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:

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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:

Copying variables

The following exercises show different ways of copying variables from one sub-directory to another. Using the FILES menu To copy a variable from one directory to another you can use the FILES menu. For example, within the sub-directory {HOME MANS INTRO}, we have

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variables p1, z1, R, Q, A12, , and A. Suppose that we want to copy variable A and place a copy in sub-directory {HOME MANS}. Also, we will copy variable R and place a copy in the HOME directory. Here is how to do it: Press @@OK@@ to produce the following list of variables:
Use the down-arrow key to select variable A (the last in the list), then press @@COPY@. The calculator will respond with a screen labeled PICK DESTINATION:
Use the up arrow key to select the sub-directory MANS and press @@OK@@. If you now press , the screen will show the contents of sub-directory MANS (notice that variable A is shown in this list, as expected):
Press $ @INTRO@ `(Algebraic mode), or $ @INTRO@ (RPN mode) to return to the INTRO directory. Press @@OK@@ to produce the list of variables in {HOME MANS INTRO}. Use the down arrow key () to select variable R, then press @@COPY@. Use the up arrow key () to select the HOME directory, and press @@OK@@. If you now press , twice, the screen will show the contents of the HOME directory, including a copy of variable R:

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Using the history in Algebraic mode Here is a way to use the history (stack) to copy a variable from one directory to another with the calculator set to the Algebraic mode. Suppose that we are within the sub-directory {HOME MANS INTRO}, and want to copy the contents of variable z1 to sub-directory {HOME MANS}. Use the following procedure: @@z1@ K@@z1@ ` This simply stores the contents of z1 into itself (no change effected on z1). Next, use ` to move to the {HOME MANS} sub-directory. The calculator screen will look like this:
Next, use the delete key three times, to remove the last three lines in the display: . At this point, the stack is ready to execute the command ANS(1) z1. Press ` to execute this command. Then, use @@z1@, to verify the contents of the variable. Using the stack in RPN mode To demonstrate the use of the stack in RPN mode to copy a variable from one sub-directory to another, we assume you are within sub-directory {HOME MANS INTRO}, and that we will copy the contents of variable z1 into the HOME directory. Use the following procedure:@@z1@ `@@z1@ ` This procedure lists the contents and the name of the variable in the stack. The calculator screen will look like this:

This menu contains the following functions:
These functions are discussed in detail in Chapter 10.
REWRITE convert menu (Option 4)
Functions I R and R I are used to convert a number from integer (I) to real (R), or vice versa. Integer numbers are shown without trailing decimal points, while real numbers representing integers will have a trailing decimal point, e.g.,

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Function NUM has the same effect as the keystroke combination (associated with the ` key). Function NUM converts a symbolic result into its floating-point value. Function Q converts a floating-point value into a fraction. Function Q converts a floating-point value into a fraction of , if a fraction of can be found for the number; otherwise, it converts the number to a fraction. Examples are of these three functions are shown next.
Out of the functions in the REWRITE menu, functions DISTRIB, EXPLN, EXP2POW, FDISTRIB, LIN, LNCOLLECT, POWEREXPAND, and SIMPLIFY apply to algebraic expressions. Many of these functions are presented in this Chapter. However, for the sake of completeness we present here the helpfacility entries for these functions. DISTRIB EXPLN

EXP2POW

FDISTRIB

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LNCOLLECT

POWEREXPAND

SIMPLIFY

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Chapter 6 Solution to single equations
In this chapter we feature those functions that the calculator provides for solving single equations of the form f(X) = 0. Associated with the 7 key there are two menus of equation-solving functions, the Symbolic SOLVer (), and the NUMerical SoLVer (). Following, we present some of the functions contained in these menus. Change CAS mode to Complex for these exercises (see Chapter 2).
Symbolic solution of algebraic equations
Here we describe some of the functions from the Symbolic Solver menu. Activate the menu by using the keystroke combination. With system flag 117 set to CHOOSE boxes, the following menu lists will be available:
Functions DESOLVE and LDEC are used for the solution of differential equations, the subject of a different chapter, and therefore will not be presented here. Similarly, function LINSOLVE relates to the solution of multiple linear equations, and it will be presented in a different chapter. Functions ISOL and SOLVE can be used to solve for any unknown in a polynomial equation. Function SOLVEVX solves a polynomial equation where the unknown is the default CAS variable VX (typically set to X). Finally, function ZEROS provides the zeros, or roots, of a polynomial. Entries for all the functions in the S.SLV menu, except ISOL, are available through the CAS help facility (IL@HELP ).

