The BASE menu, 17-1 Writing non-decimal numbers, 17-1 Reference, 17-2
Chapter 18 Using SD cards, 18-1
Storing objects in the SD card, 18-1 Recalling an object from the SD card, 18-2 Purging an object from the SD card, 18-2

Limited Warranty W-1

Service, W-2 Regulatory information, W-4

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Chapter 1 Getting started
This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aimed at familiarizing yourself with the basic operations and settings before actually performing a calculation.

Basic Operations

The following exercises are aimed at getting you acquainted with the hardware of your calculator.

Batteries

The calculator uses 3 AAA (LR03) batteries as main power and a CR2032 lithium battery for memory backup. Before using the calculator, please install the batteries according to the following procedure. To install the main batteries a. Make sure the calculator is OFF. Slide up the battery compartment cover as illustrated.
b. Insert 3 new AAA (LR03) batteries into the main compartment. Make sure each battery is inserted in the indicated direction. To install the backup battery a. Make sure the calculator is OFF. Press down the holder. Push the plate to the shown direction and lift it.

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b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facing up. c. Replace the plate and push it to the original place. After installing the batteries, press [ON] to turn the power on. Warning: When the low battery icon is displayed, you need to replace the batteries as soon as possible. However, avoid removing the backup battery and main batteries at the same time to avoid data lost.
Turning the calculator on and off
The $ key is located at the lower left corner of the keyboard. Press it once to turn your calculator on. To turn the calculator off, press the red right-shift key @ (first key in the second row from the bottom of the keyboard), followed by the $ key. Notice that the $ key has a red OFF label printed in the upper right corner as a reminder of the OFF command.
Adjusting the display contrast
You can adjust the display contrast by holding the $ key while pressing the + or - keys. The $(hold) + key combination produces a darker display The $(hold) - key combination produces a lighter display
Contents of the calculators display
Turn your calculator on once more. At the top of the display you will have two lines of information that describe the settings of the calculator. The first line shows the characters: RAD XYZ HEX R= 'X'

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For details on the meaning of these specifications see Chapter 2 in the calculators users guide. The second line shows the characters { HOME } indicating that the HOME directory is the current file directory in the calculators memory. At the bottom of the display you will find a number of labels, namely, @EDIT @VIEW @@ RCL @@ @@STO@ ! PURGE !CLEAR associated with the six soft menu keys, F1 through F6: ABCDEF The six labels displayed in the lower part of the screen will change depending on which menu is displayed. But A will always be associated with the first displayed label, B with the second displayed label, and so on.

Let's try now the expression proposed earlier:

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2.5 +e 3
3` 5` 3` 3* Y * 23` 3Q / 2.5 ! + R
Enter 3 in level 1 Enter 5 in level 1, 3 moves to level 2 Enter 3 in level 1, 5 moves to level 2, 3 to level 3 Place 3 and multiply, 9 appears in level 1 1/(33), last value in lev. 1; 5 in level 2; 3 in level - 1/(33) , occupies level 1 now; 3 in level (5 - 1/(33)), occupies level 1 now. Enter 23 in level 1, 14.66666 moves to level 2. Enter 3, calculate 233 into level 1. 14.666 in lev. 2. (3 (5-1/(33)))/233 into level 1 Enter 2.5 level 1 e2.5, goes into level 1, level 2 shows previous value. (3 (5 - 1/(33)))/233 + e2.5 = 12.18369, into lev. 1. ((3 (5 - 1/(33)))/233 + e2.5) = 3.4905156, into 1.
To select between the ALG vs. RPN operating mode, you can also set/clear system flag 95 through the following keystroke sequence: H @)FLAGS @ @CHK@@
Number Format and decimal dot or comma
Changing the number format allows you to customize the way real numbers are displayed by the calculator. You will find this feature extremely useful in operations with powers of tens or to limit the number of decimals in a result. To select a number format, first open the CALCULATOR MODES input form by pressing the H button. Then, use the down arrow key, , to select the option Number format. The default value is Std, or Standard format. In the standard format, the calculator will show floating-point numbers with no set decimal placement and with the maximum precision allowed by the calculator

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(12 significant digits).To learn more about reals, see Chapter 2 in this guide. To illustrate this and other number formats try the following exercises: Standard format: This mode is the most used mode as it shows numbers in the most familiar notation. Press the !!@@OK#@ soft menu key, with the Number format set to Std, to return to the calculator display. Enter the number 123.4567890123456 (with16 significant figures). Press the ` key. The number is rounded to the maximum 12 significant figures, and is displayed as follows:
Fixed format with decimals: Press 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 Fixed with the arrow down key.
Press the right arrow key, , to highlight the zero in front of the option Fix. Press the @CHOOS soft menu key and, using the up and down arrow keys, , select, say, 3 decimals.

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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:

To change CAS settings press the @@ CAS@@ soft menu key. The default values of the CAS setting are shown below:
To navigate through the many options in the CAS MODES input form, use the arrow keys:. To select or deselect any of the settings shown above, select the underline before the option of interest, and toggle the @ @CHK@@ soft menu key until the right setting is achieved. When an option is selected, a check mark will be shown in the underline (e.g., the Rigorous and Simp Non-Rational

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options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow options above). After having selected and unselected all the options that you want in the CAS MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more.
Explanation of CAS settings
Indep var: The independent variable for CAS applications. Typically, VX = X. Modulo: For operations in modular arithmetic this variable holds the modulus or modulo of the arithmetic ring (see Chapter 5 in the calculators users guide). Numeric: If set, the calculator produces a numeric, or floating-point result, in calculations. Approx: If set, Approximate mode uses numerical results in calculations. If unchecked, the CAS is in Exact mode, which produces symbolic results in algebraic calculations. Complex: If set, complex number operations are active. If unchecked the CAS is in Real mode, i.e., real number calculations are the default. See Chapter 4 for operations with complex numbers. Verbose: If set, provides detailed information in certain CAS operations. Step/Step: If set, provides step-by-step results for certain CAS operations. Useful to see intermediate steps in summations, derivatives, integrals, polynomial operations (e.g., synthetic division), and matrix operations. Incr Pow: Increasing Power, means that, if set, polynomial terms are shown in increasing order of the powers of the independent variable. Rigorous: If set, calculator does not simplify the absolute value function |X| to X. Simp Non-Rational: If set, the calculator will try to simplify non-rational expressions as much as possible.

