Press ` once more to keep two copies of the expression available in the stack for evaluation. We first evaluate the expression using the function EVAL, and next using the function NUM:.

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This expression is semi-symbolic in the sense that there are floating-point components to the result, as well as a 3. Next, we switch stack locations [using ] and evaluate using function NUM, i.e.,. This latter result is purely numerical, so that the two results in the stack, although representing the same expression, seem different. To verify that they are not, we subtract the two values and evaluate this difference using function EVAL: -. The result is zero (0.). For additional information on editing arithmetic expressions in the display or stack, see Chapter 2 in the calculators users guide.
Creating algebraic expressions
Algebraic expressions include not only numbers, but also variable names. As an example, we will enter the following algebraic expression:

2L 1 +

x R +2L R+ y b
We set the calculator operating mode to Algebraic, the CAS to Exact, and the display to Textbook. To enter this algebraic expression we use the following keystrokes: 2*~l*R1+~x/~r/ ~r+~y+2*~l/~b Press ` to get the following result:

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Entering this expression when the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise. For additional information on editing algebraic expressions in the calculators display or stack see Chapter 2 in the calculators users guide.
Using the Equation Writer (EQW) to create expressions
The equation writer is an extremely powerful tool that not only let you enter or see an equation, but also allows you to modify and work/apply functions on all or part of the equation. The Equation Writer is launched by pressing the keystroke combination O (the third key in the fourth row from the top in the keyboard). The resulting screen is the following. Press L to see the second menu page:
The six soft menu keys for the Equation Writer activate functions EDIT, CURS, BIG, EVAL, FACTOR, SIMPLIFY, CMDS, and HELP. Detailed information on these functions is provided in Chapter 3 of the calculators users guide.
Entering arithmetic expressions in the Equation Writer is very similar to entering an arithmetic expression in the stack enclosed in quotes. The main difference is that in the Equation Writer the expressions produced are written in textbook style instead of a line-entry style. For example, try the following keystrokes in the Equation Writer screen: 5/5+2 The result is the expression

Page 2-5

Page 2-17

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).

Here are some calculation examples using the ALG operating mode. Be warned that, when multiplying or dividing quantities with units, you must enclosed each quantity with its units between parentheses. Thus, to enter, for example, the product 12.5m 5.2 yd, type it to read (12.5_m)*(5.2_yd) `:

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which shows as 65_(myd). To convert to units of the SI system, use function UBASE (find it using the command catalog, N):
Note: Recall that the ANS(1) variable is available through the keystroke combination (associated with the ` key). To calculate a division, say, 3250 mi / 50 h, enter it as (3250_mi)/(50_h) ` which transformed to SI units, with function UBASE, produces:
Addition and subtraction can be performed, in ALG mode, without using parentheses, e.g., 5 m + 3200 mm, can be entered simply as 5_m + 3200_mm `. More complicated expression require the use of parentheses, e.g., (12_mm)*(1_cm^2)/(2_s) `: Stack calculations in the RPN mode, do not require you to enclose the different terms in parentheses, e.g., 12 @@@m@@@ 1.5 @@yd@@ * 3250 @@mi@@ 50 @@@h@@@ /

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These operations produce the following output:

Unit conversions

The UNITS menu contains a TOOLS sub-menu, which provides the following functions: CONVERT(x,y): UBASE(x): UVAL(x): UFACT(x,y): UNIT(x,y): convert unit object x to units of object y convert unit object x to SI units extract the value from unit object x factors a unit y from unit object x combines value of x with units of y
Examples of function CONVERT are shown below. Examples of the other UNIT/TOOLS functions are available in Chapter 3 of the calculators users guide. For example, to convert 33 watts to btus use either of the following entries: CONVERT(33_W,1_hp) ` CONVERT(33_W,11_hp) `
Physical constants in the calculator
The calculators physical constants are contained in a constants library activated with the command CONLIB. To launch this command you could simply type it in the stack: ~~conlib`, or, you can select the command CONLIB from the command catalog, as follows: First, launch the catalog by using: N~c. Next, use the up and down arrow keys to select CONLIB. Finally, press the F(@@OK@@) soft menu key. Press `, if needed. Use the up and down arrow keys () to navigate through the list of constants in your calculator.

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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.

Page 4-5

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):

Page 4-6

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

We notice that, at the bottom of the screen, the line See: EXPAND FACTOR suggests links to other help facility entries, the functions EXPAND and FACTOR. To move directly to those entries, press the soft menu key @SEE1! for EXPAND, and @SEE2! for FACTOR. Pressing @SEE1!, for example, shows the following information for EXPAND, while @SEE2! shows information for FACTOR:

Page 5-4

Copy the examples provided onto your stack by pressing @ECHO!. For example, for the EXPAND entry shown above, press the @ECHO! soft menu key to get the following example copied to the stack (press ` to execute the command):
Thus, we leave for the user to explore the applications of the functions in the ALG menu. This is a list of the commands:
For example, for function SUBST, we find the following CAS help facility entry:
Note: Recall that, to use these, or any other functions in the RPN mode, you must enter the argument first, and then the function. For example, the example for TEXPAND, in RPN mode will be set up as: +~x+~y` At this point, select function TEXPAND from menu ALG (or directly from the catalog N), to complete the operation.

