Exploring Asymptotes

Time required 00 minutes
Activity Overview In this activity, students will explore asymptotes and singularities, paying particular attention to the connection between the algebraic and graphical representations. Material Technology: TI-Nspire handheld, TI-Nspire CAS handheld, or TI-Nspire software Documents: Asymptotes.tns, Asymptotes_Student.doc Rational Functions Part 1 involves an exploration of asymptotes involving graphing, exploring a table of values, and algebraic manipulation. Students will use a variety of TI-Nspire tools to identify undened values of x. First, a graphic representation is explored. From the graph, students can get reasonable approximations of undened values. Ask students how they can be certain of the undened values. What tools are provided on the graphing screen to help with this?
Getting Started with TI-Nspire Developmental Algebra
2009 Texas Instruments Incorporated ti-educators@ti.com

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Developmental Algebra
Horizontal Asymptotes Students explore what was learned in Part 1 to develop an understanding of the patterns involved with asymptotes and rational functions. This part provides a great opportunity for students working in pairs or small groups to develop a deeper understanding of what was learned in Part 1.
Functions and Relations to Vertical & Horizontal Asymptotes Students apply what was learned in Parts 1 and 2 to nding singularities and asymptotes for the function 5x 7 f(x) _____________. 4x2 8x 12
Texas Instruments Incorporated
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Lead Coefficients and Relations to Asymptotes Students again apply what has been learned, but the challenge level increases. In this situation, the function is represented in two different ways, one of which helps to illustrate the vertical shift of a simpler graph. You may wish to have students graph 1 f(x) __________ and compare it to the x2 x 12 second function given. If time allows, consider having students verify algebraically that the two functions given on page 6.5 are equivalent. Also, have students label the asymptotes on the function graph with equations.

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Exploring AsymptotesStudent Solutions
1. a. No; certain values of x are skipped b. The table displays (.) for the undened x-values. c. 3 have no values d. (x 3)(x 3) e. The factors match the skipped x-values. (x 3) matches with the skipped value of 3, and (x 3) matches with the skipped value of 3. f. These values make the denominator equal to zero, causing the function to be undened at these values of x. Students might expect asymptotes here. g. Would expect asymptotes at x 3 and x 3. h. Only one asymptote at x 3. i. The x 3 cancels for everything except x 3 and so it does not go to innity. 2. a. Yes; y 0 b. The denominator gets very small making the fraction very large in magnitude. 3. Vertical asymptotes are always at x a and x b. 4. a. Horizontal asymptotes are present whenever the degree of the numerator is less than or equal to the degree of the denominator. They may or may not be present when the numerator degree exceeds that of the denominator. b. y p/q, or at y (ratio of leading coefcient of numerator to leading coefcient of denominator) c. y 0 5. a. x 1 and x 3 b. Yes; y 0 6. a. (x 3)(x 4); singularities at x 3 and x 4 b.
1 c. Since __________ approaches zero as x tends to innity, the second x2 x 12 representation clearly tends to 2.
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Reference Guide
This guidebook applies to TI-Nspire software version 3.0. To obtain the latest version of the documentation, go to education.ti.com/guides.

Important Information

Except as otherwise expressly stated in the License that accompanies a program, Texas Instruments makes no warranty, either express or implied, including but not limited to any implied warranties of merchantability and fitness for a particular purpose, regarding any programs or book materials and makes such materials available solely on an "as-is" basis. In no event shall Texas Instruments 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 whatsoever against the use of these materials by any other party.

License

Please see the complete license installed in C:\Program Files\TI Education\TI-Nspire CAS.
2006 - 2011 Texas Instruments Incorporated