Selecting Display modes

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The calculator display can be customized to your preference by selecting different display modes. To see the optional display settings use the following: First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key (D) to display the DISPLAY MODES input form.
To navigate through the many options in the DISPLAY MODES input form, use the arrow keys:. To select or deselect any of the settings shown above, that require a check mark, select the underline before the option of interest, and toggle the @ CHK@@ soft menu key until the right setting is achieved. When an option is selected, a check mark will be shown in the underline (e.g., the Textbook option in the Stack: line above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Small, _Full page, and _Indent options in the Edit: line above). To select the Font for the display, highlight the field in front of the Font: option in the DISPLAY MODES input form, and use the @CHOOS soft menu key (B). After having selected and unselected all the options that you want in the DISPLAY MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more.

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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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____________________________________________________ Prefix Name x Prefix Name x ____________________________________________________ Y yotta +24 d deci -1 Z zetta +21 c centi -2 E exa +18 m milli -3 P peta +15 micro -6 T tera +12 n nano -9 G giga +9 p pico -12 M mega +6 f femto -15 k,K kilo +3 a atto -18 h,H hecto +2 z zepto -21 D(*) deka +1 y yocto -24 _____________________________________________________ (*) In the SI system, this prefix is da rather than D. Use D for deka in the calculator, however. To enter these prefixes, simply type the prefix using the ~ keyboard. For example, to enter 123 pm (picometer), use: 123~p~m Using UBASE (type the name) to convert to the default unit (1 m) results in:

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)::
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:

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Following, we present applications of items 3. Solve poly., 5. Solve finance, and 1. Solve equation., in that order. Appendix 1-A, in the calculators users guide, contains instructions on how to use input forms with examples for the numerical solver applications. Item 6. MSLV (Multiple equation SoLVer) will be presented later in page 6-10. 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; and, (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. For example, solve the equation: 3s4 + 2s3 - s + 1 = 0. We want to place the coefficients of the equation in a vector: [3,2,0,-1,1]. To solve for this polynomial equation using the calculator, try the following: @@OK@@ 1\1@@OK@@ @SOLVE@ The screen will show the solution as follows: Select Solve poly Enter vector of coefficients Solve equation

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:
In RPN mode, you can enter a vector in the stack by opening a set of brackets and typing the vector components or elements separated by either commas

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() or spaces (#). Notice that after pressing ` , in either mode, the calculator shows the vector elements separated by spaces.
Storing vectors into variables in the stack
Vectors can be stored into variables. The screen shots below show the vectors u2 = [1, 2], u3 = [-3, 2, -2], v2 = [3,-1], v3 = [1, -5, 2] Stored into variables @@@u2@@, @@@u3@@, @@@v2@@, and @@@v3@@, respectively. First, in ALG mode:

Dot product

Function DOT (option 2 in CHOOSE box above) is used to calculate the dot product of two vectors of the same length. Some examples of application of function DOT, using the vectors A, u2, u3, v2, and v3, stored earlier, are shown next in ALG mode. Attempts to calculate the dot product of two vectors of different length produce an error message:

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Cross product
Function CROSS (option 3 in the MTH/VECTOR menu) is used to calculate the cross product of two 2-D vectors, of two 3-D vectors, or of one 2-D and one 3D vector. For the purpose of calculating a cross product, a 2-D vector of the form [Ax, Ay], is treated as the 3-D vector [Ax, Ay,0]. Examples in ALG mode are shown next for two 2-D and two 3-D vectors. Notice that the cross product of two 2-D vectors will produce a vector in the z-direction only, i.e., a vector of the form [0, 0, Cz]:
Examples of cross products of one 3-D vector with one 2-D vector, or vice versa, are presented next:
Attempts to calculate a cross product of vectors of length other than 2 or 3, produce an error message:
Additional information on operations with vectors, including applications in the physical sciences, is presented in Chapter 9 of the calculators users guide.

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Chapter 9 Matrices and linear algebra
This chapter shows examples of creating matrices and operations with matrices, including linear algebra applications.
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 8, 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:

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Press ` once more to place the matrix on the stack. The ALG mode stack is shown next, before and after pressing , once more:
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.
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. For future exercises, lets save this matrix under the name A. In ALG mode use K~a. In RPN mode, use ~a K.

The inverse matrix The inverse of a square matrix A is the matrix A-1 such that AA-1 = A-1A = I, where I is the identity matrix of the same dimensions as A. The inverse of a matrix is obtained in the calculator by using the inverse function, INV (i.e., the Y key). Examples of the inverse of some of the matrices stored earlier are presented next:
To verify the properties of the inverse matrix, we present the following multiplications:

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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.

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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

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x1 , x = x , and A= 2 x

13 b = 13. 6

Fast 3D plots

Fast 3D plots are used to visualize three-dimensional surfaces represented by equations of the form z = f(x,y). For example, if you want to visualize z = f(x,y) = x2+y2, we can use the following:

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Press , simultaneously if in RPN mode, to access to the PLOT SETUP window. Change TYPE to Fast3D. ( @CHOOS!, find Fast3D, @@OK@@). Press and type X^2+Y^2 @@@OK@@@. Make sure that X is selected as the Indep: and Y as the Depnd: variables. Press L@@@OK@@@ to return to normal calculator display. Press , simultaneously if in RPN mode, to access the PLOT WINDOW screen. Keep the default plot window ranges to read:
X-Left:-1 Y-Near:-1 Z-Low: -1 Step Indep: 10 X-Right:1 Y-Far: 1 Z-High: 1 Depnd: 8
Note: The Step Indep: and Depnd: values represent the number of gridlines to be used in the plot. The larger these number, the slower it is to produce the graph, although, the times utilized for graphic generation are relatively fast. For the time being well keep the default values of 10 and 8 for the Step data. Press @ERASE @DRAW to draw the three-dimensional surface. The result is a wireframe picture of the surface with the reference coordinate system shown at the lower left corner of the screen. By using the arrow keys () you can change the orientation of the surface. The orientation of the reference coordinate system will change accordingly. Try changing the surface orientation on your own. The following figures show a couple of views of the graph:

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When done, press @EXIT. Press @CANCL to return to the PLOT WINDOW environment. Change the Step data to read: Step Indep: 20

Depnd: 16

Press @ERASE @DRAW to see the surface plot. Sample views:
When done, press @EXIT. Press @CANCL to return to PLOT WINDOW. Press $ , or L@@@OK@@@, to return to normal calculator display.
Try also a Fast 3D plot for the surface z = f(x,y) = sin (x2+y2) Press , simultaneously if in RPN mode, to access the PLOT SETUP window. Press and type SIN(X^2+Y^2) @@@OK@@@. Press @ERASE @DRAW to draw the slope field plot. Press @EXIT @EDIT L @)LABEL @MENU to see the plot unencumbered by the menu and with identifying labels.

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Chapter 13 Vector Analysis Applications
This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vector analysis.

