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Gridmap plots, 12-42 Pr-Surface plots, 12-43 The VPAR variable, 12-44 Interactive drawing, 12-44 DOT+ and DOT-, 12-45 MARK, 12-46 LINE, 12-46 TLINE, 12-46 BOX, 12-47 CIRCL, 12-47 LABEL, 12-47 DEL, 12-47 ERASE, 12-48 MENU, 12-48 SUB, 12-48 REPL, 12-48 PICT , 12-48 X,Y , 12-48 Zooming in and out in the graphics display, 12-49 ZFACT, ZIN, ZOUT, and ZLAST, 12-49 BOXZ, 12-50 ZDFLT, ZAUTO, 12-50 HZIN, HZOUT, VZIN, and VZOUT, 12-50 CNTR, 12-50 ZDECI, 12-50 ZINTG, 12-51 ZSQR, 12-51 ZTRIG, 12-51 The SYMBOLIC menu and graphs, 12-51 The SYMB/GRAPH menu, 12-52 Function DRAW3DMATRIX, 12-54
Chapter 13 - Calculus Applications, 13-1
The CALC (Calculus) menu, 13-1 Limits and derivatives, 13-1 Function lim, 13-2

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Derivatives, 13-3 Function DERIV and DERVX,13-3 The DERIV&INTEG menu, 13-3 Calculating derivatives with ,13-4 The chain rule,13-6 Derivatives of equations,13-6 Implicit derivatives,13-7 Application of derivatives,13-7 Analyzing graphics of functions,13-7 Function DOMAIN, 13-9 Function TABVAL, 13-9 Function SIGNTAB, 13-10 Function TABVAR, 13-10 Using derivatives to calculate extreme points, 13-12 Higher-order derivatives, 13-13 Anti-derivatives and integrals, ,13-14 Functions INT, INTVX, RISCH, SIGMA, and SIGMAVX,13-14 Definite integrals,13-15 Step-by-step evaluation of derivatives and integrals,13-16 Integrating an equation, 13-18 Techniques of integration, 13-18 Substitution or change of variables, 13-18 Integration by parts and differentials,13-19 Integration by partial fractions,13-20 Improper integrals,13-21 Integration with units, 13-21 Infinite series,13-23 Taylor and Maclaurins series,13-23 Taylor polynomial and remainder,13-23 Functions TAYLR, TAYRL0, and SERIES,13-24
Chapter 14 - Multi-variate Calculus Applications, 14-1
Multi-variate functions, 14-1 Partial derivatives, 14-1 Higher-order derivatives, 14-3 The chain rule for partial derivatives, 14-4

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Total differential of a function z = z(x,y) , 14-5 Determining extrema in functions of two variables, 14-5 Using function HESS to analyze extrema, 14-6 Multiple integrals, 14-8 Jacobian of coordinate transformation, 14-9 Double integral in polar coordinates, 14-9
Chapter 15 - Vector Analysis Applications, 15-1
Definitions, 15-1 Gradient and directional derivative, 15-1 A program to calculate the gradient, 15-2 Using function HESS to obtain the gradient, 15-2 Potential of a gradient, 15-3 Divergence, 15-4 Laplacian, 15-4 Curl, 15-5 Irrotational fields and potential function, 15-5 Vector potential, 15-6

Discrete probability distributions, 17-4 Binomial distribution, 17-4 Poisson distribution, 17-5 Continuous probability distributions, 17-6 The gamma distribution, 17-6 The exponential distribution, 17-7 The beta distribution, 17-7 The Weibull distribution, 17-7 Functions for continuous distributions, 17-7 Continuous distributions for statistical inference, 17-9 Normal distribution pdf, 17-9 Normal distribution cdf, 17-10 The Student-t distribution, 17-10 The Chi-square distribution, 17-11 The F distribution, 17-12 Inverse cumulative distribution functions, 17-13
Chapter 18 - Statistical Applications, 18-1
Pre-programmed statistical features, 18-1 Entering data, 18-1 Calculating single-variable statistics, 18-2 Obtaining frequency distributions, 18-5 Fitting data to a function y = f(x) , 18-10 Obtaining additional summary statistics, 18-13 Calculation of percentiles, 18-14 The STAT soft menu, 18-15 The DATA sub-menu, 18-15 The PAR sub-menu, 18-16 The 1VAR sub-menu, 18-16 The PLOT sub-menu, 18-17 The FIT sub-menu, 18-18 Example of STAT menu operations, 18-18 Confidence intervals, 18-22 Estimation of Confidence Intervals, 18-23 Definitions, 18-23 Confidence intervals for the population mean when the

Page TOC-16

population variance is known, 18-23 Confidence intervals for the population mean when the population variance is unknown, 18-24 Confidence interval for a proportion, 18-24 Sampling distributions of differences and sums of statistics, 18-25 Confidence intervals for sums and differences of mean values, 18-26 Determining confidence intervals, 18-27 Confidence intervals for the variance, 18-33 Hypothesis testing, 18-34 Procedure for testing hypotheses, 18-35 Errors in hypothesis testing, 18-35 Inferences concerning one mean, 18-36 Inferences concerning two means, 18-39 Paired sample tests, 18-40 Inferences concerning one proportion, 18-41 Testing the difference between two proportions, 18-42 Hypothesis testing using pre-programmed features, 18-43 Inferences concerning one variance, 18-47 Inferences concerning two variances, 18-48 Additional notes on linear regression, 18-49 The method of least squares, 18-49 Additional equations for linear regression, 18-51 Prediction error, 18-51 Confidence intervals and hypothesis testing in linear regression, 18-52 Procedure for inference statistics for linear regression using the calculator, 18-53 Multiple linear fitting, 18-56 Polynomial fitting, 18-58 Selecting the best fitting, 18-62

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To use function ZEROS in RPN mode, enter first the polynomial expression, then the variable to solve for, and then function ZEROS. The following screen shots show the RPN stack before and after the application of ZEROS to the two examples above:
The Symbolic Solver functions presented above produce solutions to rational equations (mainly, polynomial equations). If the equation to be solved for has all numerical coefficients, a numerical solution is possible through the use of the Numerical Solver features of the calculator.

Numerical solver menu

The calculator provides a very powerful environment for the solution of single algebraic or transcendental equations. To access this environment we start the numerical solver (NUM.SLV) by using. This produces a dropdown menu that includes the following options:
Item 2. Solve diff eq. is to be discussed in a later chapter on differential equations. Item 4. Solve lin sys. will be discussed in a later Chapter on matrices. Item 6. MSLV (Multiple equation SoLVer) will be presented in the next chapter. Following, we present applications of items 3. Solve poly., 5. Solve finance, and 1. Solve equation., in that order. Appendix 1-A, at the end of Chapter 1, contains instructions on how to use input forms with examples for the numerical solver applications.

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Notes: 1. Whenever you solve for a value in the NUM.SLV applications, the value solved for will be placed in the stack. This is useful if you need to keep that value available for other operations. 2. There will be one or more variables created whenever you activate some of the applications in the NUM.SLV menu.

