creates Var and initializes it to Expr List or Matrix If Var already exists and is not locked or protected replaces its contents with Expr List or Matrix Hint If you plan to do symbolic computations using undefined variables avoid storing any...

Page 231: ... entering multi line program and function definitions refer to the Calculator section of your product guidebook 0b 0h 0B keys 0H keys 0b binaryNumber 0h hexadecimalNumber Denotes a binary or hexadecimal number respectively To enter a binary or hex number you must enter the 0b or 0h prefix regardless of the Base mode Without a prefix a number is treated as decimal base 10 Results are displayed acco...

Page 232: ...o enter an empty element manually in a math expression type _ or the keyword void The keyword void is automatically converted to a _ symbol when the expression is evaluated To type _ on the handheld press _ Calculations involving void elements The majority of calculations involving a void input will produce a void result See special cases below List arguments containing void elements The following...

Page 233: ...introduces a void for the corresponding element of the residual An omitted category in regressions introduces a void for the corresponding element of the residual A frequency of 0 in regressions introduces a void for the corresponding element of the residual Empty Void Elements 233 ...

Page 234: ...ome shortrcuts are useful from both the handheld and the computer keyboard Others are useful primarily from the computer keyboard From the Handheld or Computer Keyboard To enter this Type this shortcut π pi θ theta infinity logical implication logical double implication XNOR store operator absolute value abs sqrt d derivative integral Σ Sum template sumSeq Π Product template prodSeq sin cos arcsin...

Page 235: ...nts n1 n2 i imaginary constant i e natural log base e e E scientific notation E T transpose t r radians r degrees d g gradians g angle conversion Decimal approxFraction and so on Decimal approxFraction and so on Shortcuts for Entering Math Expressions 235 ...

Page 236: ...rs degrees minutes seconds factorial percentage radian r subscript transpose T 5 Exponentiation power operator 6 Negation 7 String concatenation 8 Multiplication division 9 Addition subtraction 10 Equality relations equal not equal or less than less than or equal or greater than greater than or equal or 11 Logical not 12 Logical and 13 Logical or 14 xor nor nand 15 Logical implication 16 Logical d...

Page 237: ...ample if 10 r and r s1 then s1 10 Post Operators Post operators are operators that come directly after an argument such as 5 25 or 60 15 45 Arguments followed by a post operator are evaluated at the fourth priority level For example in the expression 4 3 3 is evaluated first The result 6 then becomes the exponent of 4 to yield 4096 Exponentiation Exponentiation and element by element exponentiatio...

Page 238: ...dwhenthe If statementis executed 30 Argumentcannotbe a folder name 40 Argumenterror 50 Argumentmismatch Two or more arguments mustbe of the same type 60 Argumentmustbe a Booleanexpressionor integer 70 Argumentmustbe a decimal number 90 Argumentmustbe a list 100 Argumentmustbe a matrix 130 Argumentmustbe a string 140 Argumentmustbe a variable name Make sure thatthe name does not begin with a digit ...

Page 239: ...nerror because L1 only contains four elements 235 DimensionError Notenoughelements inthe lists 240 Dimensionmismatch Two or more arguments mustbe of the same dimension For example 1 2 1 2 3 is a dimensionmismatchbecause the matrices containa differentnumber of elements 250 Divide by zero 260 Domainerror Anargumentmustbe ina specifieddomain For example rand 0 is notvalid 270 Duplicate variable name...

Page 240: ...edmultiplicationandfunctioncalls 450 Invalidina functionor currentexpression Only certaincommands are validina user definedfunction 490 InvalidinTry EndTry block 510 Invalidlistor matrix 550 Invalidoutside functionor program A number of commands are notvalidoutside a functionor program For example Local cannotbe usedunless itis ina functionor program 560 Invalidoutside Loop EndLoop For EndFor or W...

Page 241: ...s 665 Matrix notdiagonalizable 670 LowMemory 1 Delete some data inthis document 2 Save andclose this document If 1 and2 fail pull outandre insertbatteries 672 Resource exhaustion 673 Resource exhaustion 680 Missing 690 Missing 700 Missing 710 Missing 720 Missing 730 Missing startor endof block syntax 740 Missing Theninthe If EndIf block 750 Name is nota functionor program 765 No functions selected...

Page 242: ...be appliedto data set 910 Syntax error 920 Textnotfound 930 Too fewarguments The functionor commandis missing one or more arguments 940 Too many arguments The expressionor equationcontains anexcessive number of arguments andcannotbe evaluated 950 Too many subscripts 955 Too many undefinedvariables 960 Variable is notdefined No value is assignedto variable Use one of the following commands sto Defi...