The del operator

The following operator, referred to as the del or nabla operator, is a vectorbased operator that can be applied to a scalar or vector function:

[ ] = i

[ ]+ j [ ]+ k [ x y z
When applied to a scalar function we can obtain the gradient of the function, and when applied to a vector function we can obtain the divergence and the curl of that function. A combination of gradient and divergence produces the Laplacian of a scalar function.

Gradient

The gradient of a scalar function (x,y,z) is a vector function defined by
grad = . Function HESS can be used to obtain the gradient of a
function. The function takes as input a function of n independent variables (x1, x2, ,xn), and a vector of the functions [x1 x2xn]. The function returns the Hessian matrix of the function, H = [hij] = [/xixj], the gradient of the function with respect to the n-variables, grad f = [ /x1 /x2 /xn], and the list of variables [x1, x2,,xn]. This function is easier to visualize in the RPN mode. Consider as an example the function (X,Y,Z) = X2 + XY + XZ, well apply function HESS to this scalar field in the following example:
Thus, the gradient is [2X+Y+Z, X, X].

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Alternatively, use function DERIV as follows:

Divergence

The divergence of a vector function, F(x,y,z) = f(x,y,z)i +g(x,y,z)j +h(x,y,z)k, is defined by taking a dot-product of the del operator with the function, i.e.,
divF = F. Function DIV can be used to calculate the divergence of a
vector field. For example, for F(X,Y,Z) = [XY,X2+Y2+Z2,YZ], the divergence is calculated, in ALG mode, as follows: DIV([X*Y,X^2+Y^2+Z^2,Y*Z],[X,Y,Z])
The curl of a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,is defined by a cross-product of the del operator with the vector field, i.e., curlF = F. The curl of vector field can be calculated with function CURL. For example, for the function F(X,Y,Z) = [XY,X2+Y2+Z2,YZ], the curl is calculated as follows: CURL([X*Y,X^2+Y^2+Z^2,Y*Z],[X,Y,Z])
For additional information on vector analysis applications see Chapter 15 in the calculators users guide.

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Chapter 14 Differential Equations
In this Chapter we present examples of solving ordinary differential equations (ODE) using calculator functions. A differential equation is an equation involving derivatives of the independent variable. In most cases, we seek the dependent function that satisfies the differential equation.

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 ,

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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 ().

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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

Using the RPN mode, the results are shown in the stack as a column vector in stack level 2, and a row vector of two components in stack level 1. The vector in stack level 1 is the number of outliers outside of the interval where the frequency count was performed. For this case, I get the values [ 14. 8.] indicating that there are, in the DAT vector, 14 values smaller than -8 and 8 larger than 8. Press to drop the vector of outliers from the stack. The remaining result is the frequency count of data.
The bins for this frequency distribution will be: -8 to -6, -6 to -4, , 4 to 6, and 6 to 8, i.e., 8 of them, with the frequencies in the column vector in the stack, namely (for this case): 23, 22, 22, 17, 26, 15, 20, 33. This means that there are 23 values in the bin [-8,-6], 22 in [-6,-4], 22 in [-4,2], 17 in [-2,0], 26 in [0,2], 15 in [2,4], 20 in [4,6], and 33 in [6,8]. You can also check that adding all these values plus the outliers, 14 and 8, show above, you will get the total number of elements in the sample, namely, 200.
Fitting data to a function y = f(x)
The program 3. Fit data., available as option number 3 in the STAT menu, can be used to fit linear, logarithmic, exponential, and power functions to

Page 16-4

data sets (x,y), stored in columns of the DAT matrix. For this application, you need to have at least two columns in your DAT variable. For example, to fit a linear relationship to the data shown in the table below: x y 0.5 2.3 3.6 6.7 7.2 11
First, enter the two columns of data into variable DAT by using the matrix writer, and function STO. To access the program 3. Fit data., use the following keystrokes: @@@OK@@@ The input form will show the current DAT, already loaded. If needed, change your set up screen to the following parameters for a linear fitting:
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

Page 16-5

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

You are then asked to select the alternative hypothesis:

Page 16-10

Select 150. Then, press @@@OK@@@. The result is:
Then, we reject H0: = 150, against H1: 150. The test z value is z0 = 5.656854. The P-value is 1.5410-8. The critical values of z/2 = 1.959964, corresponding to critical x range of {147.2 152.8}. This information can be observed graphically by pressing the soft-menu key @GRAPH:
Additional materials on statistical analysis, including definitions of concepts, and advanced statistical applications, are available in Chapter 18 in the users guide.

Page 16-11

Chapter 17 Numbers in Different Bases
Besides our decimal (base 10, digits = 0-9) number system, you can work with a binary system (base 2, digits = 0,1), an octal system (base 8, digits = 0-7), or a hexadecimal system (base 16, digits=0-9,A-F), among others. The same way that the decimal integer 321 means 3x102+2x101+1x100, the number 100110, in binary notation, means 1x25 + 0x24 + 0x23 + 1x22 + 1x21 + 0x20 = 32+0+0+4+2+0 = 38.

The BASE menu

The BASE menu is accessible through (the 3 key). With system flag 117 set to CHOOSE boxes (see Chapter 1 in this guide), the following entries are available:
With system flag 117 set to SOFT menus, the BASE menu shows the following:
This figure shows that the LOGIC, BIT, and BYTE entries within the BASE menu are themselves sub-menus. These menus are discussed in detail in Chapter 19 of the calculators users guide.
Writing non-decimal numbers
Numbers in non-decimal systems, referred to as binary integers, are written preceded by the # symbol () in the calculator. To select the current

Page 17-1

base to be used for binary integers, choose either HEX(adecimal), DEC(imal), OCT(al), or BIN(ary) in the BASE menu. For example, if @HEX ! is selected, binary integers will be a hexadecimal numbers, e.g., #53, #A5B, etc. As different systems are selected, the numbers will be automatically converted to the new current base. To write a number in a particular system, start the number with # and end with either h (hexadecimal), d (decimal), o (octal), or b (binary), examples: HEX DEC
For additional details on numbers from different bases see Chapter 19 in the calculators users guide.

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Chapter 18

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. 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: 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 supported, however all names longer than 62 characters will be ignored. THIS IS IMPORTANT, names longer than 62 characters cant be used with the Filer and will simply be ignored. 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.