Polynomial Equations

Using the Solve poly option in the calculators SOLVE environment you can: (1) find the solutions to a polynomial equation; (2) obtain the coefficients of the polynomial having a number of given roots; (3) obtain an algebraic expression for the polynomial as a function of X. Finding the solutions to a polynomial equation A polynomial equation is an equation of the form: anxn + an-1xn-1 + + a1x + a0 = 0. The fundamental theorem of algebra indicates that there are n solutions to any polynomial equation of order n. Some of the solutions could be complex numbers, nevertheless. As an example, solve the equation: 3s4 + 2s3 - s + 1 = 0. We want to place the coefficients of the equation in a vector [an,an-1,a1 a0]. For this example, let's use the vector [3,2,0,-1,1]. To solve for this polynomial equation using the calculator, try the following: @@OK@@ Select Solve poly 1\1@@OK@@ Enter vector of coefficients @SOLVE@ Solve equation The screen will show the solution as follows:

A = [aij ] nm

a11 a = 21 M a n1

a12 a 22 M an2

L a1m L a2m . O L a nm
A matrix is square if m = n. The transpose of a matrix is constructed by swapping rows for columns and vice versa. Thus, the transpose of matrix A, is AT = [(aT)ij] mn = [aji]mn. The main diagonal of a square matrix is the collection of elements aii. An identity matrix, Inn, is a square matrix whose main diagonal elements are all equal to 1, and all off-diagonal elements are zero. For example, a 33 identity matrix is written as

0 I = 0 1

An identity matrix can be written as Inn = [ij], where ij is a function known as Kroneckers delta, and defined as
1, if i = j ij = . 0, if i j

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Entering matrices in the stack
In this section we present two different methods to enter matrices in the calculator stack: (1) using the Matrix Writer, and (2) typing the matrix directly into the stack.

Using the Matrix Writer

As with the case of vectors, discussed in Chapter 9, matrices can be entered into the stack by using the Matrix Writer. For example, to enter the matrix:
2.5 4.2 2.0 0.3 1.9 2.8, 2 0.1 0.5
first, start the matrix writer by using. Make sure that the option @GO is selected. Then use the following keystrokes: 2.5\` 4.2` 2`.3` 1.9` 2.8 ` 2`.1\`.5` At this point, the Matrix Writer screen may look like this:
Press ` once more to place the matrix on the stack. The ALG mode stack is shown next, before and after pressing , once more:

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If you have selected the textbook display option (using H@)DISP! and checking off Textbook), the matrix will look like the one shown above. Otherwise, the display will show:
The display in RPN mode will look very similar to these. Note: Details on the use of the matrix writer were presented in Chapter 9.
Typing in the matrix directly into the stack
The same result as above can be achieved by entering the following directly into the stack: 2.5\ 4. .3 1.9 2.8 2 .1\ .5 Thus, to enter a matrix directly into the stack open a set of brackets () and enclose each row of the matrix with an additional set of brackets (). Commas (.) should separate the elements of each row, as well as the brackets between rows. (Note: In RPN mode, you can omit the inner brackets after the first set has been entered, thus, instead of typing, for example, [[3] [6] [9]], type [[3] 8 9].) For future exercises, lets save this matrix under the name A. In ALG mode use K~a. In RPN mode, use ~a K.

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Inverse functions and their graphs
Let y = f(x), if we can find a function y = g(x), such that, g(f(x)) = x, then we say that g(x) is the inverse function of f(x). Typically, the notation g(x) = f -1(x) is used to denote an inverse function. Using this notation we can write: if y = f(x), then x = f -1(y). Also, f(f -1(x)) = x, and f -1(f(x)) = x. As indicated earlier, the ln(x) and exp(x) functions are inverse of each other, i.e., ln(exp(x)) = x, and exp(ln(x)) = x. This can be verified in the calculator by typing and evaluating the following expressions in the Equation Writer: LN(EXP(X)) and EXP(LN(X)). They should both evaluate to X. When a function f(x) and its inverse f -1(x) are plotted simultaneously in the same set of axes, their graphs are reflections of each other about the line y = x. Lets check this fact with the calculator for the functions LN(X) and EXP(X) by following this procedure: Press, simultaneously if in RPN mode,. The function Y1(X) = EXP(X) should be available in the PLOT - FUNCTION window from the previous exercise. Press @@ADD@! , and type the function Y2(X) = LN(X). Also, load the function Y3(X) = X. Press L@@@OK@@@ to return to normal calculator display. Press, simultaneously if in RPN mode, range to read: H-View: -, and change the H-View
Press @AUTO to generate the vertical range. Press @ERASE @DRAW to produce the graph of y = ln(x), y = exp(x), and y =x, simultaneously if in RPN mode. You will notice that only the graph of y = exp(x) is clearly visible. Something went wrong with the @AUTO selection of the vertical range. What happens is that, when you press @AUTO in the PLOT FUNCTION WINDOW screen, the calculator produces the vertical range corresponding to the first function in the list of functions to be plotted. Which, in this case, happens to be Y1(X) = EXP(X). We will have to enter the vertical range ourselves in order to display the other two functions in the same plot.

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Press @CANCL to return to the PLOT FUNCTION WINDOW screen. Modify the vertical and horizontal ranges to read: H-View: -8 8, V-View: -By selecting these ranges we ensure that the scale of the graph is kept 1 vertical to 1 horizontal. Press @ERASE @DRAW and you will get the plots of the natural logarithm, exponential, and y = x functions. It will be evident from the graph that LN(X) and EXP(X) are reflections of each other about the line y = X. Press @CANCL to return to the PLOT WINDOW FUNCTION. Press ` to return to normal calculator display.

The first four options in this menu are actually sub-menus that apply to (1) derivatives and integrals, (2) limits and power series, (3) differential equations, and (4) graphics. The functions in entries (1) and (2) will be presented in this Chapter. Differential equations, the subject of item (3), are presented in Chapter 16. Graphic functions, the subject of item (4), were presented at the end of Chapter 12. Finally, entries 5. DERVX and 6.INTVX are the functions to obtain a derivative and a indefinite integral for a function of the default CAS variable (typically, X). Functions DERVX and INTVX are discussed in detail later.

Limits and derivatives

Differential calculus deals with derivatives, or rates of change, of functions and their applications in mathematical analysis. The derivative of a function is defined as a limit of the difference of a function as the increment in the independent variable tends to zero. Limits are used also to check the continuity of functions.

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Function lim
The calculator provides function lim to calculate limits of functions. This function uses as input an expression representing a function and the value where the limit is to be calculated. Function lim is available through the command catalog (N~l) or through option 2. LIMITS & SERIES of the CALC menu (see above). Note: The functions available in the LIMITS & SERIES menu are shown next:
Function DIVPC is used to divide two polynomials producing a series expansion. Functions DIVPC, SERIES, TAYLOR0, and TAYLOR are used in series expansions of functions and discussed in more detail in this Chapter. Function lim is entered in ALG mode as lim(f(x),x=a) to calculate the limit lim f ( x). In RPN mode, enter the function first, then the expression
x=a, and finally function lim. Examples in ALG mode are shown next, including some limits to infinity. The keystrokes for the first example are as follows (using Algebraic mode, and system flag 117 set to CHOOSE boxes): 2 @@OK@@ 2 @@OK@@ x+1 x 1`
The infinity symbol is associated with the 0 key, i.e.,.

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Derivatives
The derivative of a function f(x) at x = a is defined as the limit
f ( x + h) f ( x ) df = f ' ( x) = lim h >0 h dx
Some examples of derivatives using this limit are shown in the following screen shots:
Functions DERIV and DERVX
The function DERIV is used to take derivatives in terms of any independent variable, while the function DERVX takes derivatives with respect to the CAS default variable VX (typically X). While function DERVX is available directly in the CALC menu, both functions are available in the DERIV.&INTEG submenu within the CALCL menu ( ). Function DERIV requires a function, say f(t), and an independent variable, say, t, while function DERVX requires only a function of VX. Examples are shown next in ALG mode. Recall that in RPN mode the arguments must be entered before the function is applied.

Calculate n+1 Calculate p+1 Start a loop with j = n, n+1, , p+1. Calculate xj, as a list Convert list to array Add column to matrix Close FOR-NEXT loop Ends second IF clause. Ends first IF clause. Its result is X Convert list y to an array X and y used by program MTREG Convert to decimal format Close sub-program 2 Close sub-program 1 Close main program
Save it into a variable called POLY (POLYnomial fitting).