Function ISOL

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

Solution to simultaneous equations with MSLV
Function MSLV is available as the last option in the menu:
The help-facility entry for function MSLV is shown next:

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Example 1 Example from the help facility
As with all function entries in the help facility, there is an example attached to the MSLV entry as shown above. Notice that function MSLV requires three arguments: 1. A vector containing the equations, i.e., [SIN(X)+Y,X+SIN(Y)=1] 2. A vector containing the variables to solve for, i.e., [X,Y] 3. A vector containing initial values for the solution, i.e., the initial values of both X and Y are zero for this example. In ALG mode, press @ECHO to copy the example to the stack, press ` to run the example. To see all the elements in the solution you need to activate the line editor by pressing the down arrow key ():
In RPN mode, the solution for this example is produced by using:
Activating function MSLV results in the following screen.
You may have noticed that, while producing the solution, the screen shows intermediate information on the upper left corner. Since the solution provided by MSLV is numerical, the information in the upper left corner shows the results of the iterative process used to obtain a solution. The final solution is X = 1.8238, Y = -0.9681.

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Example 2 - Entrance from a lake into an open channel
This particular problem in open channel flow requires the simultaneous solution of two equations, the equation of energy: H o = y + Mannings equation: Q =

V2 , and 2g

Cu A 5 / 3 S o. In these equations, Ho n P2/3
represents the energy head (m, or ft) available for a flow at the entrance to a channel, y is the flow depth (m or ft), V = Q/A is the flow velocity (m/s or ft/s), Q is the volumetric discharge (m3/s or ft3/s), A is the cross-sectional area (m2 or ft2), Cu is a coefficient that depends on the system of units (Cu = 1.0 for the SI, Cu = 1.486 for the English system of units), n is the Mannings coefficient, a measure of the channel surface roughness (e.g., for concrete, n = 0.012), P is the wetted perimeter of the cross section (m or ft), So is the slope of the channel bed expressed as a decimal fraction. For a trapezoidal channel, as shown below, the area is given by A = (b + my ) y , while the wetted perimeter is given by P = b + 2 y 1 + m , where b is the bottom width (m or ft), and m is the side slope (1V:mH) of the cross section. Typically, one has to solve the equations of energy and Mannings simultaneously for y and Q. Once these equations are written in terms of the primitive variables b, m, y, g, So, n, Cu, Q, and Ho, we are left with a system of equations of the form f1(y,Q) = 0, f2(y,Q) = 0. We can build these two equations as follows. We assume that we will be using the ALG and Exact modes in the calculator, although defining the equations and solving them with MSLV is very similar in the RPN mode. Create a sub-directory, say CHANL (for open CHANneL), and within that sub-directory define the following variables:

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Functions GETI and PUTI, also available in sub-menu PRG/ ELEMENTS/, can also be used to extract and place elements in a list. These two functions, however, are useful mainly in programming. Function GETI uses the same arguments as GET and returns the list, the element location plus one, and the element at the location requested. Function PUTI uses the same arguments as GET and returns the list and the list size.
Element position in the list
To determine the position of an element in a list use function POS having the list and the element of interest as arguments. For example,

HEAD and TAIL functions

The HEAD function extracts the first element in the list. The TAIL function removes the first element of a list, returning the remaining list. Some examples are shown next:

The SEQ function

Item 2. PROCEDURES. in the PRG/LIST menu contains the following functions that can be used to operate on lists.
Functions REVLIST and SORT were introduced earlier as part of the MTH/LIST menu. Functions DOLIST, DOSUBS, NSUB, ENDSUB, and STREAM, are designed as programming functions for operating lists in RPN mode. Function

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SEQ is useful to produce a list of values given a particular expression and is described in more detail here. The SEQ function takes as arguments an expression in terms of an index, the name of the index, and starting, ending, and increment values for the index, and returns a list consisting of the evaluation of the expression for all possible values of the index. The general form of the function is SEQ(expression, index, start, end, increment). In the following example, in ALG mode, we identify expression = n2, index = n, start = 1, end = 4, and increment = 1:
The list produced corresponds to the values {12, 22, 32, 42}. In RPN mode, you can list the different arguments of the function as follows:
before applying function SEQ.