Page 1-12

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:

2.5 +e 3

To enter this expression in the calculator we will first use the equation writer, O. Please identify the following keys in the keyboard, besides the numeric keypad keys: !@.#*+-/R QO`

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The equation writer is a display mode in which you can build mathematical expressions using explicit mathematical notation including fractions, derivatives, integrals, roots, etc. To use the equation writer for writing the expression shown above, use the following keystrokes: OR3*!51/3*3 /23Q3+!2.5` After pressing `the calculator displays the expression: (3*(5-1/(3*3))/23^3+EXP(2.5)) Pressing `again will provide the following value. Accept Approx. mode on, if asked, by pressing !!@@OK#@. [Note: The integer values used above, e.g., 3, 5, 1, represent exact values. The EXP(2.5), however, cannot be expressed as an exact value, therefore, a switch to Approx mode is required]:
You could also type the expression directly into the display without using the equation writer, as follows: R!3.*!5.1./ !3.*3. /23.Q3+!2.5` to obtain the same result. Change the operating mode to RPN by first pressing the H button. Select the RPN operating mode by either using the \key, or pressing the @CHOOS soft menu key. Press the !!@@OK#@ F soft menu key to complete the operation. The display, for the RPN mode looks as follows:

Page 1-14

Notice that the display shows several levels of output labeled, from bottom to top, as 1, 2, 3, etc. This is referred to as the stack of the calculator. The different levels are referred to as the stack levels, i.e., stack level 1, stack level 2, etc. Basically, what RPN means is that, instead of writing an operation such as 3 + 2, in the calculator by using 3+2`, we write first the operands, in the proper order, and then the operator, i.e., 3`2`+. As Entering you enter the operands, they occupy different stack levels. 3`puts the number 3 in stack level 1. Next, entering 2`pushes the 3 upwards to occupy stack level 2. Finally, by pressing +, we are telling the calculator to apply the operator, or program, + to the objects occupying levels 1 and 2. The result, 5, is then placed in level 1. A simpler way to calculate this operation is by using: 3`2+. Let's try some other simple operations before trying the more complicated expression used earlier for the algebraic operating mode: 123/32 123`32/ 4`2Q 27`3@ Notice the position of the y and the x in the last two operations. The base in the exponential operation is y (stack level 2) while the exponent is x (stack level 1) before the key Q is pressed. Similarly, in the cubic root operation, y (stack level 2) is the quantity under the root sign, and x (stack level 1) is the root. Try the following exercise involving 3 factors: (5 + 3) 2 Calculates (5 +3) first. 5`3+ 2X Completes the calculation. Let's try now the expression proposed earlier:

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 sub-menus

Page 5-9

of functions that apply to specific mathematical objects. This distinction between sub-menus (options 1 through 4) and plain functions (options 5 through 9) is made clear when system flag 117 is set to SOFT menus. Activating the ARITHMETIC menu ( ), under these circumstances, produces:
Following, we present the help facility entries for the functions of options 5 through 9 in the ARITHMETIC menu: DIVIS: FACTORS:
LGCD (Greatest Common Denominator):
PROPFRAC (proper fraction)

SIMP2:

The functions associated with the ARITHMETIC submenus: INTEGER, POLYNOMIAL, MODULO, and PERMUTATION, are the following:

INTEGER menu

EULER Number of integers < n, co -prime with n

Page 5-10

IABCUV Solves au + bv = c, with a,b,c = integers IBERNOULLI n-th Bernoulli number ICHINREM Chinese reminder for integers IDIV2 Euclidean division of two integers IEGCD Returns u,v, such that au + bv = gcd(a,b) IQUOT Euclidean quotient of two integers IREMAINDER Euclidean remainder of two integers ISPRIME? Test if an integer number is prime NEXTPRIME Next prime for a given integer number PA2B2 Prime number as square norm of a complex number PREVPRIME Previous prime for a given integer number

POLYNOMIAL menu

ABCUV Bzout polynomial equation (au+bv=c) CHINREM Chinese remainder for polynomials CYCLOTOMIC n-th cyclotomic polynomial DIV2 Euclidean division of two polynomials EGDC Returns u,v, from au+bv=gcd(a,b) FACTOR Factorizes an integer number or a polynomial FCOEF Generates fraction given roots and multiplicity FROOTS Returns roots and multiplicity given a fraction GCD Greatest common divisor of 2 numbers or polynomials HERMITE n-th degree Hermite polynomial HORNER Horner evaluation of a polynomial LAGRANGE Lagrange polynomial interpolation LCM Lowest common multiple of 2 numbers or polynomials LEGENDRE n-th degree Legendre polynomial PARTFRAC Partial-fraction decomposition of a given fraction PCOEF (help-facility entry missing) PTAYL Returns Q(x-a) in Q(x-a) = P(x), Taylor polynomial QUOT Euclidean quotient of two polynomials RESULTANT Determinant of the Sylvester matrix of 2 polynomials REMAINDER Euclidean reminder of two polynomials STURM Sturm sequences for a polynomial STURMAB Sign at low bound and number of zeros between bounds

Next, we proceed to eliminate the in position (1,3) by 2 Y \#3#1@RCIJ 1 -1/33/0 -1

Page 11-36

Finally, we eliminate the 1/16 from position (1,2) by using: 16 Y # 2#1@RCIJ -0 We now have an identity matrix in the portion of the augmented matrix corresponding to the original coefficient matrix A, thus we can proceed to obtain the solution while accounting for the row and column exchanges coded in the permutation matrix P. We identify the unknown vector x, the modified independent vector b and the permutation matrix P as:
X Y , b' = 1, P = 1. x= Z
The solution is given by Px=b, or

0 X Y = 1. 0 Z 1

Which results in

Y 3 Z = 1. X 1

Step-by-step calculator procedure for solving linear systems
The example we just worked is, of course, the step-by-step, user-driven procedure to use full pivoting for Gauss-Jordan elimination solution of linear equation systems. You can see the step-by-step procedure used by the calculator to solve a system of equations, without user intervention, by setting the step-by-step option in the calculators CAS, as follows:

Page 11-37

Then, for this particular example, in RPN mode, use: [2,-1,41] ` [[1,2,3],[2,0,3],[8,16,-1]] `/ The calculator shows an augmented matrix consisting of the coefficients matrix A and the identity matrix I, while, at the same time, showing the next procedure to calculate:
L2 = L2-2L1 stands for replace row 2 (L2) with the operation L2 2L1. If we had done this operation by hand, it would have corresponded to: 2\#1#1@RCIJ. Press @@@OK@@@, and follow the operations in your calculators screen. You will see the following operations performed: L3=L3-8L1, L1 = 2L1--1L2, L1=25L1--3L3, L2 = 25L2-3L3, and finally a message indicating Reduction result showing:
When you press @@@OK@@@ , the calculator returns the final result [1]. Calculating the inverse matrix step-by-step The calculation of an inverse matrix can be considered as calculating the solution to the augmented system [A | I ]. For example, for the matrix A used in the previous example, we would write this augmented matrix as

A aug ( I )

= 1 0.