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As an example, use the following data to obtain a polynomial fitting with p = 2, 3, 4, 5, 6. x y 2.30 179.72 3.20 562.30 4.50 1969.11 1.65 65.87 9.32 31220.89 1.18 32.81 6.24 6731.48 3.45 737.41 9.89 39248.46 1.22 33.45 Because we will be using the same x-y data for fitting polynomials of different orders, it is advisable to save the lists of data values x and y into variables xx and yy, respectively. This way, we will not have to type them all over again in each application of the program POLY. Thus, proceed as follows: { 2.3 3.2 4.5 1.65 9.32 1.18 6.24 3.45 9.89 1.22 } ` xx K {179.72 562.30 1969.11 65.87 31220.89 32.81 6731.48 737.41 39248.46 33.45} ` yy K To fit @@xx@@ i.e., @@xx@@ i.e., @@xx@@ i.e., @@xx@@ i.e., @@xx@@ i.e., the data to polynomials use the following: @@yy@@ 2 @POLY, Result: [4527.73 -3958.52 742.23] y = 4527.73-39.58x+742.23x2 @@yy@@ 3 @POLY, Result: [ 998.05 1303.21 -505.27 79.23] y = -998.05+1303.21x-505.27x2+79.23x3 @@yy@@ 4 @POLY, Result: [20.92 2.61 1.52 6.05 3.51 ] y = 20.92-2.61x-1.52x2+6.05x3+3.51x4. @@yy@@ 5 @POLY, Result: [19.08 0.18 2.94 6.36 3.48 0.00 ] y = 19.08+0.18x-2.94x2+6.36x3+3.48x4+0.0011x5 @@yy@@ 6 @POLY, Result: [-16.73 67.17 48.69 21.11 1.07 0.19 0.00] y = -16.73+67.17x-48.69x2+21.11x3+1.07x4+0.19x5+0.0058x6

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Selecting the best fitting
As you can see from the results above, you can fit any polynomial to a set of data. The question arises, which is the best fitting for the data? To help one decide on the best fitting we can use several criteria: The correlation coefficient, r. This value is constrained to the range 1 < r < 1. The closer r is to +1 or 1, the better the data fitting. The sum of squared errors, SSE. This is the quantity that is to be minimized by least-square approach. A plot of residuals. This is a plot of the error corresponding to each of the original data points. If these errors are completely random, the residuals plot should show no particular trend.

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@)LIST@ @)PROC@ REVLIST SORT SEQ @)MODES @)ANGLE@ DEG RAD @)MODES @)MENU@ CST MENU BEEP @)@@IN@@
@)LIST@ @)PROC@ @REVLI@ @)LIST@ @)PROC@ L @SORT@ @)LIST@ @)PROC@ L @@SEQ@@ L@)MODES @)ANGLE@ @@DEG@@ L@)MODES @)ANGLE@ @@RAD@@ L@)MODES @)MENU@ @@CST@@ L@)MODES @)MENU@ @@MENU@ L@)MODES @)MISC@ @@BEEP@ L@)@@IN@@ L@)@@IN@@ L@)@OUT@ L@)@OUT@ LL LL LL LL LL @INFOR@ @INPUT@ @MSGBO@ @PVIEW@ @)@RUN@ @)@RUN@ @)@RUN@ @)@RUN@ @)@RUN@ @@DBG@ @@SST@ @SST@ @HALT@ @KILL
INFORM INPUT MSGBOX PVIEW DBUG SST SST HALT KILL

@)@RUN@

Programs for generating lists of numbers
Please notice that the functions in the PRG menu are not the only functions that can be used in programming. As a matter of fact, almost all functions in the calculator can be included in a program. Thus, you can use, for example, functions from the MTH menu. Specifically, you can use functions for list operations such as SORT, LIST, etc., available through the MTH/LIST menu.

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As additional programming exercises, and to try the keystroke sequences listed above, we present herein three programs for creating or manipulating lists. The program names and listings are as follows: LISC: n x 1 n FOR j x NEXT n CRLST: st en df st en FOR LIST
j j df STEP en st - df / FLOOR 1 +
SWAP FOR j DUP LIST SWAP TAIL NEXT 1 GET n PURGE
CLIST: REVLIST DUP DUP SIZE 'n' STO LIST SWAP TAIL DUP SIZE 1 - 1

LIST REVLIST 'n'

The operation of these programs is as follows: (1) LISC: creates a list of n elements all equals to a constant c. Operation: enter n, enter c, press @LISC Example: 5 ` 6.5 ` @LISC creates the list: {6.5 6.5 6.5 6.5 6.5} (2) CRLST: creates a list of numbers from n1 to n2 with increment n, i.e., {n1, n1+n, n1+2n, n1+Nn }, where N=floor((n2-n1)/n)+1. Operation: enter n1, enter n2, enter n, press @CRLST Example:.5 `3.5 `.5 ` @CRLST produces: {0.1.2.3.5} (3) CLIST: creates a list with cumulative sums of the elements, i.e., if the original list is {x1 x2 x3 xN}, then CLIST creates the list:
{x1 , x1 + x2 , x1 + x2 + x3 ,., xi }
Operation: place the original list in level 1, press @CLIST. Example: {5} `@CLIST produces {15}.

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Examples of sequential programming
In general, a program is any sequence of calculator instructions enclosed between the program containers and. Subprograms can be included as part of a program. The examples presented previously in this guide (e.g., in Chapters 3 and 8) 6 can be classified basically into two types: (a) programs generated by defining a function; and, (b) programs that simulate a sequence of stack operations. These two types of programs are described next. The general form of these programs is input process output, therefore, we refer to them as sequential programs.
Programs generated by defining a function

The PLOT menu

Commands for setting up and producing plots are available through the PLOT menu. You can access the PLOT menu by using: 81.01 L@)MODES @)MENU@ @@MENU@.
The menu thus produced provides the user access to a variety of graphics functions. For application in subsequent examples, lets user-define the C (GRAPH) key to provide access to this menu as described below.
User-defined key for the PLOT menu
Enter the following keystrokes to determine whether you have any user-defined keys already stored in your calculator: L@)MODES @)@KEYS@ @@RCLK@. Unless you have user-defined some keys, you should get in return a list containing an S, i.e., {S}. This indicates that the Standard keyboard is the only key definition stored in your calculator.

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To user-define a key you need to add to this list a command or program followed by a reference to the key (see details in Chapter 20). Type the list { S << 81.01 MENU >> 13.0 } in the stack and use function STOKEYS (L@)MODES @)@KEYS@ @@STOK@) to user-define key C as the access to the PLOT menu. Verify that such list was stored in the calculator by using L@)MODES @)@KEYS@ @@RCLK@. Note: We will not work any exercise while presenting the PLOT menu, its functions or sub-menus. This section will be more like a tour of the contents of PLOT as they relate to the different type of graphs available in the calculator. To activate a user defined key you need to press (same as the ~ key) before pressing the key or keystroke combination of interest. To activate the PLOT menu, with the key definition used above, press: C. You will get the following menu (press L to move to second menu)
Description of the PLOT menu
The following diagram shows the menus in PLOT. The number accompanying the different menus and functions in the diagram are used as reference in the subsequent description of those objects.
The soft menu key labeled 3D, STAT, FLAG, PTYPE, and PPAR, produce additional menus, which will be presented in more detail later. At this point we describe the functions directly accessible through soft menu keys for menu number 81.02. These are:

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LABEL (10) The function LABEL is used to label the axes in a plot including the variable names and minimum and maximum values of the axes. The variable names are selected from information contained in the variable PPAR. AUTO (11) The function AUTO (AUTOscale) calculates a display range for the y-axis or for both the x- and y-axes in two-dimensional plots according to the type of plot defined in PPAR. For any of the three-dimensional graphs the function AUTO produces no action. For two-dimensional plots, the following actions are performed by AUTO: FUNCTION: based on the plotting range of x, it samples the function in EQ and determines the minimum and maximum values of y. CONIC: sets the y-axis scale equal to the x-axis scale POLAR: based on the values of the independent variable (typically ), it samples the function in EQ and determines minimum and maximum values of both x and y. PARAMETRIC: produces a similar result as POLAR based on the values of the parameter defining the equations for x and y. TRUTH: produces no action. BAR: the x-axis range is set from 0 to n+1 where n is the number of elements in DAT. The range of values of y is based on the contents of DAT. The minimum and maximum values of y are determined so that the x-axis is always included in the graph. HISTOGRAM: similar to BAR. SCATTER: sets x- and y-axis range based on the contents of the independent and dependent variables from DAT.