The MAP function

The MAP function, available through the command catalog (N), takes as arguments a list of numbers and a function f(X) or a program of the form << a >>, 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}:

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] A general row vector [1.5,-2.2] A 2-D vector [3,-1,2] A 3-D vector ['t','t^2','SIN(t)'] 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:

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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 () or spaces (#). Notice that after pressing ` , in either mode, the calculator shows the vector elements separated by spaces.
Storing vectors into variables
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:
Then, in RPN mode (before pressing K, repeatedly):
Using the Matrix Writer (MTRW) to enter vectors
Vectors can also be entered by using the Matrix Writer (third key in the fourth row of keys from the top of the keyboard). This command generates a species of spreadsheet corresponding to rows and columns of a matrix (Details on using the Matrix Writer to enter matrices will be presented in a subsequent chapter). For a vector we are interested in filling only elements in the top row. By default, the cell in the top row and first column is selected. At the bottom of the spreadsheet you will find the following soft menu keys: @EDIT! @VEC

@WID @GO

The @EDIT key is used to edit the contents of a selected cell in the Matrix Writer. The @VEC@@ key, when selected, will produce a vector, as opposite to a matrix of one row and many columns.

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Vectors vs. matrices To see the @VEC@ key in action, try the following exercises: (1) Launch the Matrix Writer (). With @VEC and @GO selected, enter 3`5`2``. This produces [3. 5. 2.]. (In RPN mode, you can use the following keystroke sequence to produce the same result: 3#5#2``). (2) With @VEC@@ deselected and @GO selected,, enter 3#5#2``. This produces [[3. 5. 2.]]. Although these two results differ only in the number of brackets used, for the calculator they represent different mathematical objects. The first one is a vector with three elements, and the second one a matrix with one row and three columns. There are differences in the way that mathematical operations take place on a vector as opposite to a matrix. Therefore, for the time being, keep the soft menu key @VEC selected while using the Matrix Writer.
The WID key is used to decrease the width of the columns in the spreadsheet. Press this key a couple of times to see the column width decrease in your Matrix Writer. The @WID key is used to increase the width of the columns in the spreadsheet. Press this key a couple of times to see the column width increase in your Matrix Writer. The @GO key, when selected, automatically selects the next cell to the right of the current cell when you press `. This option is selected by default. The @GO key, when selected, automatically selects the next cell below the current cell when you press `. Moving to the right vs. moving down in the Matrix Writer Activate the Matrix Writer and enter 3`5`2`` with the @GO key selected (default). Next, enter the same sequence of numbers with the @GO key selected to see the difference. In the first case you entered a vector of three elements. In the second case you entered a matrix of three rows and one column.

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Activate the Matrix Writer again by using , and press L to check out the second soft key menu at the bottom of the display. It will show the keys: @+ROW@ @-ROW @+COL@ @-COL@ @STK@@ @GOTO@ The @+ROW@ key will add a row full of zeros at the location of the selected cell of the spreadsheet. The @-ROW key will delete the row corresponding to the selected cell of the spreadsheet. The @+COL@ key will add a column full of zeros at the location of the selected cell of the spreadsheet. The @-COL@ key will delete the column corresponding to the selected cell of the spreadsheet. The @STK@@ key will place the contents of the selected cell on the stack. The @GOTO@ key, when pressed, will request that the user indicate the number of the row and column where he or she wants to position the cursor. Pressing L once more produces the last menu, which contains only one function @@DEL@ (delete). The function @@DEL@ will delete the contents of the selected cell and replace it with a zero. To see these keys in action try the following exercise: (1) Activate the Matrix Writer by using. Make sure the @VEC and @GO keys are selected. (2) Enter the following: 1`2`3` L @GOTO@ 2@@OK@@ 1 @@OK@@ @@OK@@ 2`1`5` 4`5`6` 7`8`9` (3) Move the cursor up two positions by using. Then press @-ROW. The second row will disappear. (4) Press @+ROW@. A row of three zeroes appears in the second row.