Contents

Expression templates
Fraction template.. 1 Exponent template.. 1 Square root template.. 1 Nth root template.. 1 e exponent template.. 2 Log template.. 2 Piecewise template (2-piece).. 2 Piecewise template (N-piece).. 2 System of 2 equations template.. 3 System of N equations template.. 3 Absolute value template.. 3 ddmmss.ss template.. 3 Matrix template (2 x 2).. 3 Matrix template (1 x 2).. 4 Matrix template (2 x 1).. 4 Matrix template (m x n).. 4 Sum template (G)... 4 Product template ().. 4 First derivative template.. 5 Second derivative template. 5 Nth derivative template.. 5 Definite integral template. 5 Indefinite integral template.. 5 Limit template... 6 4Base16.. 15 binomCdf().. 15 binomPdf().. 15
ceiling().. 15 centralDiff().. 16 cFactor()... 16 char()... 17 charPoly()... 17 c22way... 17 c2Cdf()... 17 c2GOF.. 18 c2Pdf()... 18 ClearAZ.. 18 ClrErr... 19 colAugment().. 19 colDim().. 19 colNorm()... 19 comDenom()... 19 completeSquare().. 20 conj()... 21 constructMat().. 21 CopyVar.. 21 corrMat().. 22 4cos... 22 cos().. 22 cos/()... 23 cosh()... 24 cosh/()... 24 cot().. 24 cot/()... 25 coth()... 25 coth/()... 25 count()... 25 countif()... 26 cPolyRoots().. 26 crossP().. 26 csc()... 27 csc/()... 27 csch()... 27 csch/().. 27 cSolve().. 28 CubicReg.. 30 cumulativeSum().. 30 Cycle... 31 4Cylind... 31 cZeros().. 31

Alphabetical listing A

abs().. 7 amortTbl().. 7 and... 7 angle()... 8 ANOVA.. 8 ANOVA2way.. 9 Ans.. 11 approx().. 11 4approxFraction().. 11 approxRational().. 11 arccos()... 11 arccosh()... 12 arccot()... 12 arccoth()... 12 arccsc()... 12 arccsch().. 12 arcLen().. 12 arcsec()... 12 arcsech().. 12 arcsin()... 12 arcsinh().. 12 arctan().. 12 arctanh()... 12 augment()... 12 avgRC().. 13
dbd()... 33 4DD... 33 4Decimal... 33 Define.. 34 Define LibPriv.. 34 Define LibPub.. 35 deltaList()... 35 deltaTmpCnv().. 35 DelVar... 35 delVoid().. 35
bal()... 13 4Base2.. 14 4Base10.. 14
derivative()..35 deSolve()...36 det()..37 diag()...37 dim()..37 Disp...38 4DMS...38 dominantTerm()...39 dotP()...39

4Grad... 56

identity()... 56 If... 57 ifFn()... 58 imag().. 58 impDif().. 58 Indirection.. 58 inString().. 59 int()... 59 intDiv()... 59 integral.. 59 interpolate().. 60 invc2().. 60 invF()... 60 invNorm()... 60 invt()... 60 iPart()... 61 irr().. 61 isPrime().. 61 isVoid().. 61

deSolve(2ndOrderODE and initCond1 and initCond2, Var, depVar) a particular solution
Returns a particular solution that satisfies 2nd Order ODE and has a specified value of the dependent variable and its first derivative at one point. For initCond1, use the form: depVar (initialIndependentValue) = initialDependentValue For initCond2, use the form: depVar (initialIndependentValue) = initial1stDerivativeValue
deSolve(2ndOrderODE 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(squareMatrix[, Tolerance])
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 has floating-point entries and does not contain any symbolic variables that have not been assigned a value. Otherwise, Tolerance is ignored. If you use or set the Auto or Approximate mode to Approximate, computations are done using floatingpoint arithmetic. If Tolerance is omitted or not used, the default tolerance is calculated as: 5EM14 max(dim(squareMatrix)) rowNorm(squareMatrix)

diag()

diag(List) matrix diag(rowMatrix) matrix diag(columnMatrix) 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(List)

integer list
Returns the dimension of List.

dim(Matrix)

Returns the dimensions of matrix as a two-element list {rows, columns}.

dim(String)

Returns the number of characters contained in character string String.
Disp [exprOrString1] [, exprOrString2].
Displays the arguments in the Calculator history. The arguments are displayed in succession, with thin spaces as separators. Useful mainly in programs and functions to ensure the display of intermediate calculations.
at the end of each line. On the computer keyboard, instead of hold down Alt and press Enter.

Note: See also tExpand() for trigonometric angle-sum and multiple-angle expansion.

expr()

expr(String)
Returns the character string contained in String as an expression and immediately executes it.