Pixel coordinates

The figure below shows the graphic coordinates for the typical (minimum) screen of 13164 pixels. Pixels coordinates are measured from the top left corner of the screen {# 0h # 0h}, which corresponds to user-defined coordinates (xmin, ymax). The maximum coordinates in terms of pixels

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correspond to the lower right corner of the screen {# 82h #3Fh}, which in user-coordinates is the point (xmax, ymin). The coordinates of the two other corners both in pixel as well as in user-defined coordinates are shown in the figure.

Animating graphics

Herein we present a way to produce animation by using the Y-Slice plot type. Suppose that you want to animate the traveling wave, f(X,Y) = 2.5 sin(X-Y). We can treat the X as time in the animation producing plots of f(X,Y) vs. Y for different values of X. To produce this graph use the following: simultaneously. Select Y-Slice for TYPE. 2.5*SIN(X-Y) for EQ. X for INDEP. Press L@@@OK@@@. , simultaneously (in RPN mode). Use the following values:
Press @ERASE @DRAW. Allow some time for the calculator to generate all the needed graphics. When ready, it will show a traveling sinusoidal wave in your screen.

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Animating a collection of graphics
The calculator provides the function ANIMATE to animate a number of graphics that have been placed in the stack. You can generate a graph in the graphics screen by using the commands in the PLOT and PICT menus. To place the generated graph in the stack, use PICT RCL. When you have n graphs in levels n through 1 of the stack, you can simply use the command n ANIMATE to produce an animation made of the graphs you placed in the stack. Example 1 Animating a ripple in a water surface As an example, type in the following program that generates 11 graphics showing a circle centered in the middle of the graphics screen and whose radius increase by a constant value in each subsequent graph. RAD 131 R B 64 R B PDIM XRNG YRNG FOR j ERASE (50., 50.) 5*(j-1) NUM 0 2* NUM ARC PICT RCL NEXT 11 ANIMATE Begin program Set angle units to radians Set PICT to 13164 pixels Set x- and y-ranges to 0-100 Start loop with j = 1. 11 Erase current PICT Centers of circles (50,50) Draw circle center r = 5(j-1) Place current PICT on stack End FOR-NEXT loop Animate End program

Backup objects

Backup objects are used to copy data from your home directory into a memory port. The purpose of backing up objects in memory port is to preserve the contents of the objects for future usage. Backup objects have the following characteristics: Backup objects can only exist in port memory (i.e., you cannot back up an object in the HOME directory, although you can make as many copies of it as you want) You cannot modify the contents of a backup object (you can, however, copy it back to a directory in the HOME directory, modify it there, and back it up again modified) You can store either a single object or an entire directory as a single backup object. You cannot, however, create a backup object out of a number of selected objects in a directory.
When you create a backup object in port memory, the calculator obtains a cyclic redundancy check (CRC) or checksum value based on the binary data contained in the object. This value is stored with the backup object, and is used by the calculator to monitor the integrity of the backup object. When you restore a backup object into the HOME directory, the calculator obtains again the CRC value and compares it to the original value. If a discrepancy is noticed, the calculator warns the user that the restored data may be corrupted.

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Backing up objects in port memory
The operation of backing up an object from user memory into one of the memory ports is similar to the operation of copying a variable from one subdirectory to another (see details in Chapter 2 of the Users Guide). You can, for example, use the File Manager () to copy and delete backup objects as you would do with normal calculator objects. In addition, there are specific commands for manipulating back up objects, as described next.
Backing up and restoring HOME
You can back up the contents of the current HOME directory in a single back up object. This object will contain all variables, key assignments, and alarms currently defined in the HOME directory. You can also restore the contents of your HOME directory from a back up object previously stored in port memory. The instructions for these operations follow. Backing up the HOME directory To back up the current HOME directory using algebraic mode, enter the command: ARCHIVE(:Port_Number: Backup_Name) Here, Port_Number is 0, 1, 2 (or 3, if an SD memory card is available -- see below), and Backup_Name is the name of the backup object that will store the contents of HOME. The : : container is entered by using the keystroke sequence. For example, to back up HOME into HOME1 in Port 1, use:
To back up the HOME directory in RPN mode, use the command: : Port_Number : Backup_Name ` ARCHIVE Restoring the HOME directory To restore the Home directory in algebraic mode use the command:

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y function calculates the x th root of y.
The function is used to enter summations (or the upper case Greek letter sigma). The function is used to calculate derivatives. The function is used to calculate integrals. The LOG function calculates the logarithm of base 10. The ARG function calculates the argument of a complex number. The ENTRY function is used to change entry mode in editing. The NUM.SLV function launches the NUMerical SOLver menu. The TRIG function activates the trigonometric substitution menu. The TIME function activates the time menu. The ALG function activates the algebra menu. The STAT function activates the statistical operations menu. The UNITS function activates the menu for units of measurement. The CMPLX function activates the complex number functions menu. The LIB function activates the library functions. The BASE function activates the numeric base conversion menu. The OFF key turns the calculator off, the NUM key produces a numeric (or floating-point) value of an expression. The key enters a set of double-quotes used for entering text strings. The __ key enters an underscore. The << >> key enters the symbol for a program. The key enters an arrow representing an input in a program. The key enters a return character in programs or text strings. The comma (,) key enters a comma. The arrow keys, when combined with the right-shift key, move the cursor to the farthest character in the direction of the key pressed.

ALPHA characters

The following sketch shows the characters associated with the different calculator keys when the ALPHA ~ is activated. Notice that the ~ function is used mainly to enter the upper-case letters of the English alphabet

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(A through Z). The numbers, mathematical symbols (-, +), decimal point (.), and the space (SPC) are the same as the main functions of these keys. The ~ function produces an asterisk (*) when combined with the times key, i.e., ~*.
Alpha ~ functions of the calculators keyboard
Alpha-left-shift characters
The following sketch shows the characters associated with the different calculator keys when the ALPHA ~ is combined with the left-shift key.

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Notice that the ~ combination is used mainly to enter the lower-case letters of the English alphabet (A through Z). The numbers, mathematical symbols (-, +, ), decimal point (.), and the space (SPC) are the same as the main functions of these keys. The ENTER and CONT keys also work as their main function even when the ~ combination is used.

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Alpha-right-shift characters
The following sketch shows the characters associated with the different calculator keys when the ALPHA ~ is combined with the right-shift key.