Dot product

Function DOT 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:

Cross product

Function CROSS is used to calculate the cross product of two 2-D vectors, of two 3-D vectors, or of one 2-D and one 3-D 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]:

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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 (Invalid Dimension), e.g., CROSS(v3,A), etc.

Decomposing a vector

Function V is used to decompose a vector into its elements or components. If used in the ALG mode, V will provide the elements of the vector in a list, e.g.,
In the RPN mode, application of function V will list the components of a vector in the stack, e.g., V (A) will produce the following output in the RPN stack (vector A is listed in stack level 6:).
Building a two-dimensional vector
Function V2 is used in the RPN mode to build a vector with the values in stack levels 1: and 2:. The following screen shots show the stack before and after applying function V2:

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Building a three-dimensional vector
Function V3 is used in the RPN mode to build a vector with the values in stack levels 1: , 2:, and 3:. The following screen shots show the stack before and after applying function V2:
Changing coordinate system
Functions RECT, CYLIN, and SPHERE are used to change the current coordinate system to rectangular (Cartesian), cylindrical (polar), or spherical coordinates. The current system is shown highlighted in the corresponding CHOOSE box (system flag 117 unset), or selected in the corresponding SOFT menu label (system flag 117 set). In the following figure the RECTangular coordinate system is shown as selected in these two formats:
When the rectangular, or Cartesian, coordinate system is selected, the top line of the display will show an XYZ field, and any 2-D or 3-D vector entered in the calculator is reproduced as the (x,y,z) components of the vector. Thus, to enter the vector A = 3i+2j-5k, we use [3,2,-5], and the vector is shown as:

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Function SRAD
Function SRAD determines the Spectral RADius of a matrix, defined as the largest of the absolute values of its eigenvalues. For example,
Definition of eigenvalues and eigenvectors of a matrix The eigenvalues of a square matrix result from the matrix equation Ax = x. The values of that satisfy the equation are known as the eigenvalues of the matrix A. The values of x that result from the equation for each value of l are known as the eigenvectors of the matrix. Further details on calculating eigenvalues and eigenvectors are presented later in the chapter.

Function COND

Function COND determines the condition number of a matrix. Examples,
Condition number of a matrix The condition number of a square non-singular matrix is defined as the product of the matrix norm times the norm of its inverse, i.e., cond(A) = ||A||||A-1||. We will choose as the matrix norm, ||A||, the maximum of its row norm (RNRM) and column norm (CNRM), while the norm of the inverse, ||A-1||, will be selected as the minimum of its row norm and column norm. Thus, ||A|| = max(RNRM(A),CNRM(A)), and ||A-1|| = min(RNRM(A-1), CNRM(A-1)).

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The condition number of a singular matrix is infinity. The condition number of a non-singular matrix is a measure of how close the matrix is to being singular. The larger the value of the condition number, the closer it is to singularity. (A singular matrix is one for which the inverse does not exist). Try the following exercise for matrix condition number on matrix A33. The condition number is COND(A33) , row norm, and column norm for A33 are shown to the left. The corresponding numbers for the inverse matrix, INV(A33), are shown to the right:
Since RNRM(A33) > CNRM(A33), then we take ||A33|| = RNRM(A33) = 21. Also, since CNRM(INV(A33)) < RNRM(INV(A33)), then we take ||INV(A33)|| = CNRM(INV(A33)) = 0.261044. Thus, the condition number is also calculated as CNRM(A33)*CNRM(INV(A33)) = COND(A33) = 6.7871485

Function RANK

Function RANK determines the rank of a square matrix. Try the following examples:
The rank of a matrix The rank of a square matrix is the maximum number of linearly independent rows or columns that the matrix contains. Suppose that you write a square matrix Ann as A = [c1 c2 cn], where ci (i = 1, 2, , n) are vectors representing the columns of the matrix A, then, if any of those columns, say ck, can be written as c k =

The MTH/PROBABILITY. sub-menu is accessible through the keystroke sequence. With system flag 117 set to CHOOSE boxes, the following list of MTH options is provided (see left-hand side figure below). We have selected the PROBABILITY. option (option 7), to show the following functions (see right-hand side figure below):
In this section we discuss functions COMB, PERM, ! (factorial), RAND, and RDZ.
Factorials, combinations, and permutations
The factorial of an integer n is defined as: n! = n (n-1) (n-2)321. By definition, 0! = 1. Factorials are used in the calculation of the number of permutations and combinations of objects. For example, the number of permutations of r objects from a set of n distinct objects is
Pr = n( n 1)(n 1).( n r + 1) = n! /( n r )!
Also, the number of combinations of n objects taken r at a time is
n n(n 1)(n 2).(n r + 1) n! = = r r! r!(n r )!