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Operations with transcendental functions
The calculator offers a number of functions that can be used to replace expressions containing logarithmic and exponential functions (), as well as trigonometric functions ().
Expansion and factoring using log-exp functions
The produces the following menu:
Information and examples on these commands are available in the help facility of the calculator. For example, the description of EXPLN is shown in the left-hand side, and the example from the help facility is shown to the right:
Expansion and factoring using trigonometric functions
The TRIG menu, triggered by using , shows the following functions:

Page 5-6

These functions allow to simplify expressions by replacing some category of trigonometric functions for another one. For example, the function ACOS2S allows to replace the function arccosine (acos(x)) with its expression in terms of arcsine (asin(x)). Description of these commands and examples of their applications are available in the calculators help facility (IL@HELP). The user is invited to explore this facility to find information on the commands in the TRIG menu.
Functions in the ARITHMETIC menu
The ARITHMETIC menu is triggered through the keystroke combination (associated with the 1 key). With system flag 117 set to CHOOSE boxes, shows the following menu:
Out of this menu list, options 5 through 9 (DIVIS, FACTORS, LGCD, PROPFRAC, SIMP2) correspond to common functions that apply to integer numbers or to polynomials. The remaining options (1. INTEGER, 2. POLYNOMIAL, 3. MODULO, and 4. PERMUTATION) are actually submenus of functions that apply to specific mathematical objects. When system flag 117 is set to SOFT menus, the ARITHMETIC menu () produces:

To verify the properties of the inverse matrix, we present the following multiplications:

Page 9-6

Characterizing a matrix (The matrix NORM menu)
The matrix NORM (NORMALIZE) menu is accessed through the keystroke sequence . This menu is described in detail in Chapter 10 of the calculators users guide. Some of these functions are described next.

Function DET

Function DET calculates the determinant of a square matrix. For example,

Function TRACE

Function TRACE calculates the trace of square matrix, defined as the sum of the elements in its main diagonal, or

tr (A ) = aii.

Examples:
Solution of linear systems
A system of n linear equations in m variables can be written as a11x1 + a12x2 + a13x3 + + a21x1 + a22x2 + a23x3 + + a31x1 + a32x2 + a33x3 + +. an-1,1x1 + an-1,2x2 + an-1,3x3 + + an1x1 + an2x2 + an3x3 + + a1,m-1x m-1 a2,m-1x m-1 a3,m-1x m-1. an-1,m-1x m-1 an,m-1x m-1 + a1,mx m + a2,mx m + a3,mx m. + an-1,mx m + an,mx m = b1, = b2, = b3,. = bn-1, = bn.

Page 9-7

This system of linear equations can be written as a matrix equation, Anmxm1 = bn1, if we define the following matrix and vectors:

a11 a A = 21 M an1

a12 a22 M an 2
L a1m x1 b1 x b L a2 m , x= , b= M M O M L anm nm xm m1 bn n1
Using the numerical solver for linear systems
There are many ways to solve a system of linear equations with the calculator. One possibility is through the numerical solver. From the numerical solver screen, shown below (left), select the option 4. Solve lin sys., and press @@@OK@@@. The following input form will be provide (right):
To solve the linear system Ax = b, enter the matrix A, in the format [[ a11, a12, ], [.]] in the A: field. Also, enter the vector b in the B: field. When the X: field is highlighted, press @SOLVE. If a solution is available, the solution vector x will be shown in the X: field. The solution is also copied to stack level 1. Some examples follow. The system of linear equations 2x1 + 3x2 5x3 = 13, x1 3x2 + 8x3 = -13, 2x1 2x2 + 4x3 = -6, can be written as the matrix equation Ax = b, if

Page 9-8

x1 , x = x , and A= 2 x

13 b = 13. 6

This system has the same number of equations as of unknowns, and will be referred to as a square system. In general, there should be a unique solution to the system. The solution will be the point of intersection of the three planes in the coordinate system (x1, x2, x3) represented by the three equations. To enter matrix A you can activate the Matrix Writer while the A: field is selected. The following screen shows the Matrix Writer used for entering matrix A, as well as the input form for the numerical solver after entering matrix A (press ` in the Matrix Writer):

Press to select the B: field. The vector b can be entered as a row vector with a single set of brackets, i.e., [13,-13,-6] @@@OK@@@. After entering matrix A and vector b, and with the X: field highlighted, we can press @SOLVE! to attempt a solution to this system of equations:
A solution was found as shown next.

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Solution with the inverse matrix
The solution to the system Ax = b, where A is a square matrix is x = A-1 b. For the example used earlier, we can find the solution in the calculator as follows (First enter matrix A and vector b once more):
Solution by division of matrices
While the operation of division is not defined for matrices, we can use the calculators / key to divide vector b by matrix A to solve for x in the matrix equation Ax = b. The procedure for the case of dividing b by A is illustrated below for the example above. The procedure is shown in the following screen shots (type in matrices A and vector b once more):
Additional information on creating matrices, matrix operations, and matrix applications in linear algebra is presented in Chapters 10 and 11 of the calculators users guide.

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Chapter 10 Graphics
In this chapter we introduce some of the graphics capabilities of the calculator. We will present graphics of functions in Cartesian coordinates and polar coordinates, parametric plots, graphics of conics, bar plots, scatterplots, and fast 3D plots.
Graphs options in the calculator
To access the list of graphic formats available in the calculator, use the keystroke sequence (D) Please notice that if you are using the RPN mode these two keys must be pressed simultaneously to activate any of the graph functions. After activating the 2D/3D function, the calculator will produce the PLOT SETUP window, which includes the TYPE field as illustrated below.
Right in front of the TYPE field you will, most likely, see the option Function highlighted. This is the default type of graph for the calculator. To see the list of available graph types, press the soft menu key labeled @CHOOS. This will produce a drop down menu with the following options (use the up- and downarrow keys to see all the options):

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.

Page 11-2

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.

Compare these expressions with the one given earlier in the definition of the Laplace transform, i.e.,
L{ f (t )} = F ( s ) = f (t ) e st dt ,

Page 14-5

and you will notice that the CAS default variable X in the equation writer screen replaces the variable s in this definition. Therefore, when using the function LAP you get back a function of X, which is the Laplace transform of f(X). Example 2 Determine the inverse Laplace transform of F(s) = sin(s). Use: 1/(X+1)^2 ` ILAP The calculator returns the result: Xe-X, meaning that L -1{1/(s+1)2} = xe-x.