NDIST, 17-10

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NEG, 4-6 Nested IF.THEN.ELSE.END, 21-49 NEW, 2-33 NEXt eQuation, 12-6 NEXTPRIME, 5-11 Non-CAS commands, C-13 Non-linear differential equations, 16-4 Non-verbose CAS mode, C-7 NORM menu, 11-6 Normal distribution, 17-10 Normal distribution standard, 17-17 Normal distribution cdf, 17-10 NOT, 19-6 NSUB, 8-12 NUM, 23-1 NUM.SLV, 6-14 NUM.SLV input forms, A-1 Number in bases, 19-1 Number format, 1-17 Numeric CAS mode, C-3 Numeric solver menu, F-3 Numeric vs. symbolic CAS mode, C-3 Numerical solution of ODEs, 16-60 Numerical solution to stiff ODEs, 16-68 Numerical solver, 6-5 NUMX, 22-10 NUMY, 22-10
Objects, 2-1 objects, 24-1 OCT, 19-2 Octal numbers, 3-2 ODEs Laplace transform applications, 16-17 ODEs Fourier series, 16-46 ODEs Graphical solution, 16-60 ODEs Numerical solution, 16-60 ODEs (ordinary differential equations), 16-1 ODETYPE, 16-8 OFF, 1-2 ON, 1-2 OPER menu, 11-14 Operating mode, 1-13 Operations with units, 3-25 Operators, 3-7 OR, 19-5 ORDER, 2-33 Organizing data, 2-32 Orthogonal matrices, 11-49 Other characters, D-3 Output tagging, 21-33
PA2B2, 5-11 Paired sample tests, 18-40 Parametric plots, 12-23 PARTFRAC, 5-5 Partial derivatives, 14-1
Partial derivatives higher-order 14-3

OBJ-->, 9-19

Partial derivatives chain rule 14-4 Partial fractions integration, 13-20 Partial pivoting, 11-33 PASTE, 2-26

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PCAR, 11-44 PCOEF, 5-11, 5-22 PDIM, 22-20 Percentiles, 18-14, PERIOD, 2-35 16-35 PERM, 17-2 Permutation matrix, 11-34 Permutations, 17-1 PEVAL, 5-23 PGDIR, 2-43 Physical constants, 3-28 PICT, 12-8 Pivoting, 11-33 PIX?, 22-22 Pixel coordinates, 22-25 Pixel references, 19-7 PIXOFF, 22-22 PIXON, 22-22 Plane in space, 9-18 PLOT, 12-52 PLOT environment, 12-3 Plot functions menu, F-1 PLOT menu, 22-1 PLOT menu interactive plots, 22-17 PLOT menu (menu 81), G-3 PLOT operations, 12-5 Plot setup, 12-52 PLOT SETUP environment, 12-3
PLOT WINDOW environment, 12-4
PLOT/FLAG menu, 22-14 PLOT/STAT menu, 22-11 PLOT/STAT/DATA menu, 22-12 PLOTADD, 12-52 Plots program-generated, 22-17 Poisson distribution, 17-5 Polar coordinate plot, 12-19
Polar coordinates double integrals, 14-9 Polar plot, 12-19 Polar representation, 4-1 POLY sub-menu, 6-30 Polynomial equations, 6-6 Polynomial fitting, 18-56 Polynomials, 5-18 Population, 18-3 POS, 8-11 POTENTIAL, 15-3 Potential function, 15-3 15-6 Potential of a gradient, 15-3 Power units, 3-20 POWEREXPAND, 5-29 POWMOD, 5-12 PPAR, 12-2, 12-11 Prediction error linear regression, 18-51 Pressure units, 3-20 PREVAL, 13-15 PREVPRIME, 5-11 PRG menu, 21-5 PRG menu shortcuts, 21-10 PRG/MODES/KEYS menu, 20-5 PRG/MODES/MENU menu, 20-1 PRIMIT, 2-35 Probability, 17-1 Probability density function, 17-6 Probability distributions discrete 17-4 Probability distributions continuous, 17-6 Probability distributions for statistical inference, 17-9 Probability mass function, 17-4

Pressing the L key will show the original TOOL menu. Another way to recover the TOOL menu is to press the I key (third key from the left in the second row of keys from the top of the keyboard).

Setting time and date

See Chapter 1 in the calculators users guide to learn how to set time and date.
Introducing the calculators keyboard
The figure below shows a diagram of the calculators keyboard with the numbering of its rows and columns. Each key has three, four, or five functions. The main key function correspond to the most prominent label in the key. Also, the green left-shift key, key (8,1), the red right-shift key, key (9,1), and

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the blue ALPHA key, key (7,1), can be combined with some of the other keys to activate the alternative functions shown in the keyboard.
For example, the P key, key(4,4), has the following six functions associated with it: P N Main function, to activate the SYMBolic menu Left-shift function, to activate the MTH (Math) menu Right-shift function, to activate the CATalog function

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~p ~p ~p
ALPHA function, to enter the upper-case letter P ALPHA-Left-Shift function, to enter the lower-case letter p ALPHA-Right-Shift function, to enter the symbol
Of the six functions associated with a key only the first four are shown in the keyboard itself. The figure in next page shows these four labels for the P key. Notice that the color and the position of the labels in the key, namely, SYMB, MTH, CAT and P, indicate which is the main function (SYMB), and which of the other three functions is associated with the left-shift (MTH), right-shift (CAT ), and ~ (P) keys.
For detailed information on the calculator keyboard operation refer to Appendix B in the calculators users guide.
Selecting calculator modes
This section assumes that you are now at least partially familiar with the use of choose and dialog boxes (if you are not, please refer to appendix A in the users guide). Press the H button (second key from the left on the second row of keys from the top) to show the following CALCULATOR MODES input form:

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Press the !!@@OK#@ ( F) soft menu key to return to normal display. Examples of selecting different calculator modes are shown next.

Operating Mode

The calculator offers two operating modes: the Algebraic mode, and the Reverse Polish Notation (RPN) mode. The default mode is the Algebraic mode (as indicated in the figure above), however, users of earlier HP calculators may be more familiar with the RPN mode. To select an operating mode, first open the CALCULATOR MODES input form by pressing the H button. The Operating Mode field will be highlighted. Select the Algebraic or RPN operating mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOS soft menu key ( B). If using the latter approach, use up and down arrow keys, , to select the mode, and press the !!@@OK#@ soft menu key to complete the operation. To illustrate the difference between these two operating modes we will calculate the following expression in both modes:

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Press the !!@@OK#@ soft menu key to complete the selection:
Press the !!@@OK#@ soft menu key return to the calculator display. number now is shown as:
Notice how the number is rounded, not truncated. Thus, the number 123.4567890123456, for this setting, is displayed as 123.457, and not as 123.456 because the digit after 6 is > 5. Scientific format To set this format, start by pressing the H button. Next, use the down arrow key, , to select the option Number format. Press the @CHOOS soft menu key ( B), and select the option Scientific with the arrow down key. Keep the number 3 in front of the Sci. (This number can be changed in the same fashion that we changed the Fixed number of decimals in the example above).

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This result, 1.23E2, is the calculators version of powers-of-ten notation, i.e., 1.235 102. In this, so-called, scientific notation, the number 3 in front of the Sci number format (shown earlier) represents the number of significant figures after the decimal point. Scientific notation always includes one integer figure as shown above. For this case, therefore, the number of significant figures is four. Engineering format The engineering format is very similar to the scientific format, except that the powers of ten are multiples of three. To set this format, start by pressing the H button. Next, use the down arrow key, , to select the option Number format. Press the @CHOOS soft menu key ( B), and Keep the select the option Engineering with the arrow down key. number 3 in front of the Eng. (This number can be changed in the same fashion that we changed the Fixed number of decimals in an earlier example).
Because this number has three figures in the integer part, it is shown with four significative figures and a zero power of ten, while using the Engineering format. For example, the number 0.00256, will be shown as:

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Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with such notation. To replace decimal points for commas, change the FM option in the CALCULATOR MODES input form to commas, as follows (Notice that we have changed the Number Format to Std): Press the H button. Next, use the down arrow key, , once, and the right arrow key, , highlighting the option __FM,. To select commas, press the @ @CHK@@ soft menu key (i.e., the B key). The input form will look as follows:
Press the !!@@OK#@ soft menu key return to the calculator display. number 123.4567890123456, entered earlier, now is shown as:

Angle Measure

Trigonometric functions, for example, require arguments representing plane angles. The calculator provides three different Angle Measure modes for working with angles, namely: Degrees: There are 360 degrees (360o) in a complete circumference. Radians: There are 2 radians (2 r) in a complete circumference.