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To see the intermediate steps in calculating and inverse, just enter the matrix A from above, and press Y, while keeping the step-by-step option active in the calculators CAS. Use the following: [[ 1,2,3],[3,-2,1],[4,2,-1]] `Y After going through the different steps, the solution returned is:
What the calculator showed was not exactly a Gauss-Jordan elimination with full pivoting, but a way to calculate the inverse of a matrix by performing a Gauss-Jordan elimination, without pivoting. This procedure for calculating the inverse is based on the augmented matrix (Aaug)nn = [A nn |Inn]. The calculator showed you the steps up to the point in which the left-hand half of the augmented matrix has been converted to a diagonal matrix. From there, the final step is to divide each row by the corresponding main diagonal pivot. In other words, the calculator has transformed (Aaug)nn = [A nn |Inn], into [I |A-1]. Inverse matrices and determinants Notice that all the elements in the inverse matrix calculated above are divided by the value 56 or one of its factors (28, 7, 8, 4 or 1). If you calculate the determinant of the matrix A, you get det(A) = 56. We could write, A-1 = C/det(A), where C is the matrix

Plots in polar coordinates
First of all, you may want to delete the variables used in previous examples (e.g., X, EQ, Y1, PPAR) using function PURGE (I @PURGE). By doing this, all parameters related to graphics will be cleared. Press J to check that the variables were indeed purged. We will try to plot the function f() = 2(1-sin()), as follows: First, make sure that your calculators angle measure is set to radians. Press , simultaneously if in RPN mode, to access to the PLOT SETUP window. Change TYPE to Polar, by pressing @CHOOS @@@OK@@@. Press and type: 2* 1-S~t @@@OK@@@.

Page 12-19

The cursor is now in the Indep field. Press ~t @@@OK@@@ to change the independent variable to. Press L@@@OK@@@ to return to normal calculator display. Press , simultaneously if in RPN mode, to access the PLOT window (in this case it will be called PLOT POLAR window). Change the H-VIEW range to 8 to 8, by using 8\@@@OK@@@8@@@OK@@@, and the V-VIEW range to -6 to 2 by using 6\@@@OK@@@2@@@OK@@@. Note: the H-VIEW and V-VIEW determine the scales of the display window only, and their ranges are not related to the range of values of the independent variable in this case. Change the Indep Low value to 0, and the High value to 6.28 ( 2), by using: 0@@@OK@@@ 6.28@@@OK@@@. Press @ERASE @DRAW to plot the function in polar coordinates. The result is a curve shaped like a hearth. This curve is known as a cardiod (cardios, Greek for heart).
Press @EDIT L @LABEL @MENU to see the graph with labels. Press L to recover the menu. Press L @)PICT to recover the original graphics menu. Press @TRACE @x,y@ to trace the curve. The data shown at the bottom of the display is the angle and the radius r, although the latter is labeled Y (default name of dependent variable). Press L@CANCL to return to the PLOT WINDOW screen. Press L@@@OK@@@ to return to normal calculator display.
In this exercise we entered the equation to be plotted directly in the PLOT SETUP window. We can also enter equations for plotting using the PLOT window, i.e., simultaneously if in RPN mode, pressing. For example, when you press after finishing the previous exercise, you

x = cos( ) r sin( ) = r y sin( ) r cos( )
With this result, integrals in polar coordinates are written as

Page 14-9

(r , )dA =

(r , )rdrd

where the region R in polar coordinates is R = { < < , f() < r < g()}. Double integrals in polar coordinates can be entered in the calculator, making sure that the Jacobian |J| = r is included in the integrand. The following is an example of a double integral calculated in polar coordinates, shown step-bystep:

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Chapter 15 Vector Analysis Applications
In this Chapter we present a number of functions from the CALC menu that apply to the analysis of scalar and vector fields. The CALC menu was presented in detail in Chapter 13. In particular, in the DERIV&INTEG menu we identified a number of functions that have applications in vector analysis, namely, CURL, DIV, HESS, LAPL. For the exercises in this Chapter, change your angle measure to radians.
A function defined in a region of space such as (x,y,z) is known as a scalar field, examples are temperature, density, and voltage near a charge. If the function is defined by a vector, i.e., F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, it is referred to as a vector field. The following operator, referred to as the del or nabla operator, is a vectorbased operator that can be applied to a scalar or vector function:

[ ] = i

[ ]+ j [ ]+ k [ x y z
When this operator is applied to a scalar function we can obtain the gradient of the function, and when applied to a vector function we can obtain the divergence and the curl of that function. A combination of gradient and divergence produces another operator, called the Laplacian of a scalar function. These operations are presented next.
Gradient and directional derivative
The gradient of a scalar function (x,y,z) is a vector function defined by

grad = = i

+ j +k x y z
The dot product of the gradient of a function with a given unit vector represents the rate of change of the function along that particular vector. This rate of change is called the directional derivative of the function, Du(x,y,z) = u.

Page 15-1

At any particular point, the maximum rate of change of the function occurs in the direction of the gradient, i.e., along a unit vector u = /||. The value of that directional derivative is equal to the magnitude of the gradient at any point Dmax(x,y,z) = /|| = || The equation (x,y,z) = 0 represents a surface in space. It turns out that the gradient of the function at any point on this surface is normal to the surface. Thus, the equation of a plane tangent to the curve at that point can be found by using a technique presented in Chapter 9. The simplest way to obtain the gradient is by using function DERIV, available in the CALC menu, e.g.,

Page 16-46

Fs { f (t )} = F ( ) =

Inverse sine transform

2 f (t ) sin( t ) dt 0

Fourier cosine transform

Fs1{F ( )} = f (t ) = F ( ) sin( t ) dt

Fc { f (t )} = F ( ) =

Inverse cosine transform

2 f (t ) cos( t ) dt 0

Fourier transform (proper)
Fc1 {F ( )} = f (t ) = F ( ) cos( t ) dt

F { f (t )} = F ( ) =

Inverse Fourier transform (proper)

1 f (t ) e it dt 2

F 1{F ( )} = f (t ) = F ( ) e it dt
Example 1 Determine the Fourier transform of the function f(t) = exp(- t), for t >0, and f(t) = 0, for t<0. The continuous spectrum, F(), is calculated with the integral:

e (1+i ) t dt = lim

(1+ i ) t
exp((1 + i ) ) = + i. + i
This result can be rationalized by multiplying numerator and denominator by the conjugate of the denominator, namely, 1-i. The result is now:

Page 16-47

F ( ) =
1 i = + i + i 1 i = i 1 + 1+ 2
which is a complex function. The absolute value of the real and imaginary parts of the function can be plotted as shown below
Notes: The magnitude, or absolute value, of the Fourier transform, |F()|, is the frequency spectrum of the original function f(t). For the example shown above, |F()| = 1/[2(1+2)]1/2. The plot of |F()| vs. was shown earlier. Some functions, such as constant values, sin x, exp(x), x2, etc., do not have Fourier transform. Functions that go to zero sufficiently fast as x goes to infinity do have Fourier transforms.
Properties of the Fourier transform
Linearity: If a and b are constants, and f and g functions, then F{af + bg} = a F{f }+ b F{g}. Transformation of partial derivatives. Let u = u(x,t). If the Fourier transform transforms the variable x, then F{u/x} = i F{u}, F{2u/x2} = -2 F{u}, F{u/t} = F{u}/t, F{2u/t2} = 2F{u}/t2

Page 16-48

convolution: For Fourier transform applications, the operation of convolution is defined as

( f * g )( x) =

f ( x ) g ( ) d.
The following property holds for convolution: F{f*g} = F{f}F{g}.
Fast Fourier Transform (FFT)
The Fast Fourier Transform is a computer algorithm by which one can calculate very efficiently a discrete Fourier transform (DFT). This algorithm has applications in the analysis of different types of time-dependent signals, from turbulence measurements to communication signals. The discrete Fourier transform of a sequence of data values {xj}, j = 0, 1, 2, , n-1, is a new finite sequence {Xk}, defined as
1 n 1 x j exp(i 2kj / n), n j =0

k = 0,1,2,., n 1

The direct calculation of the sequence Xk involves n2 products, which would involve enormous amounts of computer (or calculator) time particularly for large values of n. The Fast Fourier Transform reduces the number of operations to the order of nlog2n. For example, for n = 100, the FFT requires about 664 operations, while the direct calculation would require 10,000 operations. Thus, the number of operations using the FFT is reduced by a factor of 10000/664 15. The FFT operates on the sequence {xj} by partitioning it into a number of shorter sequences. The DFTs of the shorter sequences are calculated and later combined together in a highly efficient manner. For details on the algorithm refer, for example, to Chapter 12 in Newland, D.E., 1993, An

The PLOT sub-menu

The PLOT sub-menu contains functions that are used to produce plots with the data in the DATA matrix.
The functions included are: BARPL: produces a bar plot with data in Xcol column of the DATA matrix. HISTP: produces histogram of the data in Xcol column in the DATA matrix, using the default width corresponding to 13 bins unless the bin size is modified using function BINS in the 1VAR sub-menu (see above). SCATR: produces a scatterplot of the data in Ycol column of the DATA matrix vs. the data in Xcol column of the DATA matrix. Equation fitted will be stored in the variable EQ.

Page 18-17

The FIT sub-menu
The FIT sub-menu contains functions used to fit equations to the data in columns Xcol and Ycol of the DATA matrix.
The functions available in this sub-menu are: LINE: provides the equation corresponding to the most recent fitting. LR: provides intercept and slope of most recent fitting. PREDX: used as y @PREDX, given y find x for the fitting y = f(x). PREDY: used as x @PREDY, given x find y for the fitting y = f(x). CORR: provides the correlation coefficient for the most recent fitting. COV: provides sample co-variance for the most recent fitting PCOV: shows population co-variance for the most recent fitting.

The SUMS sub-menu

The SUMS sub-menu contains functions used to obtain summary statistics of the data in columns Xcol and Ycol of the DATA matrix.
X : provides the sum of values in Xcol column. Y : provides the sum of values in Ycol column. X^2 : provides the sum of squares of values in Xcol column. Y^2 : provides the sum of squares of values in Ycol column. X*Y: provides the sum of xy, i.e., the products of data in columns Xcol and Ycol. N : provides the number of columns in the DATA matrix.
Example of STAT menu operations
Let DATA be the matrix shown in next page. Type the matrix in level 1 of the stack by using the Matrix Writer. To store the matrix into DATA, use: @)DATA @DAT

Page 18-18

Calculate statistics of each column: @)STAT @)1VAR: @TOT @MEAN @SDEV @MAX @MIN L @VAR @PSDEV @PVAR Data: produces produces produces produces produces produces produces produces [38.5 87.5 82799.8] [5.5. 12.5 11828.54] [3.39 6.78 21097.01] [10 21.5 55066] [1.1 3.7 7.8] [11.52 46.08 445084146.33] [3.142 6.284 19532.04] [9.87 39.49 381500696.85]

Inferences concerning two means
The null hypothesis to be tested is Ho: 1-2 = , at a level of confidence (1)100%, or significance level , using two samples of sizes, n1 and n2, mean values x1 and x2, and standard deviations s1 and s2. If the populations standard deviations corresponding to the samples, 1 and 2, are known, or if n1 > 30 and n2 > 30 (large samples), the test statistic to be used is

( x1 x2 )

2 + n1 n2
If n1 < 30 or n2 < 30 (at least one small sample), use the following test statistic:
( x1 x2 ) (n1 1) s + (n2 1) s

n1n2 (n1 + n2 2) n1 + n2

Page 18-39
Two-sided hypothesis If the alternative hypothesis is a two-sided hypothesis, i.e., H1: 1-2 , The P-value for this test is calculated as If using z, If using t, P-value = 2UTPN(0,1, |zo|) P-value = 2UTPT(,|to|)
with the degrees of freedom for the t-distribution given by = n1 + n2 - 2. The test criteria are Reject Ho if P-value < Do not reject Ho if P-value >.
One-sided hypothesis If the alternative hypothesis is a two-sided hypothesis, i.e., H1: 1-2 < , or, H1: 1-2 < ,, the P-value for this test is calculated as: If using z, If using t, P-value = UTPN(0,1, |zo|) P-value = UTPT(,|to|)
The criteria to use for hypothesis testing is: Reject Ho if P-value < Do not reject Ho if P-value >.

Paired sample tests

When we deal with two samples of size n with paired data points, instead of testing the null hypothesis, Ho: 1-2 = , using the mean values and standard deviations of the two samples, we need to treat the problem as a single sample of the differences of the paired values. In other words, generate a new random variable X = X1-X2, and test Ho: = , where represents the mean of the population for X. Therefore, you will need to obtainx and s for the sample of values of x. The test should then proceed as a one-sample test using the methods described earlier.