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

Random numbers

The calculator provides a random number generator that returns a uniformly distributed, random real number between 0 and 1. The generator is able to produce sequences of random numbers. However, after a certain number of times (a very large number indeed), the sequence tends to repeat itself. For that reason, the random number generator is more properly referred to as a pseudo-random number generator. To generate a random number with your calculator use function RAND from the MTH/PROBABILITY sub-menu. The following screen shows a number of random numbers produced using RAND. The numbers in the left-hand side figure are produced with calling function RAND without an argument. If you place an argument list in function RAND, you get back the list of numbers plus an additional random number attached to it as illustrated in the right-hand side figure.

The complement of is called the power of the test of the null hypothesis H0 vs. the alternative H1. The power of a test is used, for example, to determine a minimum sample size to restrict errors. Selecting values of and A typical value of the level of significance (or probability of Type I error) is = 0.05, (i.e., incorrect rejection once in 20 times on the average). If the consequences of a Type I error are more serious, choose smaller values of , say 0.01 or even 0.001. The value of , i.e., the probability of making an error of Type II, depends on , the sample size n, and on the true value of the parameter tested. Thus, the value of is determined after the hypothesis testing is performed. It is customary to draw graphs showing , or the power of the test (1- ), as a function of the true value of the parameter tested. These graphs are called operating characteristic curves or power function curves, respectively.
Inferences concerning one mean
Two-sided hypothesis The problem consists in testing the null hypothesis Ho: = o, against the alternative hypothesis, H1: at a level of confidence (1-)100%, or significance level , using a sample of size n with a mean x and a standard deviation s. This test is referred to as a two-sided or two-tailed test. The procedure for the test is as follows:

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First, we calculate the appropriate statistic for the test (to or zo) as follows: If n < 30 and the standard deviation of the population, , is known, use the z-statistic:
If n > 30, and is known, use zo as above. replace s for in zo, i.e., use z o =

x o s/ n

If is not known,
If n < 30, and is unknown, use the t-statistic to = n - 1 degrees of freedom.

x o , with = s/ n

Then, calculate the P-value (a probability) associated with either z or t , and compare it to to decide whether or not to reject the null hypothesis. The Pvalue for a two-sided test is defined as either P-value = P(|z|>|zo|), or, P-value = P(|t|>|to|). The criteria to use for hypothesis testing is: Reject Ho if P-value < Do not reject Ho if P-value >.
The P-value for a two-sided test can be calculated using the probability functions in the calculator as follows: If using z, If using t, P-value = 2UTPN(0,1,|zo|) P-value = 2UTPT(,|to|)
Example 1 -- Test the null hypothesis Ho: = 22.5 ( = o), against the alternative hypothesis, H1: 22.5, at a level of confidence of 95% i.e., = 0.05, using a sample of size n = 25 with a mean x = 22.0 and a standard

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deviation s = 3.5. We assume that we don't know the value of the population standard deviation, therefore, we calculate a t statistic as follows:
x o 22.0 22.5 = = 0.7142 s/ n 3.5 / 25
The corresponding P-value, for n = 25 - 1 = 24 degrees of freedom is P-value = 2UTPT(24,-0.7142) = 20.7590 = 1.5169, since 1.5169 > 0.05, i.e., P-value > , we cannot reject the null hypothesis Ho: = 22.0. One-sided hypothesis The problem consists in testing the null hypothesis Ho: = o, against the alternative hypothesis, H1: > or H1: < at a level of confidence (1)100%, or significance level , using a sample of size n with a mean x and a standard deviation s. This test is referred to as a one-sided or one-tailed test. The procedure for performing a one-side test starts as in the two-tailed test by calculating the appropriate statistic for the test (to or zo) as indicated above. Next, we use the P-value associated with either z or t , and compare it to to decide whether or not to reject the null hypothesis. The P-value for a twosided test is defined as either P-value = P(z > |zo|), or, P-value = P(t > |to|). The criteria to use for hypothesis testing is: Reject Ho if P-value < Do not reject Ho if P-value >.