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p1: ~r* ~rQ2 ~p1` K. The screen, at this point, will look as follows:
Checking variables contents
The simplest way to check a variable content is by pressing the soft menu key label for the variable. For example, for the variables listed above, press the following keys to see the contents of the variables: Algebraic mode Type these keystrokes: J@@z1@@ ` @@@R@@ `@@@Q@@@ `. At this point, the screen looks as follows:
RPN mode In RPN mode, you only need to press the corresponding soft menu key label to get the contents of a numerical or algebraic variable. For the case under consideration, we can try peeking into the variables z1, R, Q, A12, , created above, as follows: J@@z1@@ @@@R@@ @@@Q@@ @@A12@@ @@@@ At this point, the screen looks like this:

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Using the right-shift key followed by soft menu key labels This approach for viewing the contents of a variable works the same in both Algebraic and RPN modes. Try the following examples in either mode: J@@p1@@ @@z1@@ @@@R@@ @@@Q@@ @@A12@@ This produces the following screen (Algebraic mode in the left, RPN in the right)
Notice that this time the contents of program p1 are listed in the screen. To see the remaining variables in this directory, use: @@@@@ L @@@A@@ Listing the contents of all variables in the screen Use the keystroke combination to list the contents of all variables in the screen. For example:
Press $ to return to normal calculator display.

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Deleting variables
The simplest way of deleting variables is by using function PURGE. This function can be accessed directly by using the TOOLS menu (I), or by using the FILES menu @@OK@@.
Using function PURGE in the stack in Algebraic mode
Our variable list contains variables p1, z1, Q, R, and. We will use command PURGE to delete variable p1. Press I @PURGE@ J@@p1@@ `. The screen will now show variable p1 removed:
You can use the PURGE command to erase more than one variable by placing their names in a list in the argument of PURGE. For example, if now we wanted to purge variables R and Q, simultaneously, we can try the following exercise. Press : I @PURGE@ J@@@R!@@ J@@@Q!@@ At this point, the screen will show the following command ready to be executed:
To finish deleting the variables, press `. The screen will now show the remaining variables:

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Using function PURGE in the stack in RPN mode
Assuming that our variable list contains the variables p1, z1, Q, R, and. We will use command PURGE to delete variable p1. Press @@p1@@ ` I @PURGE@. The screen will now show variable p1 removed:
To delete two variables simultaneously, say variables R and Q, first create a list (in RPN mode, the elements of the list need not be separated by commas as in Algebraic mode): J @@@R!@@ @@@Q!@@ ` Then, press I@PURGE@ use to purge the variables. Additional information on variable manipulation is available in Chapter 2 of the calculators users guide.

UNDO and CMD functions

Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made. These functions are associated with the HIST key: UNDO results from the keystroke sequence , while CMD results from the keystroke sequence.
CHOOSE boxes vs. Soft MENU
In some of the exercises presented in this chapter we have seen menu lists of commands displayed in the screen. This menu lists are referred to as CHOOSE boxes. Herein we indicate the way to change from CHOOSE boxes to Soft MENUs, and vice versa, through an exercise. Although not applied to a specific example, the present exercise shows the two options for menus in the calculator (CHOOSE boxes and soft MENUs). In

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this exercise, we use the ORDER command to reorder variables in a directory, we use, in ALG mode: Show PROG menu list and select MEMORY

@@OK@@

Show the MEMORY menu list and select DIRECTORY
Show the DIRECTORY menu list and select ORDER

@@OK@@

activate the ORDER command
There is an alternative way to access these menus as soft MENU keys, by setting system flag 117. (For information on Flags see Chapters 2 and 24 in the calculators users guide). To set this flag try the following: H @FLAGS! The screen shows flag 117 not set (CHOOSE boxes), as shown here:

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Press the @CHECK! soft menu key to set flag 117 to soft MENU. The screen will reflect that change:
Press @@OK@@ twice to return to normal calculator display. Now, well try to find the ORDER command using similar keystrokes to those used above, i.e., we start with. Notice that instead of a menu list, we get soft menu labels with the different options in the PROG menu, i.e.,
Press B to select the MEMORY soft menu ()@@MEM@@). The display now shows:
Press E to select the DIRECTORY soft menu ()@@DIR@@)

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The ORDER command is not shown in this screen. To find it we use the L key to find it:

To activate the ORDER command we press the C(@ORDER) soft menu key.
For additional information on entering and manipulating expressions in the display or in the Equation Writer see Chapter 2 of the calculators users guide. For CAS (Computer Algebraic System) settings, see Appendix C in the calculators users guide. For information on Flags see, Chapter 24 in the calculators users guide.

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Chapter 3 Calculations with real numbers
This chapter demonstrates the use of the calculator for operations and functions related to real numbers. The user should be acquainted with the keyboard to identify certain functions available in the keyboard (e.g., SIN, COS, TAN, etc.). Also, it is assumed that the reader knows how to change the calculators operating system (Chapter 1), use menus and choose boxes (Chapter 1), and operate with variables (Chapter 2).
Examples of real number calculations
To perform real number calculations it is preferred to have the CAS set to Real (as opposed to Complex) mode. Exact mode is the default mode for most operations. Therefore, you may want to start your calculations in this mode. Some operations with real numbers are illustrated next: Use the \ key for changing sign of a number. For example, in ALG mode, \2.5`. In RPN mode, e.g., 2.5\. Use the Ykey to calculate the inverse of a number. For example, in ALG mode, Y2`. In RPN mode use 4`Y. For addition, subtraction, multiplication, division, use the proper operation key, namely, + - * /. Examples in ALG mode: 3.7 6.3 4.2 2.3 Examples in RPN mode: 3.7` 5.2 + + * / 5.2 8.5 2.5 4.5 ` ` ` `

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6.3` 8.5 4.2` 2.5 * 2.3` 4.5 / Alternatively, in RPN mode, you can separate the operands with a space (#) before pressing the operator key. Examples: 3.7#5.2 6.3#8.5 4.2#2.5 2.3#4.5 + * /
Parentheses () can be used to group operations, as well as to enclose arguments of functions. In ALG mode: 5+3.2/72.2` In RPN mode, you do not need the parenthesis, calculation is done directly on the stack: 5`3.2`+7`2.2`-/ In RPN mode, typing the expression between single quotes will allow you to enter the expression like in algebraic mode: 5+3.2/ 7-2.2` For both, ALG and RPN modes, using the Equation Writer: O5+3.2/7-2.2 The expression can be evaluated within the Equation writer, by using @EVAL@ or, @EVAL@
The absolute value function, ABS, is available through. Example in ALG mode:

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\2.32` Example in RPN mode: 2.32\ The square function, SQ, is available through. Example in ALG mode: \2.3` Example in RPN mode: 2.3\ The square root function, , is available through the R key. When calculating in the stack in ALG mode, enter the function before the argument, e.g., R123.4` In RPN mode, enter the number first, then the function, e.g., 123.4R The power function, ^, is available through the Q key. When calculating in the stack in ALG mode, enter the base (y) followed by the Q key, and then the exponent (x), e.g., 5.2Q1.25` In RPN mode, enter the number first, then the function, e.g., 5.2`1.25Q The root function, XROOT(y,x), is available through the keystroke combination. When calculating in the stack in ALG mode,

The user will recognize most of these units (some, e.g., dyne, are not used very often nowadays) from his or her physics classes: N = newtons, dyn = dynes, gf = grams force (to distinguish from gram-mass, or plainly gram, a unit of mass), kip = kilo-poundal (1000 pounds), lbf = pound-force (to distinguish from pound-mass), pdl = poundal. To attach a unit object to a number, the number must be followed by an underscore. Thus, a force of 5 N will be entered as 5_N. For extensive operations with units SOFT menus provide a more convenient way of attaching units. Change system flag 117 to SOFT menus (see Chapter 2), and use the keystroke combination to get the following menus. Press L to move to the next menu page.

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Pressing on the appropriate soft menu key will open the sub-menu of units for that particular selection. For example, for the @)SPEED sub-menu, the following units are available:
Pressing the soft menu key @)UNITS will take you back to the UNITS menu. Recall that you can always list the full menu labels in the screen by using , e.g., for the @)ENRG set of units the following labels will be listed:
Note: Use the L key or the keystroke sequence to navigate through the menus.