Page 18-40

Inferences concerning one proportion
Suppose that we want to test the null hypothesis, H0: p = p0, where p represents the probability of obtaining a successful outcome in any given repetition of a Bernoulli trial. To test the hypothesis, we perform n repetitions of the experiment, and find that k successful outcomes are recorded. Thus, an estimate of p is given by p = k/n. The variance for the sample will be estimated as sp2 = p(1-p)/n = k(n-k)/n3. Assume that the Z score, Z = (p-p0)/sp, follows the standard normal distribution, i.e., Z ~ N(0,1). The particular value of the statistic to test is z0 = (p-p0)/sp. Instead of using the P-value as a criterion to accept or not accept the hypothesis, we will use the comparison between the critical value of z0 and the value of z corresponding to or /2. Two-tailed test If using a two-tailed test we will find the value of z /2, from Pr[Z> z/2] = 1-(z/2) = /2, or (z

To see this version of the program in action do the following: Store the program back into variable p by using [ ][ p ]. Run the program by pressing [ p ]. Enter values of V = 0.01, T = 300, and n = 0.8, when prompted (no units required now).
Before pressing ` for input, the stack will look like this:
Press ` to run the program. The output is a message box containing the string:
Press @@@OK@@@ to cancel message box output.

Page 21-42

Message box output without units Lets modify the program @@@p@@@ once more to eliminate the use of units throughout it. The unit-less program will look like this:
Enter V,T,n [S.I.]: { :V: :T: :n: {2 0} V } INPUT OBJ V T n V DTAG T DTAG n DTAG V T n V= V STR + + T= T STR + + n= n STR + + 8.31451*n*T/V EVAL STR p= SWAP + + + + MSGBOX
And when run with the input data V = 0.01, T = 300, and n = 0.8, produces the message box output:
Press @@@OK@@@ to cancel the message box output.
Relational and logical operators
So far we have worked mainly with sequential programs. The User RPL language provides statements that allow branching and looping of the program flow. Many of these make decisions based on whether a logical statement is true or not. In this section we present some of the elements used to construct such logical statements, namely, relational and logical operators.

Relational operators

Relational operators are those operators used to compare the relative position of two objects. For example, dealing with real numbers only, relational operators are used to make a statement regarding the relative position of two or more real numbers. Depending on the actual numbers used, such a

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statement can be true (represented by the numerical value of 1. in the calculator), or false (represented by the numerical value of 0. in the calculator). The relational operators available for programming the calculator are: ____________________________________________________ Operator Meaning Example ____________________________________________________ == is equal to x==2 is not equal to < is less than m<n > is greater than 10>a is greater than or equal to p q is less than or equal to 712 _____________________________________________________ All of the operators, except == (which can be created by typing ), are available in the keyboard. They are also available in @)TEST@. Two numbers, variables, or algebraics connected by a relational operator form a logical expression that can take value of true (1.), false (0.), or could simply not be evaluated. To determine whether a logical statement is true or not, place the statement in stack level 1, and press EVAL (). Examples: 2<10 , result: 1. (true) 2>10 , result: 0. (false) In the next example it is assumed that the variable m is not initialized (it has not been given a numerical value): 2==m , result: 2==m The fact that the result from evaluating the statement is the same original statement indicates that the statement cannot be evaluated uniquely.

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INDEP (a) The command INDEP specifies the independent variable and its plotting range. These specifications are stored as the third parameter in the variable PPAR. The default value is 'X'. The values that can be assigned to the independent variable specification are: A variable name, e.g., 'Vel' A variable name in a list, e.g., { Vel } A variable name and a range in a list, e.g., { Vel } A range without a variable name, e.g., { } Two values representing a range, e.g., 0 20
In a program, any of these specifications will be followed by the command INDEP. DEPND (b) The command DEPND specifies the name of the dependent variable. For the case of TRUTH plots it also specifies the plotting range. The default value is Y. The type of specifications for the DEPND variable are the same as those for the INDEP variable. XRNG (c) and YRNG (d) The command XRNG specifies the plotting range for the x-axis, while the command YRNG specifies the plotting range for the y-axis. The input for any of these commands is two numbers representing the minimum and maximum values of x or y. The values of the x- and y-axis ranges are stored as the ordered pairs (xmin, ymin) and (xmax, ymax) in the two first elements of the variable PPAR. Default values for xmin and xmax are -6.5 and 6.5, respectively. Default values for xmin and xmax are 3.1 and 3.2, respectively. RES (e) The RES (RESolution) command specifies the interval between values of the independent variable when producing a specific plot. The resolution can be expressed in terms of user units as a real number, or in terms of pixels as a binary integer (numbers starting with #, e.g., #10). The resolution is stored as the fourth item in the PPAR variable.

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CENTR (g) The command CENTR takes as argument an ordered pair (x,y) or a value x, and adjusts the first two elements in the variable PPAR, i.e., (xmin, ymin) and (xmax, ymax), so that the center of the plot is (x,y) or (x,0), respectively. SCALE (h) The SCALE command determines the plotting scale represented by the number of user units per tick mark. The default scale is 1 user-unit per tick mark. When the command SCALE is used, it takes as arguments two numbers, xscale and yscale, representing the new horizontal and vertical scales. The effect of the SCALE command is to adjust the parameters (xmin, ymin) and (xmax, ymax) in PPAR to accommodate the desired scale. The center of the plot is preserved. SCALEW (i) Given a factor xfactor, the command SCALEW multiplies the horizontal scale by that factor. The W in SCALEW stands for 'width.' The execution of SCALEW changes the values of xmin and xmax in PPAR. SCALEH (j) Given a factor yfactor, the command SCALEH multiplies the vertical scale by that factor. The H in SCALEH stands for 'height.' The execution of SCALEW changes the values of ymin and ymax in PPAR. Note: Changes introduced by using SCALE, SCALEW, or SCALEH, can be used to zoom in or zoom out in a plot. ATICK (l) The command ATICK (Axes TICK mark) is used to set the tick-mark annotations for the axes. The input value for the ATICK command can be one of the following: A real value x : sets both the x- and y-axis tick annotations to x units A list of two real values { x y }: sets the tick annotations in the x- and yaxes to x and y units, respectively. A binary integer #n: sets both the x- and y-axis tick annotations to #n pixels

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L produces no additional menu items). The soft menu key commands are the following: @EXIT @ECHO @@ SEE1@@ @@SEE2@ !@@SEE3@ @!MAIN A B C D E F EXIT the help facility Copy the example command to the stack and exit See the first link (if any) in the list of references See the second link (if any) of the list of references See the third link (if any) of the list of references Return to the MAIN command list in the help facility
In this case we want to ECHO the example into the stack by pressing @ECHO B. The resulting display is the following:
There are now four lines of the display occupied with output. The first two lines from the top correspond to the first exercise with the HELP facility in which we cancel the request for help. The third line from the top shows the most recent call to the HELP facility, while the last line shows the ECHO of the example command. To activate the command press the ` key. The result is:
Notice that, as new lines of output are produced, the display (or stack) pushes the existing lines upwards and fills the bottom of the screen with more output. The HELP facility, described in this section, will be very useful to refer to the definition of the many CAS commands available in the calculator. Each entry in the CAS help facility, whenever appropriate, will have an example of application of the command, as well as references as shown in this example.