Page 243: ...licationdoes notsupportAns 1090 Functionis notdefined Use one of the following commands Define sto to define a function 1100 Non real calculation For example if the software is inthe Real setting 1 is invalid To allowcomplex results change the Real or Complex Mode Setting to RECTANGULAR or POLAR 1110 Invalidbounds 1120 No signchange 1130 Argumentcannotbe a listor matrix 1140 Argumenterror The firs...

Page 244: ...racters See the Library sectioninthe documentationfor more details 1190 Library documentnotfound Verify library is in the MyLib folder Refresh Libraries See the Library sectioninthe documentationfor more details 1200 Library variable notfound Verify library variable exists in the first problem in the library Make sure library variable has been defined as LibPub or LibPriv Refresh Libraries See the...

Page 245: ...variable Itcannot containa non valuedvariable other thanthe variable of interest 1270 ArgumentError Order of the derivative mustbe equal to 1 or 2 1280 ArgumentError Use a polynomial inexpandedform inone variable 1290 ArgumentError Use a polynomial inone variable 1300 ArgumentError The coefficients of the polynomial mustevaluate to numeric values 1310 Argumenterror A functioncouldnotbe evaluatedfo...

Page 246: ...7 More solutions may exist Try specifying appropriate lower andupper bounds and or a guess Examples using solve solve Equation Var Guess lowBound Var upBound solve Equation Var lowBound Var upBound solve Equation Var Guess 10008 Domainof the resultmightbe smaller thanthe domainof the input 10009 Domainof the resultmightbe larger thanthe domainof the input 10012 Non real calculation 10013 0 or unde...

Page 247: ...s beenmultipliedby the identity matrix 10024 Resultobtainedusing approximate arithmetic 10025 Equivalence cannotbe verifiedinEXACT mode 10026 Constraintmightbe ignored Specify constraintinthe form Variable MathTestSymbol Constant or a conjunctof these forms for example x 3 andx 12 Warning Codes and Messages 247 ...

Page 248: ...248 ...

Page 249: ... not toll free 972 917 8324 For Product Hardware Service Customers in the U S Canada Mexico and U S territories Always contact Texas Instruments Customer Support before returning a product for service For All Other Countries For general information For more information about TI products and services contact TI by e mail or visit the TI Internet address E mail inquiries ti cares ti com Home Page ed...

Page 250: ...250 ...

Page 251: ...second notation 224 indirection 222 indirection operator 237 percent 212 append 216 multiply 207 dot subtraction 210 dot multiplication 211 dot division 211 dot power 211 dot addition 210 divide 208 assign 230 reciprocal 228 Index 251 ...

Page 252: ...ation 226 constraint operator 228 minute notation 224 prime 226 add 206 not equal 213 less than or equal 214 greater than or equal 215 greater than 214 equal 212 product 219 sum 220 Int 221 Prn 221 square root 219 angle 225 integral 217 ...

Page 253: ... display as degree minute second 57 exp display in terms of e 66 Grad convert to gradian angle 85 Polar display as polar vector 131 Rad convert to radian angle 141 Rect display as rectangular vector 144 sin display in terms of sine 162 Sphere display as sphericalvector 171 logicalimplication 215 234 store variable 229 logicaldouble implication 216 234 comment 231 degree notation 224 degrees minute...

Page 254: ...4 ANOVA2way two way variance analysis 15 Ans last answer 17 answer last Ans 17 append 216 approx approximate 17 18 approximate approx 17 18 approxRational 18 arc length arcLen 19 arccos cos 18 arccosh cosh 18 arccot cot 18 arccoth coth 18 arccsc csc 18 arccsch csch 18 arcLen arc length 19 arcsec sec 19 arcsech csech 19 arcsin sin 19 arcsinh sinh 19 arctan tan 19 arctanh tanh 19 arguments in TVM fu...

Page 255: ... 71 ceiling ceiling 24 ceiling ceiling 24 25 40 centralDiff 25 cFactor complex factor 25 char character string 26 character string char 26 characters numeric code ord 128 string char 26 charPoly 27 χ 2way 27 clear error ClrErr 29 ClearAZ 29 ClrErr clear error 29 colAugment 30 colDim matrix column dimension 30 colNorm matrix column norm 30 combinations nCr 118 comDenom common denominator 30 comment...