ExpReg

ExpReg X, Y [, [Freq] [, Category, Include]]
Computes the exponential regression y = a(b)x on lists X and Y with frequency Freq. A summary of results is stored in the stat.results variable. (See page 119.) 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 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 included in this list are included in the calculation. For information on the effect of empty elements in a list, see Empty (void) elements on page 159.
Output variable stat.RegEqn stat.a, stat.b stat.r2
Description Regression equation: a(b)x Regression coefficients Coefficient of linear determination for transformed data
Output variable stat.r stat.Resid stat.ResidTrans stat.XReg
Description Correlation coefficient for transformed data (x, ln(y)) Residuals associated with the exponential model Residuals associated with linear fit of transformed data List of data points in the modified X List actually used in the regression based on restrictions of Freq, Category List, and Include Categories List of data points in the modified Y List actually used in the regression based on restrictions of Freq, Category List, and Include Categories List of frequencies corresponding to stat.XReg and stat.YReg

factor() Catalog >

expression factor(List1[,Var]) list factor(Matrix1[,Var]) matrix

factor(Expr1[, Var])

factor(Expr1) returns Expr1 factored with respect to all of its variables over a common denominator. Expr1 is factored as much as possible toward linear rational factors without introducing new non-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 sorted with Var as the main variable. Similar powers of Var are collected in each factor. Include Var if factorization is needed with respect to only that variable and you are willing to accept irrational 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 permits approximation with floating-point coefficients where irrational coefficients cannot be explicitly expressed concisely in terms of the built-in functions. Even when there is only one variable, including Var might yield more complete factorization.

Description Regression Equation: y = mx+b Regression coefficients Coefficient of determination Correlation coefficient Residuals from the regression List of data points in the modified X List actually used in the regression based on restrictions of Freq, Category List, and Include Categories List of data points in the modified Y List actually used in the regression based on restrictions of Freq, Category List, and Include Categories List of frequencies corresponding to stat.XReg and stat.YReg

LinRegtIntervals

LinRegtIntervals X,Y[,F[,0[,CLev]]]
For Slope. Computes a level C confidence interval for the slope.
LinRegtIntervals X,Y[,F[,1,Xval[,CLev]]]
For Response. 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 119.) All the lists must have equal dimension. X and Y are lists of independent and dependent variables. F 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 on page 159.
Output variable stat.RegEqn stat.a, stat.b stat.df stat.r2 stat.r stat.Resid
Description Regression Equation: a+bx Regression coefficients Degrees of freedom Coefficient of determination Correlation coefficient Residuals from the regression

For Slope type only

Output variable [stat.CLower, stat.CUpper] stat.ME stat.SESlope stat.s
Description Confidence interval for the slope
Confidence interval margin of error Standard error of slope Standard error about the line

For Response type only

Output variable [stat.CLower, stat.CUpper] stat.ME stat.SE [stat.LowerPred, stat.UpperPred] stat.MEPred stat.SEPred stat.
Description Confidence interval for the mean response
Confidence interval margin of error Standard error of mean response Prediction interval for a single observation
Prediction interval margin of error Standard error for prediction a + bXVal

LinRegtTest

LinRegtTest X,Y[,Freq[,Hypoth]]
Computes a linear regression on the X and Y lists and a t test on the value of slope b and the correlation coefficient r for the equation y=a+bx. It tests the null hypothesis H0:b=0 (equivalently, r=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 specifies the frequency of occurrence for each corresponding X and Y data point. The default value is 1. All elements must be integers | 0. Hypoth is an optional value specifying one of three alternative hypotheses against which the null hypothesis (H0:b=r=0) will be tested. For Ha: b0 and r0 (default), set Hypoth=0 For Ha: b<0 and r<0, set Hypoth<0 For Ha: b>0 and r>0, set Hypoth>0 A summary of results is stored in the stat.results variable. (See page 119.) For information on the effect of empty elements in a list, see Empty (void) elements on page 159.