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 ().

Page 14-7

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.

Page 14-8

Chapter 15 Probability Distributions

To obtain the data fitting press @@OK@@. The output from this program, shown below for our particular data set, consists of the following three lines in RPN mode:
3: '0.195238095238 + 2.00857242857*X' 2: Correlation: 0.983781424465 1: Covariance: 7.03

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Level 3 shows the form of the equation. Level 2 shows the sample correlation coefficient, and level 1 shows the covariance of x-y. For definitions of these parameters see Chapter 18 in the users guide. For additional information on the data-fit feature of the calculator see Chapter 18 in the users guide.
Obtaining additional summary statistics
The application 4. Summary stats. in the STAT menu can be useful in some calculations for sample statistics. To get started, press once more, move to the fourth option using the down-arrow key , and press @@@OK@@@. The resulting input form contains the following fields:

DAT: X-Col, Y-Col:

the matrix containing the data of interest. these options apply only when you have more than two columns in the matrix DAT. By default, the x column is column 1, and the y column is column 2. If you have only one column, then the only setting that makes sense is to have X-Col: 1. summary statistics that you can choose as results of this program by checking the appropriate field using [ CHK] when that field is selected.
Many of these summary statistics are used to calculate statistics of two variables (x,y) that may be related by a function y = f(x). Therefore, this program can be thought off as a companion to program 3. Fit data. As an example, for the x-y data currently in DAT, obtain all the summary statistics. To access the summary stats option, use: @@@OK@@@ Select the column numbers corresponding to the x- and y-data, i.e., X-Col: 1, and Y-Col: 2. Using the @ CHK@ key select all the options for outputs, i.e., _X, _Y, etc.

Page 16-6

Press @@@OK@@@ to obtain the following results:

Confidence intervals

The application 6. Conf Interval can be accessed by using @@@OK@@@. The application offers the following options:
These options are to be interpreted as follows: 1. Z-INT: 1.: Single sample confidence interval for the population mean, , with known population variance, or for large samples with unknown population variance. 2. Z-INT: 12.: Confidence interval for the difference of the population means, 1- 2, with either known population variances, or for large samples with unknown population variances. 3. Z-INT: 1 p.: Single sample confidence interval for the proportion, p, for large samples with unknown population variance.

Page 16-7

4. Z-INT: p1 p2.: Confidence interval for the difference of two proportions, p1-p2, for large samples with unknown population variances. 5. T-INT: 1.: Single sample confidence interval for the population mean, , for small samples with unknown population variance. 6. T-INT: 12.: Confidence interval for the difference of the population means, 1- 2, for small samples with unknown population variances. Example 1 Determine the centered confidence interval for the mean of a population if a sample of 60 elements indicate that the mean value of the sample is x = 23.2, and its standard deviation is s = 5.2. Use = 0.05. The confidence level is C = 1- = 0.95. Select case 1 from the menu shown above by pressing @@@OK@@@. Enter the values required in the input form as shown:

Press @HELP to obtain a screen explaining the meaning of the confidence interval in terms of random numbers generated by a calculator. To scroll down the resulting screen use the down-arrow key. Press @@@OK@@@ when done with the help screen. This will return you to the screen shown above. To calculate the confidence interval, press @@@OK@@@. calculator is: The result shown in the
Press @GRAPH to see a graphical display of the confidence interval information:

Page 16-8

The graph shows the standard normal distribution pdf (probability density function), the location of the critical points z/2, the mean value (23.2) and the corresponding interval limits (21.88424 and 24.51576). Press @TEXT to return to the previous results screen, and/or press @@@OK@@@ to exit the confidence interval environment. The results will be listed in the calculators display. Additional examples of confidence interval calculations are presented in Chapter 18 in the calculators users guide.

Hypothesis testing

A hypothesis is a declaration made about a population (for instance, with respect to its mean). Acceptance of the hypothesis is based on a statistical test on a sample taken from the population. The consequent action and decision-making are called hypothesis testing. The calculator provides hypothesis testing procedures under application 5. Hypoth. tests. can be accessed by using @@@OK@@@. As with the calculation of confidence intervals, discussed earlier, this program offers the following 6 options:
These options are interpreted as in the confidence interval applications:

Page 16-9

1. Z-Test: 1.: Single sample hypothesis testing for the population mean, , with known population variance, or for large samples with unknown population variance. 2. Z-Test: 12.: Hypothesis testing for the difference of the population means, 1- 2, with either known population variances, or for large samples with unknown population variances. 3. Z-Test: 1 p.: Single sample hypothesis testing for the proportion, p, for large samples with unknown population variance. 4. Z-Test: p1 p2.: Hypothesis testing for the difference of two proportions, p1-p2, for large samples with unknown population variances. 5. T-Test: 1.: Single sample hypothesis testing for the population mean, , for small samples with unknown population variance. 6. T-Test: 12.: Hypothesis testing for the difference of the population means, 1- 2, for small samples with unknown population variances. Try the following exercise: Example 1 For 0 = 150, = 10, x = 158, n = 50, for = 0.05, test the hypothesis H0: = 0, against the alternative hypothesis, H1: 0. Press @@@OK@@@ to access the confidence interval feature in the calculator. Press @@@OK@@@ to select option 1. Z-Test: 1. Enter the following data and press @@@OK@@@:

Storing objects in the SD card
You can only store an object at the root of the SD, i.e., no sub-directory tree can be build into Port 3 (This feature may be enhanced in a future flash ROM upgrade). To store an object, use function STO as follows: In algebraic mode: Enter object, press K, type the name of the stored object using port 3 (e.g., :3:VAR1), press `. In RPN mode:

Page 18-1

Enter object, type the name of the stored object using port 3 (e.g., :3:VAR1), press K.
Recalling an object from the SD card
To recall an object from the SD card onto the screen, use function RCL, as follows: In algebraic mode: Press , type the name of the stored object using port 3 (e.g., :3:VAR1), press `. In RPN mode: Type the name of the stored object using port 3 (e.g., :3:VAR1), press. With the RCL command, it is possible to recall variables by specifying a path in the command, e.g., in RPN mode: :3: {path} ` RCL. The path, like in a DOS drive, is a series of directory names that locate the position of the variable within a directory tree. However, some variables stored within a backup object cannot be recalled by specifying a path. In this case, the full backup object (e.g., a directory) will have to be recalled, and the individual variables then accessed in the screen.
Purging an object from the SD card
To purge an object from the SD card onto the screen, use function PURGE, as follows: In algebraic mode: Press I @PURGE, type the name of the stored object using port 3 (e.g., :3:VAR1), press `. In RPN mode: Type the name of the stored object using port 3 (e.g., :3:VAR1), press I @PURGE.

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Limited Warranty
hp 49g+ graphing calculator; Warranty period: 12 months HP warrants to you, the end-user customer, that HP hardware, accessories and supplies will be free from defects in materials and workmanship after the date of purchase, for the period specified above. If HP receives notice of such defects during the warranty period, HP will, at its option, either repair or replace products which prove to be defective. Replacement products may be either new or like-new. HP warrants to you that HP software will not fail to execute its programming instructions after the date of purchase, for the period specified above, due to defects in material and workmanship when properly installed and used. If HP receives notice of such defects during the warranty period, HP will replace software media which does not execute its programming instructions due to such defects. HP does not warrant that the operation of HP products will be uninterrupted or error free. If HP is unable, within a reasonable time, to repair or replace any product to a condition as warranted, you will be entitled to a refund of the purchase price upon prompt return of the product with proof of purchase. HP products may contain remanufactured parts equivalent to new in performance or may have been subject to incidental use.

(VCCI)

Disposal of Waste Equipment by Users in Private Household in the European Union
This symbol on the product or on its packaging indicates that this product must not be disposed of with your other household waste. Instead, it is your responsibility to dispose of your waste equipment by handing it over to a designated collection point for the recycling of waste electrical and electronic equipment. The separate collection and recycling of your waste equipment at the time of disposal will help to conserve natural resources and ensure that it is recycled in a manner that protects human health and the environment. For more information about where you can drop off your waste equipment for recycling, please contact your local city office, your household waste disposal service or the shop where you purchased the product.

Page W-4

Page TOC-10

Matrix Quadratic Forms, 11-51 The QUADF menu, 11-52 Linear Applications, 11-54 Function IMAGE, 11-54 Function ISOM, 11-54 Function KER, 11-55 Function MKISOM, 11-55
Chapter 12 - Graphics, 12-1
Graphs options in the calculator, 12-1 Plotting an expression of the form y = f(x) , 12-2 Some useful PLOT operations for FUNCTION plots, 12-5 Saving a graph for future use, 12-8 Graphics of transcendental functions, 12-10 Graph of ln(X) , 12-8 Graph of the exponential function, 12-10 The PPAR variable, 12-11 Inverse functions and their graphs, 12-12 Summary of FUNCTION plot operation, 12-13 Plots of trigonometric and hyperbolic functions and their inverses, 12-16 Generating a table of values for a function, 12-17 The TPAR variable, 12-18 Plots in polar coordinates, 12-19 Plotting conic curves, 12-21 Parametric plots, 12-23 Generating a table of parametric equations, 12-26 Plotting the solution to simple differential equations, 12-26 Truth plots, 12-29 Plotting histograms, bar plots, and scatterplots, 12-30 Bar plots, 12-30 Scatter plots, 12-32 Slope fields, 12-34 Fast 3D plots, 12-35 Wireframe plots, 12-37 Ps-Contour plots, 12-39 Y-Slice plots, 12-41

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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

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subdirectories, in a hierarchy of directories similar to folders in modern computers. The subdirectories will be given names that may reflect the contents of each subdirectory, or any arbitrary name that you can think of.

The CASDIR sub-directory

The CASDIR sub-directory contains a number of variables needed by the proper operation of the CAS (Computer Algebraic System, see appendix C). To see the contents of the directory, we can use the keystroke combination: which opens the File Manager once more:
This time the CASDIR is highlighted in the screen. To see the contents of the directory press the @@OK@@ (F) soft menu key or `, to get the following screen:
The screen shows a table describing the variables contained in the CASDIR directory. These are variables pre-defined in the calculator memory that establish certain parameters for CAS operation (see appendix C). The table above contains 4 columns: The first column indicate the type of variable (e.g., EQ means an equation-type variable, |R indicates a real-value variable, { } means a list, nam means a global name, and the symbol represents a graphic variable. The second column represents the name of the variables, i.e., PRIMIT, CASINFO, MODULO, REALASSUME, PERIOD, VX, and EPS. Column number 3 shows another specification for the variable type, e.g., ALG means an algebraic expression, GROB stands for graphics object, INTG means an integer numeric variable, LIST means a list of data,