Page 256: ...construct matrix constructMat 32 constructMat construct matrix 32 convert Grad 85 Rad 141 units 227 copy variable or function CopyVar 33 correlation matrix corrMat 33 corrMat correlation matrix 33 cos arccosine 35 cos cosine 34 cosh hyperbolic arccosine 36 cosh hyperbolic cosine 36 cosine display expression in terms of 33 cosine cos 34 cot arccotangent 37 cot cotangent 37 cotangent cot 37 coth hyp...

Page 257: ...ative sum 45 cycle Cycle 46 Cycle cycle 46 cylindricalvector display Cylind 46 cZeros complex zeros 46 D d first derivative 217 days between dates dbd 49 dbd days between dates 49 decimal angle display DD 49 integer display Base10 22 Define 50 Define LibPriv 51 Define LibPub 52 define Define 50 Define define 50 defining private function or program 51 public function or program 52 definite integral...

Page 258: ...terminant 55 diag matrix diagonal 56 dim dimension 56 dimension dim 56 Disp display data 57 155 display as binary Base2 21 cylindricalvector Cylind 46 decimalangle DD 49 decimalinteger Base10 22 degree minute second DMS 57 hexadecimal Base16 23 polar vector Polar 131 rectangular vector Rect 144 sphericalvector Sphere 171 display data Disp 57 155 distribution functions binomCdf 24 binomPdf 24 invNo...

Page 259: ...6 e to a power e 60 66 e display expression in terms of 66 E exponent 223 e e to a power 60 eff convert nominalto effective rate 61 effective rate eff 61 eigenvalue eigVl 61 eigenvector eigVc 61 eigVc eigenvector 61 eigVl eigenvalue 61 else if ElseIf 62 else Else 86 ElseIf else if 62 empty void elements 232 end for EndFor 75 function EndFunc 78 if EndIf 86 loop EndLoop 108 program EndPrgm 135 try ...

Page 260: ...xact exact 65 exclusion with operator 228 exit Exit 65 Exit exit 65 exp e to a power 66 exp list expression to list 67 expand expand 67 expand expand 67 exponent E 223 exponentialregession ExpReg 69 exponents template for 5 expr string to expression 68 105 ExpReg exponentialregession 69 expressions expression to list exp list 67 string to expression expr 68 105 F factor factor 70 factor factor 70 ...

Page 261: ...m norm 122 Func function 78 Func program function 78 functions maximum fMax 73 minimum fMin 74 part fpart 76 program function Func 78 user defined 50 functions and variables copying 33 G g gradians 223 gcd greatest common divisor 79 geomCdf 79 geomPdf 80 Get 80 get return denominator getDenom 81 number getNum 83 variables injformation getVarInfo 81 84 getDenom get return denominator 81 getLangInfo...

Page 262: ...locking 104 193 groups testing lock status 82 H hexadecimal display Base16 23 indicator 0h 231 hyperbolic arccosine cosh 36 arcsine sinh 165 arctangent tanh 180 cosine cosh 36 sine sinh 164 tangent tanh 180 I identity matrix identity 86 identity identity matrix 86 if If 86 If if 86 ifFn 87 imag imaginary part 88 imaginary part imag 88 ImpDif implicit derivative 88 implicit derivative Impdif 88 ind...

Page 263: ...ternalrate of return internalrate of return irr 91 isPrime prime test 92 isVoid test for void 92 L label Lbl 93 language get language information 81 Lbl label 93 lcm least common multiple 93 least common multiple lcm 93 left left 93 left left 93 length of string 56 less than or equal 214 LibPriv 51 LibPub 52 library create shortcuts to objects 94 libShortcut create shortcuts to library objects 94 ...

Page 264: ... 232 expression to list exp list 67 list to matrix list mat 101 matrix to list mat list 109 maximum max 109 mid string mid 112 minimum min 113 new newList 119 product product 136 sort ascending SortA 170 sort descending SortD 171 summation sum 176 ln naturallogarithm 102 LnReg logarithmic regression 102 localvariable Local 104 local Local 104 Local localvariable 104 Lock lock variable or variable ...

Page 265: ...ion 211 dot multiplication 211 dot power 211 dot subtraction 210 eigenvalue eigVl 61 eigenvector eigVc 61 filling Fill 72 identity identity 86 list to matrix list mat 101 lower upper decomposition LU 108 matrix to list mat list 109 maximum max 109 minimum min 113 new newMat 119 product product 136 QR factorization QR 137 random randMat 142 reduced row echelon form rref 153 row addition rowAdd 152 ...