Returns the first argument modulo the second argument as defined by the identities: mod(x,0) = x mod(x,y) = x - y floor(x/y) When the second argument is non-zero, the result is periodic in that argument. 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 99

mRow()

mRow(Expr, Matrix1, Index)
Returns a copy of Matrix1 with each element in row Index of Matrix1 multiplied by Expr.

mRowAdd()

mRowAdd(Expr, Matrix1, Index1, Index2)
Returns a copy of Matrix1 with each element in row Index2 of Matrix1 replaced with: Expr row Index1 + row Index2

MultReg

MultReg Y, X1[,X2[,X3,[,X10]]]
Calculates multiple linear regression of list Y on lists X1, X2, , X10. A summary of results is stored in the stat.results variable. (See page 119.) All the lists must have equal dimension. For information on the effect of empty elements in a list, see Empty (void) elements on page 159.
Output variable stat.RegEqn stat.b0, stat.b1,. stat.R2 stat. List stat.Resid
Description Regression Equation: b0+b1x1+b2x2+. Regression coefficients Coefficient of multiple determination

yList = b0+b1x1+.

Residuals from the regression

MultRegIntervals

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 119.) All the lists must have equal dimension. For information on the effect of empty elements in a list, see Empty (void) elements on page 159.
Output variable stat.RegEqn stat.
Description Regression Equation: b0+b1x1+b2x2+. A point estimate:
y = b0 + b1 xl +. for XValList
stat.dfError stat.CLower, stat.CUpper stat.ME stat.SE stat.LowerPred, stat.UpperrPred stat.MEPred stat.SEPred stat.bList stat.Resid
Error degrees of freedom Confidence interval for a mean response Confidence interval margin of error Standard error of mean response Prediction interval for a single observation
Prediction interval margin of error Standard error for prediction List of regression coefficients, {b0,b1,b2,.} Residuals from the regression

MultRegTests

MultRegTests Y, X1[,X2[,X3,[,X10]]]
Multiple linear regression 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 119.) For information on the effect of empty elements in a list, see Empty (void) elements on page 159. Outputs
Output variable stat.RegEqn stat.F stat.PVal stat.R2
Description Regression Equation: b0+b1x1+b2x2+. Global F test statistic P-value associated with global F statistic Coefficient of multiple determination

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. 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 error dialog box will be displayed as normal.
Note: See also ClrErr, page 19, and Try, page 129. Note for entering the example: In the Calculator application
instead of at the end of each line. On the computer keyboard, hold down Alt and press Enter. piecewise()
piecewise(Expr1 [, Cond1 [, Expr2 [, Cond2 [, ]]]])
Returns definitions for a piecewise function in the form of a list. You can also create piecewise definitions by using a template.
Note: See also Piecewise template, page 2.

poissCdf()

poissCdf(l,lowBound,upBound) number if lowBound and upBound are numbers, list if lowBound and upBound are lists poissCdf(l,upBound) for P(0{X{upBound) upBound is a number, list if upBound is a list
Computes a cumulative probability for the discrete Poisson distribution with specified mean l. For P(X { upBound), set lowBound=0 poissPdf()

poissPdf(l,XVal) a list

Computes a probability for the discrete Poisson distribution with the specified mean l.

4Polar

Vector 4Polar
Note: You can insert this operator from the computer keyboard by typing @>Polar.
Displays vector in polar form [r q]. The vector must be of dimension 2 and can be a row or a column.
Note: 4Polar 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 4Rect, page 98.
complexValue 4Polar Displays complexVector in polar form. Degree angle mode returns (rq). Radian angle mode returns reiq.
complexValue can have any complex form. However, an reiq entry causes an error in Degree angle mode.
Note: You must use the parentheses for an (rq) polar entry.

polyCoeffs()

polyCoeffs(Poly [,Var])
Returns a list of the coefficients of polynomial Poly with respect to variable Var. Poly must be a polynomial expression in Var. We recommend that you do not omit Var unless Poly is an expression in a single variable.