Available units

For a complete list of available units see Chapter 3 in the calculators users guide.

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Attaching units to numbers
To attach a unit object to a number, the number must be followed by an underscore (, key(8,5)). Thus, a force of 5 N will be entered as 5_N. Here is the sequence of steps to enter this number in ALG mode, system flag 117 set to CHOOSE boxes: 5 8@@OK@@ @@OK@@ ` Note: If you forget the underscore, the result is the expression 5*N, where N here represents a possible variable name and not Newtons. To enter this same quantity, with the calculator in RPN mode, use the following keystrokes: 58@@OK@@ @@OK@@ Notice that the underscore is entered automatically when the RPN mode is active. The keystroke sequences to enter units when the SOFT menu option is selected, in both ALG and RPN modes, are illustrated next. For example, in ALG mode, to enter the quantity 5_N use: 5 L @)@FORCE @ @@N@@ ` The same quantity, entered in RPN mode uses the following keystrokes: 5L @)@FORCE @ @@N@@ Note: You can enter a quantity with units by typing the underline and units with the ~keyboard, e.g., 5~n will produce the entry: 5_N Unit prefixes You can enter prefixes for units according to the following table of prefixes from the SI system. The prefix abbreviation is shown first, followed by its name, and by the exponent x in the factor 10x corresponding to each prefix:

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Examples of applications of these functions are shown next in RECT coordinates. Recall that, for ALG mode, the function must precede the argument, while in RPN mode, you enter the argument first, and then select the function. Also, recall that you can get these functions as soft menu labels by changing the setting of system flag 117 (See Chapter 2).

CMPLX menu in keyboard

A second CMPLX menu is accessible by using the right-shift option associated with the 1 key, i.e.,. With system flag 117 set to CHOOSE boxes, the keyboard CMPLX menu shows up as the following screens:
The resulting menu include some of the functions already introduced in the previous section, namely, ARG, ABS, CONJ, IM, NEG, RE, and SIGN. It also includes function i which serves the same purpose as the keystroke combination.

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Functions applied to complex numbers
Many of the keyboard-based functions and MTH menu functions defined in Chapter 3 for real numbers (e.g., SQ, ,LN, ex, etc.), can be applied to complex numbers. The result is another complex number, as illustrated in the following examples.
Note: When using trigonometric functions and their inverses with complex numbers the arguments are no longer angles. Therefore, the angular measure selected for the calculator has no bearing in the calculation of these functions with complex arguments.
Function DROITE: equation of a straight line
Function DROITE takes as argument two complex numbers, say, x1+iy1 and x2+iy2, and returns the equation of the straight line, say, y = a+bx, that contains the points (x1,y1) and (x2,y2). For example, the line between points A(5,-3) and B(6,2) can be found as follows (example in Algebraic mode):

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Function DROITE is found in the command catalog (N). If the calculator is in APPROX mode, the result will be Y = 5.*(X-5.)-3.
Additional information on complex number operations is presented in Chapter 4 of the calculators users guide.

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Chapter 5 Algebraic and arithmetic operations
An algebraic object, or simply, algebraic, is any number, variable name or algebraic expression that can be operated upon, manipulated, and combined according to the rules of algebra. Examples of algebraic objects are the following: A number: A variable name: An expression: An equation: 12.3, 15.2_m, , e, i a, ux, width, etc. p*D^2/4,f*(L/D)*(V^2/(2*g)), p*V = n*R*T, Q=(Cu/n)*A(y)*R(y)^(2/3)*So
Entering algebraic objects
Algebraic objects can be created by typing the object between single quotes directly into stack level 1 or by using the equation writer [EQW]. For example, to enter the algebraic object *D^2/4 directly into stack level 1 use: *~dQ2/4` An algebraic object can also be built in the Equation Writer and then sent to the stack, or operated upon in the Equation Writer itself. The operation of the Equation Writer was described in Chapter 2. As an exercise, build the following algebraic object in the Equation Writer:

The PEVAL function

The function PEVAL (Polynomial EVALuation) can be used to evaluate a polynomial p(x) = anxn+an-1x n-1+ + a2x2+a1x+ a0, given an array of coefficients [an, an-1, a2, a1, a0] and a value of x0. The result is the evaluation p(x0). Function PEVAL is not available in the ARITHMETIC menu, instead use the CALC/DERIV&INTEG Menu. Example: PEVAL([1,5,6,1],5) = 281. Additional applications of polynomial functions are presented in Chapter 5 in the calculators users guide.

Fractions

Fractions can be expanded and factored by using functions EXPAND and FACTOR, from the ALG menu (). For example: EXPAND((1+X)^3/((X-1)* (X+3))) = (X^3+3*X^2+3*X+1)/(X^2+2*X-3) EXPAND((X^2*(X+Y)/(2*X-X^2)^2) = (X+Y)/(X^2-4*X+4) FACTOR((3*X^3-2*X^2)/(X^2-5*X+6)) = X^2*(3*X-2)/((X-2)*(X-3)) FACTOR((X^3-9*X)/(X^2-5*X+6) ) = X*(X+3)/(X-2)

The SIMP2 function

Function SIMP2, in the ARITHMETIC menu, takes as arguments two numbers or polynomials, representing the numerator and denominator of a rational fraction, and returns the simplified numerator and denominator. For example: SIMP2(X^3-1,X^2-4*X+3) = { X^2+X+1,X-3}

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The PROPFRAC function
The function PROPFRAC converts a rational fraction into a proper fraction, i.e., an integer part added to a fractional part, if such decomposition is possible. For example: PROPFRAC(5/4) = 1+1/4 PROPFRAC((x^2+1)/x^2) = 1+1/x^2

The PARTFRAC function

The function PARTFRAC decomposes a rational fraction into the partial fractions that produce the original fraction. For example: PARTFRAC((2*X^6-14*X^5+29*X^4-37*X^3+41*X^2-16*X+5)/(X^57*X^4+11*X^3-7*X^2+10*X)) = 2*X+(1/2/(X-2)+5/(X-5)+1/2/X+X/(X^2+1))

The FCOEF function

The function FCOEF, available through the ARITHMETIC/POLYNOMIAL menu, is used to obtain a rational fraction, given the roots and poles of the fraction. Note: If a rational fraction is given as F(X) = N(X)/D(X), the roots of the fraction result from solving the equation N(X) = 0, while the poles result from solving the equation D(X) = 0. The input for the function is a vector listing the roots followed by their multiplicity (i.e., how many times a given root is repeated), and the poles followed by their multiplicity represented as a negative number. For example, if we want to create a fraction having roots 2 with multiplicity 1, 0 with multiplicity 3, and -5 with multiplicity 2, and poles 1 with multiplicity 2 and 3 with multiplicity 5, use: FCOEF([2,1,0,3,5,2,1,-2,-3,-5]) = (X--5)^2*X^3*(X-2)/(X--3)^5*(X-1)^2 If you press (or, simply , in RPN mode) you will get: (X^6+8*X^5+5*X^4-50*X^3)/(X^7+13*X^6+61*X^5+105*X^4-45*X^3297*X62-81*X+243)

Function ISOL

Function ISOL(Equation, variable) will produce the solution(s) to Equation by isolating variable. For example, with the calculator set to ALG mode, to solve for t in the equation at3-bt = 0 we can use the following:

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Using the RPN mode, the solution is accomplished by entering the equation in the stack, followed by the variable, before entering function ISOL. Right before the execution of ISOL, the RPN stack should look as in the figure to the left. After applying ISOL, the result is shown in the figure to the right:
The first argument in ISOL can be an expression, as shown above, or an equation. For example, in ALG mode, try:
Note: To type the equal sign (=) in an equation, use (associated with the \ key). The same problem can be solved in RPN mode as illustrated below (figures show the RPN stack before and after the application of function ISOL):