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GNAME means a global name, and REAL means a real (or floating-point) numeric variable. The fourth and last column represents the size, in bytes, of the variable truncated, without decimals (i.e., nibbles). Thus, for example, variable PERIOD takes 12.5 bytes, while variable REALASSUME takes 27.5 bytes (1 byte = 8 bits, 1 bit is the smallest unit of memory in computers and calculators).
CASDIR Variables in the stack Pressing the $ key closes the previous screen and returns us to normal calculator display. By default, we get back the TOOL menu: We can see the variables contained in the current directory, CASDIR, by pressing the J key (first key in the second row from the top of the keyboard). This produces the following screen: Pressing the L key shows one more variable stored in this directory: To see the contents of the variable EPS, for example, use @EPS@. This shows the value of EPS to be.0000000001 To see the value of a numerical variable, we need to press only the soft menu key for the variable. For example, pressing cz followed by `, shows the same value of the variable in the stack, if the calculator is set to Algebraic. If the calculator is set to RPN mode, you need only press the soft menu key for `. To see the full name of a variable, press the apostrophe first , and then the soft menu key corresponding to the variable. For example, for the variable listed in the stack as PERIO, we use: @PERIO@, which produces as output the string: 'PERIOD'. This procedure applies to both the Algebraic and RPN calculator operating modes.

Creating variables

To create a variable, we can use the FILES menu, along the lines of the examples shown above for creating a sub-directory. For example, within the

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sub-directory {HOME MANS INTRO}, created in an earlier example, we want to store the following variables with the values shown: Name Contents A 12.5 -0.25 AQ r/(m+r)' R [3,2,1] z1 3+5i p1 << r '*r^2' >> Type real real real algebraic vector complex program
Using the FILES menu We will use the FILES menu to enter the variable A. We assume that we are in the sub-directory {HOME MANS INTRO}. To get to this sub-directory, use the following: and select the INTRO sub-directory as shown in this screen:
Press @@OK@@ to enter the directory. You will get a files listing with no entries (the INTRO sub-directory is empty at this point)
Press the L key to move to the next set of soft menu keys, and press the @@NEW@@ soft menu key. This will produce the NEW VARIABLE input form:

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To enter variable A (see table above), we first enter its contents, namely, the number 12.5, and then its name, A, as follows: 12.5 @@OK@@ ~a@@OK@@. Resulting in the following screen:
Press @@OK@@ once more to create the variable. The new variable is shown in the following variable listing:
The listing indicates a real variable (|R), whose name is A, and that occupies 10.5 bytes of memory. To see the contents of the variable in this screen, press L@VIEW@. Press the @GRAPH (A) soft menu key to see the contents in a graphical format.
Press the @TEXT (A) soft menu key to see the contents in text format. Press @@OK@@ to return to the variable list Press $ once more to return to normal display. Variable A should now be featured in the soft menu key labels:
Using the STO command A simpler way to create a variable is by using the STO command (i.e., the K key). We provide examples in both the Algebraic and RPN modes, by creating the remaining of the variables suggested above, namely:

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Name Contents -0.25 AQ r/(m+r)' R [3,2,1] z1 3+5i p1 << r '*r^2' >>
Type real real algebraic vector complex program
Algebraic mode Use the following keystrokes to store the value of 0.25 into variable : 0.25\ K ~a. AT this point, the screen will look as follows: This expression means that the value 0.25 is being stored into (the symbol suggests the operation). Press ` to create the variable. The variable is now shown in the soft menu key labels:
The following are the keystrokes required to enter the remaining variables: A12: 3V5K~a12` Q: ~r/ ~m+~r K~q` R: 321 K~r` z1: 3+5* K~z1` (if needed, accept change to Complex mode) p1: ~r* ~rQ2 K~p1`. The screen, at this point, will look as follows:

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You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12,. RPN mode Use the following keystrokes to store the value of 0.25 into variable : 0.25\` ~a`. At this point, the screen will look as follows:

Function CSWP

Function CSWP (Column SWaP) takes as arguments two indices, say, i and j, (representing two distinct columns in a matrix), and a matrix, and produces a new matrix with columns i and j swapped. The following example, in ALG mode, shows an application of this function. We use the matrix stored in variable A for the example. This matrix is listed first.
In RPN mode, function CSWP lets you swap the columns of a matrix listed in stack level 3, whose indices are listed in stack levels 1 and 2. For example, the following figure shows the RPN stack before and after applying function CSWP to matrix A in order to swap columns 2 and 3:
As you can see, the columns that originally occupied positions 2 and 3 have been swapped. Swapping of columns, and of rows (see below), is commonly used when solving systems of linear equations with matrices. Details of these operations will be given in a subsequent Chapter.
Manipulating matrices by rows
The calculator provides a menu with functions for manipulating matrices by operating in their rows. This menu is available through the

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MTH/MATRIX/ROW. sequence: () shown in the figure below with system flag 117 set to CHOOSE boxes:
or through the MATRICES/CREATE/ROW sub-menu:
When system flag 117 is set to SOFT menus, the ROW menu is accessible through !)MATRX !)@MAKE@ !)@@ROW@ , or through !)@CREAT@ !)@@ROW@. Both approaches will show the same set of functions:

Function ROW

Function ROW takes as argument a matrix and decomposes it into vectors corresponding to its rows. An application of function ROW in ALG mode is shown below. The matrix used has been stored earlier in variable A. The matrix is shown in the figure to the left. The figure to the right shows the matrix decomposed in rows. To see the full result, use the line editor (triggered by pressing ).

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In RPN mode, you need to list the matrix in the stack, and the activate function ROW, i.e., @@@A@@@ ROW. The figure below shows the RPN stack before and after the application of function ROW.
In this result, the first row occupies the highest stack level after decomposition, and stack level 1 is occupied by the number of rows of the original matrix. The matrix does not survive decomposition, i.e., it is no longer available in the stack.