Returns the angle whose secant is Expr1 or returns a list containing the inverse secants of each element of List1.

arcsec(.).

sech()
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 > In Radian angle and Rectangular complex mode:
expression sech/ (List1) list

sech /(Expr1)

Returns the inverse hyperbolic secant of Expr1 or returns a list containing the inverse hyperbolic secants of each element of List1.
Note: You can insert this function from the keyboard by typing arcsech(.).
seq(Expr, Var, Low, High[, Step])
Increments Var from Low through High by an increment of Step, evaluates Expr, and returns the results as a list. The original contents of Var are still there after seq() is completed. The default value for Step
Press Ctrl+Enter evaluate:

/ (Macintosh: +Enter) to

seqGen()
seqGen(Expr, Var, depVar, {Var0, VarMax}[, ListOfInitTerms [, VarStep [, CeilingValue]]]) list
Catalog > Generate the first 5 terms of the sequence u(n) = u(n-1)2/2, with u(1)=2 and VarStep=1.
Generates a list of terms for sequence depVar(Var)=Expr as follows: Increments independent variable Var from Var0 through VarMax by 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 Example in which Var0=2: ListOfDepVars(Var)=ListOrSystemOfExpr 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. 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()

(Summary stats input) Performs a z test with frequency freqlist. A summary of results is stored in the stat.results variable. (See page 119.) Test H0: m = m0, against one of the following: For Ha: m < m0, set Hypoth<0 For Ha: m m0 (default), set Hypoth=0 For Ha: m > m0, set Hypoth>0 For information on the effect of empty elements in a list, see Empty (void) elements on page 159.
Output variable stat.z stat.P Value stat.x stat.sx stat.n
Description (x N m0) / (s / sqrt(n)) Least probability at which the null hypothesis can be rejected Sample mean of the data sequence in List Sample standard deviation of the data sequence. Only returned for Data input. Size of the sample

zTest_1Prop

zTest_1Prop p0,x,n[,Hypoth]
Computes a one-proportion z test. A summary of results is stored in the stat.results variable. (See page 119.) x is a non-negative integer. Test H0: p = p0 against one of the following: For Ha: p > p0, set Hypoth>0 For Ha: p p0 (default), set Hypoth=0 For Ha: p < p0, set Hypoth<0 For information on the effect of empty elements in a list, see Empty (void) elements on page 159.
Output variable stat.p0 stat.z stat.PVal stat. stat.n
Description Hypothesized population proportion Standard normal value computed for the proportion Smallest level of significance at which the null hypothesis can be rejected Estimated sample proportion Size of the sample

zTest_2Prop

zTest_2Prop x1,n1,x2,n2[,Hypoth]
Computes a two-proportion z test. A summary of results is stored in the stat.results variable. (See page 119.) x1 and x2 are non-negative integers. Test H0: p1 = p2, against one of the following: For Ha: p1 > p2, set Hypoth>0 For Ha: p1 p2 (default), set Hypoth=0 For Ha: p < p0, set Hypoth<0 For information on the effect of empty elements in a list, see Empty (void) elements on page 159.
Output variable stat.z stat.PVal stat.1 stat.2 stat. stat.n1, stat.n2
Description Standard normal value computed for the difference of proportions Smallest level of significance at which the null hypothesis can be rejected First sample proportion estimate Second sample proportion estimate Pooled sample proportion estimate Number of samples taken in trials 1 and 2

zTest_2Samp

s1,s2 ,List1,List2[,Freq1[,Freq2[,Hypoth]]] s1,s2,v1,n1,v2,n2[,Hypoth]
(Summary stats input) Computes a two-sample z test. A summary of results is stored in the stat.results variable. (See page 119.) Test H0: m1 = m2, against one of the following: For Ha: m1 < m2, set Hypoth<0 For Ha: m1 m2 (default), set Hypoth=0 For Ha: m1 > m2, Hypoth>0 For information on the effect of empty elements in a list, see Empty (void) elements on page 159.

[text] processes text as a comment line, allowing you to annotate

/t keys

functions and programs that you create.
can be at the beginning or anywhere in the line. Everything to the right of , to the end of the line, is the comment. Note for entering the example: In the Calculator application