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To navigate quickly to a particular command in the help facility list without having to use the arrow keys all the time, we can use a shortcut consisting of typing the first letter in the commands name. Suppose that we want to find information on the command IBP (Integration By Parts), once the help facility list is available, use the ~ key (first key in the fourth row from the bottom of the keyboard) followed by the key for the letter i (the same as the key I) , i.e., ~i. This will take you automatically to the first command that starts with an i, namely, IBASIS. Then, you can use the down arrow key , twice, to find the command IBP. Pressing the !!@@OK#@ F key, we activate the help facility for this command. Press @!MAIN F to recover the main list of commands, or @EXIT A to exit the facility.
References for non-CAS commands
The help facility contains entries for all the commands developed for the CAS (Computer Algebraic System). There is a large number of other functions and commands that were originally developed for the HP 48G series calculators that are not included in the help facility. Good references for those commands are the HP 48G Series users guide (HP Part No. 00048-90126) and the HP 48G Series Advanced Users Reference Manual (HP Part No. 00048-90136) both published by Hewlett-Packard Company, Corvallis, Oregon, in 1993.

These functions are available in the CALC/SOLVE menu (start with ). The functions are described in Chapters 6, 11, and 16.
The CMPLX menu includes the following functions:
The CMPLX menu is also available in the keyboard (). Some of the functions in CMPLX are also available in the MTH/COMPLEX menu (start with ). Complex number functions are presented in Chapter 4.

The ARIT sub-menu

The ARIT menu includes the following sub-menus:
The sub-menus INTEGER, MODULAR, and POLYNOMIAL are presented in detail in Appendix J.

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The EXP&LN sub-menu
The EXP&LN menu contains the following functions:
This menu is also accessible through the keyboard by using. The functions in this menu are presented in Chapter 5.

The MATR sub-menu

The MATR menu contains the following functions:
These functions are also available through the MATRICES menu in the keyboard (). The functions are described in Chapters 10 and 11.

The REWRITE sub-menu

The REWRITE menu contains the following functions:

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These functions are available through the CONVERT/REWRITE menu (start with ). The functions are presented in Chapter 5, except for functions XNUM and XQ, which are described next using the corresponding entries in the CAS help facility (IL@HELP ):

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Appendix L Line editor commands
When you trigger the line editor by using in the RPN stack or in ALG mode, the following soft menu functions are provided (press L to see the remaining functions):
The functions are briefly described as follows: SKIP: Skips characters to beginning of word. SKIP : Skips characters to end of word. DEL: Delete characters to beginning of word. DEL : Delete characters to end of word. DEL L: Delete characters in line. INS: When selected inserts characters at cursor location. If not selected, the cursor replaces characters (overwrites) instead of inserting characters. EDIT: Edits selection. BEG: Move to beginning of word. END: Mark end of selection. INFO: Provides information on Command Line editor, e.g.,

Saddle point, 14-5, Sample correlation coefficient, 18-11 Sample covariance, 18-11 Sample vs. population, 18-5 Saving a graph, 12-7 Scalar field, 15-1 SCALE, 22-7 SCALEH, 22-7 SCALEW, 22-7 Scatterplots, 12-32 Scientific format, 1-20 Scope global variable 21-4 SEARCH menu, L-2 L-3 Selection tree in Equation Writer, E-1 SEND, 2-34 SEQ, 8-11 Sequential programming, 21-15

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Series, 13-23 Series Maclaurin, 13-23 Series Taylor, 13-23 SERIES, 13-23 Series Fourier, 16-27 Setting time and date, 25-2 SHADE in plots, 12-6 Shortcuts, G-1 SI, 3-29 SIGMA, 13-14 SIGMAVX, 13-14 SIGN, 3-14 SIGN, 4-6 SIGNTAB, 12-52 13-10 SIMP2, 5-10, 5-24 SIMPLIFY, 5-29 Simplify non-rational CAS setting, C10 Simplifying an expression, 2-23 SIN, 3-7 Single-variable statistics, 18-2
Singular value decomposition, 11-8
SINH, 3-9 SIZE, 8-10 SIZE, 10-7 SKIP , L-1 SL, 19-6 , SLB, 19-7 Slope fields, 12-34 Slope fields for differential equations, 16-3 SLOPE in plots, 12-7 SNRM, 11-7 SOFT menus, 1-3 SOLVE, 5-5 SOLVE, 6-2, 7-1,
SOLVE menu, 6-27 SOLVE menu (menu 74), G-3 SOLVE/DIFF menu, 16-69 SOLVEVX, 6-4 SOLVR menu, 6-28 SORT, 2-34 Special characters, G-2 Speed units, 3-19 SPHERE, 9-15 SQ, 3-5 Square root, 3-5 Square wave Fourier series 16-39 SR, 19-6 SRAD, 11-9 SRB, 19-7 SREPL, 23-3 SST, 21-35 Stack properties, 1-27 Standard deviation, 18-4 Standard format, 1-17 Standard normal distribution, 1717, START.STEP construct, 21-58 START.NEXT construct, 21-54 STAT menu, 18-15 STAT menu (menu 96), G-3 Statistical inference probability distributions, 17-9 Statistics, 18-1 Step function (Heaviside's ), 16-15 Step-by-step CAS mode, C-7 Step-by-step derivatives, 13-16 Step-by-step integrals, 13-16 STEQ, 6-14 Stiff differential equations, 16-71 Stiff ODE, 16-68

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Stiff ODEs numerical solution, 16-69
Strings, 23-1 STO, 2-46 STOALARM, 25-4 STOKEYS, 20-6 STREAM, 8-12 String concatenation, 23-2 Student t distribution, 17-11 STURM, 5-11 STURMAB, 5-11 STWS, 19-4 Style menu, L-4 SUB, 10-11 Sub-directories creating, 2-38 Sub-directories deleting, 2-42 Sub-expressions, 2-17 SUBST, 5-5 SUBTMOD, 5-12 SUBTMOD, 5-16 Sum of squared errors (SSE), 18-62 Sum of squared totals (SST), 18-62, Summary statistics, 18-13 SVD, 11-50 SVL, 11-50 SYLVESTER, 11-53 SYMB/GRAPH menu, 12-52 Symbolic CAS mode, C-3 SYMBOLIC menu, 12-51 Synthetic division, 5-26, SYST2MAT, 11-42, System flag (EXACT/APPROX), G-1, System flag 117 (CHOOSE/SOFT), 1-4, G-2, System flag 95 (ALG/RPN), G-1