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Under-determined system The system of linear equations 2x1 + 3x2 5x3 = -10, x1 3x2 + 8x3 = 85, can be written as the matrix equation Ax = b, if
x A= , x = x2 , and 8 x3

10 b=. 85

This system has more unknowns than equations, therefore, it is not uniquely determined. We can visualize the meaning of this statement by realizing that each of the linear equations represents a plane in the three-dimensional Cartesian coordinate system (x1, x2, x3). The solution to the system of equations shown above will be the intersection of two planes in space. We know, however, that the intersection of two (non-parallel) planes is a straight line, and not a single point. Therefore, there is more than one point that satisfy the system. In that sense, the system is not uniquely determined. Lets use the numerical solver to attempt a solution to this system of equations: @@OK@@. Enter matrix A and vector b as illustrated in the previous example, and press @SOLVE when the X: field is highlighted:

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To see the details of the solution vector, if needed, press the @EDIT! button. This will activate the Matrix Writer. Within this environment, use the rightand left-arrow keys to move about the vector, e.g.,
Thus, the solution is x = [15.373, 2.4626, 9.6268]. To return to the numerical solver environment, press `. The procedure that we describe next can be used to copy the matrix A and the solution vector X into the stack. To check that the solution is correct, try the following: Press Press Press Press Press Press , to highlight the A: field. L @CALC@ `, to copy matrix A onto the stack. @@@OK@@@ to return to the numerical solver environment. @CALC@ `, to copy solution vector X onto the stack. @@@OK@@@ to return to the numerical solver environment. ` to return to the stack.
In ALG mode, the stack will now look like this:

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Lets store the latest result in a variable X, and the matrix into variable A, as follows: Press K~x` to store the solution vector into variable X Press to clear three levels of the stack Press K~a` to store the matrix into variable A Now, lets verify the solution by using: @@@A@@@ * @@@X@@@ `, which results in (press to see the vector elements): [-9.99999999999 85. ], close enough to the original vector b = [-10 85]. Try also this, @@A@@@ * [15,10/3,10] ` `, i.e.,
This result indicates that x = [15,10/3,10] is also a solution to the system, confirming our observation that a system with more unknowns than equations is not uniquely determined (under-determined). How does the calculator came up with the solution x = [15.37 2.46 9.62] shown earlier? Actually, the calculator minimizes the distance from a point, which will constitute the solution, to each of the planes represented by the equations in the linear system. The calculator uses a least-square method, i.e., minimizes the sum of the squares of those distances or errors. Over-determined system The system of linear equations x1 + 3x2 = 15, 2x1 5x2 = 5, -x1 + x2 = 22,

Graph of the exponential function
First, load the function exp(X), by pressing, simultaneously if in RPN mode, the left-shift key and the (V) key to access the PLOT-FUNCTION window. Press @@DEL@@ to remove the function LN(X), if you didnt delete Y1 as suggested in the previous note. Press @@ADD@! and type ~x` to enter EXP(X) and return to the PLOT-FUNCTION window. Press L@@@OK@@@ to return to normal calculator display. Next, press, simultaneously if in RPN mode, the left-shift key and the (B) key to produce the PLOT WINDOW - FUNCTION window. Change the H-View values to read: H-View: -by using 8\@@@OK@@ @2@@@OK@@@. Next, press @AUTO. After the vertical range is calculated, press @ERASE @DRAW to plot the exponential function.

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To add labels to the graph press @EDIT L@)LABEL. Press @MENU to remove the menu labels, and get a full view of the graph. Press LL@)PICT! @CANCL to return to the PLOT WINDOW FUNCTION. Press ` to return to normal calculator display.

The PPAR variable

Press J to recover your variables menu, if needed. In your variables menu you should have a variable labeled PPAR. Press @PPAR to get the contents of this variable in the stack. Press the down-arrow key, , to launch the stack editor, and use the up- and down-arrow keys to view the full contents of PPAR. The screen will show the following values:
PPAR stands for Plot PARameters, and its contents include two ordered pairs of real numbers, (-8.,-1.10797263281) and (2.,7.38905609893), which represent the coordinates of the lower left corner and the upper right corner of the plot, respectively. Next, PPAR lists the name of the independent variable, X, followed by a number that specifies the increment of the independent variable in the generation of the plot. The value shown here is the default value, zero (0.), which specifies increments in X corresponding to 1 pixel in the graphics display. The next element in PPAR is a list containing first the coordinates of the point of intersection of the plot axes, i.e., (0.,0.), followed by a list that specifies the tick mark annotation on the x- and y-axes, respectively {# 10d # 10d}. Next, PPAR lists the type of plot that is to be generated, i.e., FUNCTION, and, finally, the y-axis label, i.e., Y. The variable PPAR, if non-existent, is generated every time you create a plot. The contents of the function will change depending on the type of plot and on the options that you select in the PLOT window (the window generated by the simultaneous activation of the and (B) keys.

This command produces a circle. Mark the center of the circle with a MARK command, then move the cursor to a point that will be part of the periphery of the circle, and press @CIRCL. To deactivate CIRCL, return the cursor to the MARK position and press @LINE. Try this command by moving the cursor to a clear part of the graph, press @MARK. Move the cursor to another point, then press @CIRCL. A circle centered at the MARK, and passing through the last point will be drawn.
Pressing @LABEL places the labels in the x- and y-axes of the current plot. This feature has been used extensively through this chapter.
This command is used to remove parts of the graph between two MARK positions. Move the cursor to a point in the graph, and press @MARK. Move the cursor to a different point, press @MARK again. Then, press @@DEL@. The section of the graph boxed between the two marks will be deleted.