0b, 0h

0b binaryNumber 0h hexadecimalNumber

0B keys, 0H keys

Denotes a binary or hexadecimal number, respectively. To enter a binary or hex number, you must enter the 0b or 0h prefix regardless of In Bin base mode: the Base mode. Without a prefix, a number is treated as decimal (base 10). Results are displayed according to the Base mode. In Hex base mode:
When analyzing real-world data, you might not always have a complete data set. TI-Nspire CAS Software allows empty, or void, data elements so you can proceed with the nearly complete data rather than having to start over or discard the incomplete cases. You can find an example of data involving empty elements in the Lists & Spreadsheet chapter, under Graphing spreadsheet data. The delVoid() function lets you remove empty elements from a list. The isVoid() function lets you test for an empty element. For details, see delVoid(), page 35, and isVoid(), page 61. Note: To 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 functions and commands ignore (skip) void elements found in list arguments. count, countIf, cumulativeSum, freqTable4list, frequency, max, mean, median, product, stDevPop, stDevSamp, sum, sumIf, varPop, and varSamp, as well as regression calculations, OneVar, TwoVar, and FiveNumSummary statistics, confidence intervals, and stat tests
SortA and SortD move all void elements within the first argument to the bottom.
List arguments containing void elements(continued) In regressions, a void in an X or Y list 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.
Shortcuts for entering math expressions
Shortcuts let you enter elements of math expressions by typing instead of using the Catalog or Symbol Palette. For example, to enter the expression 6, you can type sqrt(6) on the entry line. When you press , the expression sqrt(6) is changed to 6. Some shortrcuts are useful from both the handheld and the computer keyboard. Others are useful primarily from the computer keyboard.

The indirection operator (#) converts a string to a variable or function name. For example, #(x&y&z) creates the variable name xyz. Indirection also allows the creation and modification of variables from inside a program. For example, 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 6015' 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 exponentiation (.^) are evaluated from right to left. For example, the expression 2^3^2 is evaluated the same as 2^(3^2) to produce 512. This is different from (2^3)^2, which is 64.

Negation

To enter a negative number, press v followed by the number. Post operations and exponentiation are performed before negation. For example, the result of Lx2 is a negative number, and L92 = L81. Use parentheses to square a negative number such as (L9)2 to produce 81.

Constraint (|)

The argument following the with (|) operator provides a set of constraints that affect the evaluation of the argument preceding the with operator.

Error codes and messages

When an error occurs, its code is assigned to variable errCode. User-defined programs and functions can examine errCode to determine the cause of an error. For an example of using errCode, See Example 2 under the Try command, page 129. Note: Some error conditions apply only to TI-Nspire CAS products, and some apply only to TI-Nspire products.
Error code Description A function did not return a value A test did not resolve to TRUE or FALSE. Generally, undefined variables cannot be compared. For example, the test If a<b will cause this error if either a or b is undefined when the If statement is executed. Argument cannot be a folder name. Argument error Argument mismatch Two or more arguments must be of the same type. Argument must be a Boolean expression or integer Argument must be a decimal number Argument must be a list Argument must be a matrix Argument must be a string Argument must be a variable name. Make sure that the name: does not begin with a digit does not contain spaces or special characters does not use underscore or period in invalid manner does not exceed the length limitations See the Calculator section in the documentation for more details. Argument must be an expression Batteries too low for sending or receiving Install new batteries before sending or receiving. Bound The lower bound must be less than the upper bound to define the search interval. Break The 190

130 140

160 165
d or c key was pressed during a long calculation or during program execution.

Circular definition This message is displayed to avoid running out of memory during infinite replacement of variable values during simplification. For example, a+1->a, where a is an undefined variable, will cause this error. Constraint expression invalid For example, solve(3x^2-4=0,x) | x<0 or x>5 would produce this error message because the constraint is separated by or instead of and. Invalid Data type An argument is of the wrong data type. Dependent limit