Page 266: ...dMed medium medium line regression 111 mid string mid 112 mid mid string 112 min minimum 113 minimum min 113 minute notation 224 mirr modified internalrate of return 113 mixed fractions using propFrac with 136 mod modulo 114 mode settings getMode 82 modes setting setMode 158 modified internalrate of return mirr 113 modulo mod 114 mRow matrix row operation 114 mRowAdd matrix row multiplication and ...

Page 267: ...e nom 121 nor Boolean operator 121 norm Frobenius norm 122 normaldistribution probability normCdf 123 normalline normalLine 122 normalLine 122 normCdf 123 normPdf 123 not equal 213 not Boolean operator 123 nPr permutations 124 npv net present value 124 nSolve numeric solution 125 nth root template for 5 numeric derivative nDeriv 119 120 derivative nDerivative 119 integral nInt 120 solution nSolve ...

Page 268: ...130 poissCdf 130 poissPdf 130 polar coordinate R Pr 141 coordinate R Pθ 140 vector display Polar 131 polyCoef 131 polyDegree 132 polyEval evaluate polynomial 132 polyGcd 133 polynomials evaluate polyEval 132 random randPoly 143 PolyRoots 134 power of ten 10 227 power regression PowerReg 134 146 147 183 power 209 PowerReg power regression 134 Prgm define program 135 prime number test isPrime 92 pri...

Page 269: ...factorization QR 137 QR QR factorization 137 quadratic regression QuadReg 138 QuadReg quadratic regression 138 quartic regression QuartReg 139 QuartReg quartic regression 139 R R radian 223 R Pr polar coordinate 141 R Pθ polar coordinate 140 radian R 223 rand random number 141 randBin random number 142 randInt random integer 142 randMat random matrix 142 randNorm random norm 143 random matrix rand...

Page 270: ...istic 106 logistic Logistic 107 medium medium line MedMed 111 MultReg 115 power regression PowerReg 134 146 147 183 quadratic QuadReg 138 quartic QuartReg 139 sinusoidal SinReg 165 remain remainder 146 remainder remain 146 remove void elements from list 53 Request 146 RequestStr 147 result display in terms of cosine 33 display in terms of e 66 display in terms of sine 162 result values statistics ...

Page 271: ...ation 224 seq sequence 155 seqGen 156 seqn 156 sequence seq 155 156 series series 157 series series 157 set mode setMode 158 setMode set mode 158 settings get current 82 shift shift 160 shift shift 160 sign sign 161 sign sign 161 simult simultaneous equations 162 simultaneous equations simult 162 sin arcsine 164 sin sine 163 sine display expression in terms of 162 sine sin 163 sinh hyperbolic arcs...

Page 272: ...0 median median 110 one variable statistics OneVar 126 permutations nPr 124 random norm randNorm 143 random number seed RandSeed 143 standard deviation stdDev 174 194 two variable results TwoVar 191 variance variance 194 stdDevPop population standard deviation 174 stdDevSamp sample standard deviation 174 Stop command 175 store variable 229 storing symbol 230 string dimension dim 56 length 56 strin...

Page 273: ...206 sum of interest payments 221 sum of principalpayments 221 sum summation 176 sum 220 template for 9 sumIf 176 summation sum 176 sumSeq 177 system of equations 2 equation template for 7 system of equations N equation template for 7 T t test tTest 188 T transpose 178 tan arctangent 179 tan tangent 178 tangent line tangentLine 179 tangent tan 178 tangentLine 179 tanh hyperbolic arctangent 180 tanh...

Page 274: ...of equations N equation 7 test for void isVoid 92 Test_2S 2 sample F test 78 tExpand trigonometric expansion 182 Text command 183 time value of money Future Value 189 time value of money Interest 189 time value of money number of payments 190 time value of money payment amount 190 time value of money present value 190 tInterval t confidence interval 183 tInterval_2Samp twosample t confidence inter...

Page 275: ...es and variable groups 193 user defined functions 50 user defined functions and programs 51 52 V variable creating name from a character string 237 variable and functions copying 33 variables clear allsingle letter 29 delete DelVar 53 local Local 104 variables locking and unlocking 82 104 193 variance variance 194 varPop 194 varSamp sample variance 194 vectors cross product crossP 40 cylindricalve...

Page 276: ...oolean exclusive or 197 Z zeroes zeroes 198 zeroes zeroes 198 zInterval z confidence interval 200 zInterval_1Prop one proportion z confidence interval 201 zInterval_2Prop two proportion z confidence interval 201 zInterval_2Samp two sample z confidence interval 202 zTest 202 zTest_1Prop one proportion z test 203 zTest_2Prop two proportion z test 203 zTest_2Samp two sample z test 204 Χ χ Cdf 28 χ GO...

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