Function SOLVE

Function SOLVE has the same syntax as function ISOL, except that SOLVE can also be used to solve a set of polynomial equations. The help-facility entry for function SOLVE, with the solution to equation X^= 3 , is shown next:

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The following examples show the use of function SOLVE in ALG and RPN modes (Use Complex mode in the CAS):
The screen shot shown above displays two solutions. In the first one, 4-5 =125, SOLVE produces no solutions { }. In the second one, 4 - 5 = 6, SOLVE produces four solutions, shown in the last output line. The very last solution is not visible because the result occupies more characters than the width of the calculators screen. However, you can still see all the solutions by using the down arrow key (), which triggers the line editor (this operation can be used to access any output line that is wider than the calculators screen):
The corresponding RPN screens for these two examples, before and after the application of function SOLVE, are shown next:

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Function SOLVEVX
The function SOLVEVX solves an equation for the default CAS variable contained in the reserved variable name VX. By default, this variable is set to X. Examples, using the ALG mode with VX = X, are shown below:
In the first case SOLVEVX could not find a solution. In the second case, SOLVEVX found a single solution, X = 2. The following screens show the RPN stack for solving the two examples shown above (before and after application of SOLVEVX):

Function ZEROS

The function ZEROS finds the solutions of a polynomial equation, without showing their multiplicity. The function requires having as input the expression for the equation and the name of the variable to solve for. Examples in ALG mode are shown next:

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To use function ZEROS in RPN mode, enter first the polynomial expression, then the variable to solve for, and then function ZEROS. The following screen shots show the RPN stack before and after the application of ZEROS to the two examples above (Use Complex mode in the CAS)::

SORT and REVLIST can be combined to sort a list in decreasing order:

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The SEQ function
The SEQ function, available through the command catalog (N), takes as arguments an expression in terms of an index, the name of the index, and starting, ending, and increment values for the index, and returns a list consisting of the evaluation of the expression for all possible values of the index. The general form of the function is SEQ(expression, index, start, end, increment) For example:
The list produced corresponds to the values {12, 22, 32, 42}.

The MAP function

The MAP function, available through the command catalog (N), takes as arguments a list of numbers and a function f(X), and produces a list consisting of the application of function f or the program to the list of numbers. For example, the following call to function MAP applies the function SIN(X) to the list {1,2,3}:
For additional references, examples, and applications of lists see Chapter 8 in the calculators users guide.

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Chapter 8 Vectors
This Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as well as physical vectors of 2 and 3 components.

Entering vectors

In the calculator, vectors are represented by a sequence of numbers enclosed between brackets, and typically entered as row vectors. The brackets are generated in the calculator by the keystroke combination , associated with the * key. The following are examples of vectors in the calculator: [3.5, 2.2, -1.3, 5.6, 2.3] [1.5,-2.2] [3,-1,2] ['t','t^2','SIN(t)'] A general row vector A 2-D vector A 3-D vector A vector of algebraics
Typing vectors in the stack
With the calculator in ALG mode, a vector is typed into the stack by opening a set of brackets () and typing the components or elements of the vector separated by commas (). The screen shots below show the entering of a numerical vector followed by an algebraic vector. The figure to the left shows the algebraic vector before pressing `. The figure to the right shows the calculators screen after entering the algebraic vector:

Functions DERIV and DERVX
The function DERIV is used to take derivatives in terms of any independent variable, while the function DERVX takes derivatives with respect to the CAS default variable VX (typically X). While function DERVX is available directly in the CALC menu, both functions are available in the DERIV.&INTEG submenu within the CALCL menu ( ). Function DERIV requires a function, say f(t), and an independent variable, say, t, while function DERVX requires only a function of VX. Examples are shown next in ALG mode. Recall that in RPN mode the arguments must be entered before the function is applied.

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Anti-derivatives and integrals
An anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx. One way to represent an anti-derivative is as a indefinite integral, i.e.,

f ( x)dx = F ( x) + C

if and only if, f(x) = dF/dx, and C = constant.
Functions INT, INTVX, RISCH, SIGMA and SIGMAVX
The calculator provides functions INT, INTVX, RISCH, SIGMA and SIGMAVX to calculate anti-derivatives of functions. Functions INT, RISCH, and SIGMA work with functions of any variable, while functions INTVX, and SIGMAVX utilize functions of the CAS variable VX (typically, x). Functions INT and RISCH require, therefore, not only the expression for the function being integrated, but also the independent variable name. Function INT, requires also a value of x where the anti-derivative will be evaluated. Functions INTVX and SIGMAVX require only the expression of the function to integrate in terms of VX. Functions INTVX, RISCH, SIGMA and SIGMAVX are available in the CALC/DERIV&INTEG menu, while INT is available in the command catalog. Some examples are shown next in ALG mode (type the function names to activate them):

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Please notice that functions SIGMAVX and SIGMA are designed for integrands that involve some sort of integer function like the factorial (!) function shown above. Their result is the so-called discrete derivative, i.e., one defined for integer numbers only.

Fourier series

A complex Fourier series is defined by the following expression

f (t ) =

2int ), T
1 T 2 i n 0 f (t ) exp( T t ) dt , n = ,.,2,1,0,1,2,. T

Function FOURIER

Function FOURIER provides the coefficient cn of the complex-form of the Fourier series given the function f(t) and the value of n. The function FOURIER requires you to store the value of the period (T) of a T-periodic function into the CAS variable PERIOD before calling the function. The function FOURIER is available in the DERIV sub-menu within the CALC menu ().
Fourier series for a quadratic function
Determine the coefficients c0, c1, and c2 for the function g(t) = (t-1)2+(t-1), with period T = 2.

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Using the calculator in ALG mode, first we define functions f(t) and g(t):
Next, we move to the CASDIR sub-directory under HOME to change the value of variable PERIOD, e.g., (hold) `J @)CASDI `2 K @PERIOD `
Return to the sub-directory where you defined functions f and g, and calculate the coefficients. Set CAS to Complex mode (see chapter 2) before trying the exercises. Function COLLECT is available in the ALG menu ().

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c0 = 1/3, c1 = (i+2)/2, c2 = (i+1)/(22).
The Fourier series with three elements will be written as g(t) Re[(1/3) + (i+2)/2exp(it)+ (i+1)/(22)exp(2it)].
For additional definitions, applications, and exercises on solving differential equations, using Laplace transform, and Fourier series and transforms, as well as numerical and graphical methods, see Chapter 16 in the calculators users guide.

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Chapter 15 Probability Distributions
In this Chapter we provide examples of applications of the pre-defined probability distributions in the calculator.
The MTH/PROBABILITY. sub-menu - part 1
The MTH/PROBABILITY. sub-menu is accessible through the keystroke sequence. With system flag 117 set to CHOOSE boxes, the following functions are available in the PROBABILITY. menu:
In this section we discuss functions COMB, PERM, ! (factorial), and RAND.
Factorials, combinations, and permutations
The factorial of an integer n is defined as: n! = n (n-1) (n-2)321. By definition, 0! = 1. Factorials are used in the calculation of the number of permutations and combinations of objects. For example, the number of permutations of r objects from a set of n distinct objects is
Pr = n( n 1)(n 1).( n r + 1) = n! /( n r )!
Also, the number of combinations of n objects taken r at a time is
n n(n 1)(n 2).(n r + 1) n! = = r r! r!(n r )!

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We can calculate combinations, permutations, and factorials with functions COMB, PERM, and ! from the MTH/PROBABILITY. sub-menu. The operation of those functions is described next: COMB(n,r): Calculates the number of combinations of n items taken r at a time PERM(n,r): Calculates the number of permutations of n items taken r at a time n!: Factorial of a positive integer. For a non-integer, x! returns (x+1), where (x) is the Gamma function (see Chapter 3). The factorial symbol (!) can be entered also as the keystroke combination ~2.

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