Error code 230

Description Dimension A list or matrix index is not valid. For example, if the list {1,2,3,4} is stored in L1, then L1[5] is a dimension error because L1 only contains four elements. Dimension Error. Not enough elements in the lists. Dimension mismatch Two or more arguments must be of the same dimension. For example, [1,2]+[1,2,3] is a dimension mismatch because the matrices contain a different number of elements. Divide by zero Domain error An argument must be in a specified domain. For example, rand(0) is not valid. Duplicate variable name Else and ElseIf invalid outside of If.EndIf block EndTry is missing the matching Else statement Excessive iteration Expected 2 or 3-element list or matrix The first argument of nSolve must be an equation in a single variable. It cannot contain a non-valued variable other than the variable of interest. First argument of solve or cSolve must be an equation or inequality For example, solve(3x^2-4,x) is invalid because the first argument is not an equation. Inconsistent units Index out of range Indirection string is not a valid variable name Undefined Ans Either the previous calculation did not create Ans, or no previous calculation was entered. Invalid assignment Invalid assignment value Invalid command Invalid for the current mode settings Invalid guess Invalid implied multiply For example, x(x+1) is invalid; whereas, x*(x+1) is the correct syntax. This is to avoid confusion between implied multiplication and function calls. Invalid in a function or current expression Only certain commands are valid in a user-defined function. Invalid in Try.EndTry block Invalid list or matrix Invalid outside function or program A number of commands are not valid outside a function or program. For example, Local cannot be used unless it is in a function or program. Invalid outside Loop.EndLoop, For.EndFor, or While.EndWhile blocks For example, the Exit command is valid only inside these loop blocks. Invalid outside program

235 240

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435 440

label, Lbl 62 language get language information 53 Lbl, label 62 lcm, least common multiple 62 least common multiple, lcm 62 left, left( ) 62 left( ), left 62 length of string 37 less than or equal, { 148 less than, 147 LibPriv 34 LibPub 35 library create shortcuts to objects 63 libShortcut( ), create shortcuts to library objects 63 limit lim( ) 63 limit( ) 63 template for 6 limit( ) or lim( ), limit 63 linear regression, LinRegAx 64 linear regression, LinRegBx 64, 65 LinRegBx, linear regression 64 LinRegMx, linear regression 64 LinRegtIntervals, linear regression 65
LinRegtTest 66 linSolve() 67 list to matrix, list4mat( ) 68 list, conditionally count items in 26 list, count items in 25 list4mat( ), list to matrix 68 lists augment/concatenate, augment( ) 12 cross product, crossP( ) 26 cumulative sum, cumulativeSum( ) 30 difference, @list( ) 67 differences in a list, @list( ) 67 dot product, dotP( ) 39 empty elements in 159 expression to list, exp4list( ) 44 list to matrix, list4mat( ) 68 matrix to list, mat4list( ) 74 maximum, max( ) 75 mid-string, mid( ) 76 minimum, min( ) 77 new, newList( ) 81 product, product( ) 92 sort ascending, SortA 117 sort descending, SortD 117 summation, sum( ) 122 ln( ), natural logarithm 68 LnReg, logarithmic regression 69 local variable, Local 70 local, Local 70 Local, local variable 70 Lock, lock variable or variable group 70 locking variables and variable groups 70 Log template for 2 logarithmic regression, LnReg 69 logarithms 68 logistic regression, Logistic 72 logistic regression, LogisticD 72 Logistic, logistic regression 72 LogisticD, logistic regression 72 Loop, loop 73 loop, Loop 73 LU, matrix lower-upper decomposition 74
mat4list( ), matrix to list 74 matrices augment/concatenate, augment( ) 12 column dimension, colDim( ) 19 column norm, colNorm( ) 19 cumulative sum, cumulativeSum( ) 30 determinant, det( ) 37 diagonal, diag( ) 37 dimension, dim( ) 37 dot addition,.+ 145 dot division,.P 146 dot multiplication,.* 145 dot power,.^ 146 dot subtraction,.N 145 eigenvalue, eigVl( ) 41 eigenvector, eigVc( ) 40 filling, Fill 47 identity, identity( ) 56 list to matrix, list4mat( ) 68 lower-upper decomposition, LU 74 matrix to list, mat4list( ) 74 maximum, max( ) 75 minimum, min( ) 77 new, newMat( ) 81 product, product( ) 92 QR factorization, QR 93 random, randMat( ) 97 reduced row echelon form, rref( ) 104 row addition, rowAdd( ) 104 row dimension, rowDim( ) 104 row echelon form, ref( ) 99 row multiplication and addition, mRowAdd( ) 78 row norm, rowNorm( ) 104 row operation, mRow( ) 78 row swap, rowSwap( ) 104 submatrix, subMat( ) 121, 122 summation, sum( ) 122 transpose, T 123 matrix (1 Q 2) template for 4 matrix (2 Q 1)