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The function ERASE clears the entire graphics window. This command is available in the PLOT menu, as well as in the plotting windows accessible through the soft menu keys.
Pressing @MENU will remove the soft key menu labels to show the graphic unencumbered by those labels. To recover the labels, press L.
Use this command to extract a subset of a graphics object. The extracted object is automatically placed in the stack. Select the subset you want to extract by placing a MARK at a point in the graph, moving the cursor to the diagonal corner of the rectangle enclosing the graphics subset, and press @@SUB@. This feature can be used to move parts of a graphics object around the graph.
This command places the contents of a graphic object currently in stack level 1 at the cursor location in the graphics window. The upper left corner of the graphic object being inserted in the graph will be placed at the cursor position. Thus, if you want a graph from the stack to completely fill the graphic window, make sure that the cursor is placed at the upper left corner of the display.
This command places a copy of the graph currently in the graphics window on to the stack as a graphic object. The graphic object placed in the stack can be saved into a variable name for storage or other type of manipulation.

TABVAR(LN(X)/X) produces the following table of variation:
A detailed interpretation of the table of variation is easier to follow in RPN mode:

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The output is in a graphical format, showing the original function, F(X), the derivative F(X) right after derivation and after simplification, and finally a table of variation. The table consists of two rows, labeled in the right-hand side. Thus, the top row represents values of X and the second row represents values of F. The question marks indicates uncertainty or non-definition. For example, for X<0, LN(X) is not defined, thus the X lines shows a question mark in that interval. Right at zero (0+0) F is infinite, for X = e, F = 1/e. F increases before reaching this value, as indicated by the upward arrow, and decreases after this value (X=e) becoming slightly larger than zero (+:0) as X goes to infinity. A plot of the graph is shown below to illustrate these observations:

Function DRAW3DMATRIX

This function takes as argument a nm matrix, Z, = [ zij ], and minimum and maximum values for the plot. You want to select the values of vmin and vmax so that they contain the values listed in Z. The general call to the function is, therefore, DRAW3DMATRIX(Z,vmin,vmax). To illustrate the use of this function we first generate a 65 matrix using RANM({6,5}), and then call function DRAW3DMATRIX, as shown below:
The plot is in the style of a FAST3DPLOT. Different views of the plot are shown below:

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Chapter 13 Calculus Applications
In this Chapter we discuss applications of the calculators functions to operations related to Calculus, e.g., limits, derivatives, integrals, power series, etc.

The CALC (Calculus) menu

Many of the functions presented in this Chapter are contained in the calculators CALC menu, available through the keystroke sequence (associated with the 4 key). The CALC menu shows the following entries:
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.

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At this point, you can press ` to return the integral to the stack, which will show the following entry (ALG mode shown):
This is the general format for the definite integral when typed directly into the stack, i.e., (lower limit, upper limit, integrand, variable of integration) Pressing ` at this point will evaluate the integral in the stack:
The integral can be evaluated also in the Equation Writer by selecting the entire expression an using the soft menu key @EVAL.
Step-by-step evaluation of derivatives and integrals
With the Step/Step option in the CAS MODES windows selected (see Chapter 1), the evaluation of derivatives and integrals will be shown step by step. For example, here is the evaluation of a derivative in the Equation Writer:

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Notice the application of the chain rule in the first step, leaving the derivative of the function under the integral explicitly in the numerator. In the second step, the resulting fraction is rationalized (eliminating the square root from the denominator), and simplified. The final version is shown in the third step. Each step is shown by pressing the @EVAL menu key, until reaching the point where further application of function EVAL produce no more changes in the expression. The following example shows the evaluation of a definite integral in the Equation Writer, step-by-step:
Notice that the step-by-step process provides information on the intermediate steps followed by the CAS to solve this integral. First, CAS identifies a square root integral, next, a rational fraction, and a second rational expression, to come up with the final result. Notice that these steps make a lot of sense to the calculator, although not enough information is provided to the user on the individual steps.

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Integrating an equation
Integrating an equation is straightforward, the calculator simply integrates both sides of the equation simultaneously, e.g.,
Techniques of integration
Several techniques of integration can be implemented in the calculators, as shown in the following examples. Substitution or change of variables Suppose we want to calculate the integral

x 1 x2

dx. If we use step-by-
step calculation in the Equation Writer, this is the sequence of variable substitutions:
This second step shows the proper substitution to use, u = x2-1.

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The last four steps show the progression of the solution: a square root, followed by a fraction, a second fraction, and the final result. This result can be simplified by using function @SIMP, to read:
Integration by parts and differentials A differential of a function y = f(x), is defined as dy = f(x) dx, where f(x) is the derivative of f(x). Differentials are used to represent small increments in the variables. The differential of a product of two functions, y = u(x)v(x), is given by dy = u(x)dv(x) +du(x)v(x), or, simply, d(uv) = udv - vdu. Thus, the integral of udv = d(uv) - vdu, is written as udv = the definition of a differential, dy = y, we write the previous expression as

Example 1 -- Let X represent the time (hours) required by a specific manufacturing process to be completed. Given the following sample of values of X: 2.2 2.5 2.1 2.3 2.2. The population from where this sample is taken is the collection of all possible values of the process time, therefore, it is an infinite population. Suppose that the population parameter we are trying

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to estimate is its mean value,. We will use as an estimator the mean value of the sample, X, defined by (a rule): X =

1 n Xi. n i =1

For the sample under consideration, the estimate of is the sample statistic x = (2.2+2.5+2.1+2.3+2.2)/5 = 2.36. This single value of X, namely x = 2.36, constitutes a point estimation of the population parameter.
Estimation of Confidence Intervals
The next level of inference from point estimation is interval estimation, i.e., instead of obtaining a single value of an estimator we provide two statistics, a and b, which define an interval containing the parameter with a certain level of probability. The end points of the interval are known as confidence limits, and the interval (a,b) is known as the confidence interval.
Let (Cl,Cu) be a confidence interval containing an unknown parameter. Confidence level or confidence coefficient is the quantity (1-), where 0 < < 1, such that P[Cl < < Cu] = 1 - , where P[ ] represents a probability (see Chapter 17). The previous expression defines the socalled two-sided confidence limits. A lower one-sided confidence interval is defined by Pr[Cl < ] = 1 -. An upper one-sided confidence interval is defined by Pr[ < Cu] = 1 -. The parameter is known as the significance level. Typical values of are 0.01, 0.05, 0.1, corresponding to confidence levels of 0.99, 0.95, and 0.90, respectively.
Confidence intervals for the population mean when the population variance is known
Let X be the mean of a random sample of size n, drawn from an infinite population with known standard deviation. The 100(1-) % [i.e., 99%, 95%, 90%, etc.], central, two-sided confidence interval for the population mean is (Xz/2/n , X+z/2/n ), where z/2 is a standard normal variate that is exceeded with a probability of /2. The standard error of the sample mean, X, is /n.

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The one-sided upper and lower 100(1-) % confidence limits for the population mean are, respectively, X+z/n , and Xz/n. Thus, a lower, one-sided, confidence interval is defined as (- , X+z/n), and an upper, one-sided, confidence interval as (Xz/n,+). Notice that in these last two intervals we use the value z, rather than z/2. In general, the value zk in the standard normal distribution is defined as that value of z whose probability of exceedence is k, i.e., Pr[Z>zk] = k, or Pr[Z<zk] = 1 k. The normal distribution was described in Chapter 17.

With the cursor in front of the IF statement prompting the user for the logical statement that will activate the IF construct when the program is executed. Example: Type in the following program:
IF x<3 THEN x^2 EVAL END Done MSGBOX

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and save it under the name f1. Press J and verify that variable @@@f1@@@ is indeed available in your variable menu. Verify the following results: 0 @@@f1@@@ Result: 0 3.5 @@@f1@@@ Result: no action 1.2 @@@f1@@@ Result: 1.@@@f1@@@ Result: no action
These results confirm the correct operation of the IFTHENEND construct. The program, as written, calculates the function f1(x) = x2, if x < 3 (and not output otherwise). The IFTHENELSEEND construct The IFTHENELSEEND construct permits two alternative program flow paths based on the truth value of the logical_statement. The general format of this construct is: IF logical_statement THEN program_statements_if_true ELSE program_statements_if_false END. The operation of this construct is as follows: 1. Evaluate logical_statement. 2. If logical_statement is true, perform program statements_if_true and continue program flow after the END statement. 3. If logical_statement is false, perform program statements_if_false and continue program flow after the END statement. To produce an IFTHENELSEEND construct directly on the stack, use: @)@BRCH@ @)@IF@@ This will create the following input in the stack:
Example: Type in the following program:

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x IF x<3 THEN x^2 ELSE 1-x END EVAL Done MSGBOX and save it under the name f2. Press J and verify that variable @@@f2@@@ is indeed available in your variable menu. Verify the following results: 0 @@@f2@@@ Result: 0 1.2 @@@f2@@@ Result: 1.44 3.5 @@@f2@@@ Result: -2.@@@f2@@@ Result: -9 These results confirm the correct operation of the IFTHENELSEEND construct. The program, as written, calculates the function
x 2 , if x < 3 f 2 ( x) = 1 x, otherwise
Note: For this particular case, a valid alternative would have been to use an IFTE function of the form: f2(x) = IFTE(x<3,x^2,1-x) Nested IFTHENELSEEND constructs In most computer programming languages where the IFTHENELSEEND construct is available, the general format used for program presentation is the following: IF logical_statement THEN program_statements_if_true ELSE program_statements_if_false END In designing a calculator program that includes IF constructs, you could start by writing by hand the pseudo-code for the IF constructs as shown above. For example, for program @@@f2@@@, you could write IF x<3 THEN x2 ELSE 1-x END

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When a backup object is restored, the calculator performs an integrity check on the restored object by calculating its CRC value. Any discrepancy between the calculated and the stored CRC values result in an error message indicating a corrupted data.
Using data in backup objects
Although you cannot directly modify the contents of backup objects, you can use those contents in calculator operations. For example, you can run programs stored as backup objects or use data from backup objects to run programs. To run backup-object programs or use data from backup objects you can use the File Manager () to copy backup object contents to the screen. Alternatively, you can use function EVAL to run a program stored in a backup object, or function RCL to recover data from a backup object as follows: In algebraic mode: To evaluate a back up object, enter: EVAL(argument(s), : Port_Number : Backup_Name ) To recall a backup object to the command line, enter: RCL(: Port_Number : Backup_Name) In RPN mode: To evaluate a back up object, enter: Argument(s) ` : Port_Number : Backup_Name EVAL To recall a backup object to the command line, enter: : Port_Number : Backup_Name ` RCL

Using SD cards

The calculator provides a memory card port where you can insert an SD flash card for backing up calculator objects, or for downloading objects from other sources. Insert the card with the label side down. The SD card in the calculator will appear as port number 3. Accessing an object from the SD card is performed similarly as if the object were located in ports 0, 1, or 2. However, Port 3 will not appear in the menu when using the LIB function (). The SD files can only be managed using the Filer, or File Manager (). When starting the Filer, the Tree view will show:

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0: IRAM 1: ERAM 2: FLASH 3: SD HOME |-- sub-directories When you enter in the SD tree, all objects will appear as backup objects. Therefore, it is not possible to tell what type a given objects by just looking at its name in the Filer. Long names are not supported in the Filer. Thus, all names must be in the form 8.3 characters, similar as used in DOS, i.e., names will have a maximum of 8 characters with 3 characters in the suffix. As an alternative to using the File Manager operations, you can use functions STO and RCL to store and recall objects from the SD card, as shown below. You can also use the PURGE command to erase backup objects in the SD card. Long names can be used with these commands (namely, STO, RCL, and PURGE).