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An example of a program using GROB The following program produces the graph of the sine function including a frame drawn with the function BOX and a GROB to label the graph. Here is the listing of the program: RAD 131 R B 64 R B PDIM -6.28 6.28 XRNG 2. 2. YRNG FUNCTION SIN(X) STEQ ERASE DRAX LABEL DRAW (-6.28,-2.) (6.28,2.) BOX PICT RCL SINE FUNCTION 1 GROB (-6., 1.5) SWAP GOR PICT STO { } PVIEW Begin program Set angle units to radians Set PICT screen to 13164 pixels Set x- and y-ranges Select FUNCTION type for graphs Store the function sine into EQ Clear, draw axes, labels, graph Draw a frame around the graph Place contents of PICT on stack Place graph label string in stack Convert string into a small GROB Coordinates to place label GROB Combine PICT with the label GROB Save combined GROB into PICT Bring PICT to the stack End program
Save the program under the name GRPR (GROB PRogram). Press @GRPR to run the program. The output will look like this:
A program with plotting and drawing functions
In this section we develop a program to produce, draw and label Mohrs circle for a given condition of two-dimensional stress. The left-hand side figure below shows the given state of stress in two-dimensions, with xx and yy being normal stresses, and xy = yx being shear stresses. The right-hand

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side figure shows the state of stresses when the element is rotated by an angle. In this case, the normal stresses are xx and yy, while the shear stresses are xy and yx.
The relationship between the original state of stresses (xx, yy, xy, yx) and the state of stress when the axes are rotated counterclockwise by f (xx, yy, xy, yx), can be represented graphically by the construct shown in the figure below. To construct Mohrs circle we use a Cartesian coordinate system with the xaxis corresponding to the normal stresses (), and the y-axis corresponding to the shear stresses (). Locate the points A(xx,xy) and B (yy, xy), and draw the segment AB. The point C where the segment AB crosses the n axis will be the center of the circle. Notice that the coordinates of point C are ((yy + xy), 0). When constructing the circle by hand, you can use a compass to trace the circle since you know the location of the center C and of two points, A and B. Let the segment AC represent the x-axis in the original state of stress. If you want to determine the state of stress for a set of axes x-y, rotated counterclockwise by an angle with respect to the original set of axes x-y, draw segment AB, centered at C and rotated clockwise by and angle 2

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with respect to segment AB. The coordinates of point A will give the values (xx,xy), while those of B will give the values (yy,xy).

Example - Using input forms in the NUM.SLV menu
Before discussing these items in detail we will present some of the characteristics of the input forms by using input forms from the financial calculation application in the numerical solver. Launch the numerical solver by using (associated with the 7 key). This produces a choose box that includes the following options:
To get started with financial calculations use the down arrow key () to select item 5. Solve finance. Press @@OK@@, to launch the application. The

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resulting screen is an input form with input fields for a number of variables (n, I%YR, PV, PMT, FV).
In this particular case we can give values to all but one of the variables, say, n = 10, I%YR = 8.5, PV = 10000, FV = 1000, and solve for variable PMT (the meaning of these variables will be presented later). Try the following: 10 @@OK@@ 8.5 @@OK@@ 10000 @@OK@@ 1000 @@OK@@ @SOLVE! The resulting screen is: Enter n = 10 Enter I%YR = 8.5 Enter PV = 10000 Enter FV = 1000 Select and solve for PMT
In this input form you will notice the following soft menu key labels: @EDIT !)AMOR @SOLVE Press to edit highlighted field Amortization menu - option specific to this application Press to solve for highlighted field
Pressing L we see the following soft menu key labels:

!RESET

Reset fields to default values

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!CALC !TYPES !CANCL @@OK@@
Press to access the stack for calculations Press to determine the type of object in highlighted field Cancel operation Accept entry
If you press !RESET you will be asked to select between the two options:
If you select Reset value only the highlighted value will be reset to the default value. If, instead, you select Rest all, all the fields will be reset to their default values (typically, 0). At this point you can accept your choice (press @@OK@@), or cancel the operation (press !CANCL). Press !CANCL in this instance. Press !CALC to access the stack. The resulting screen is the following:
At this point, you have access to the stack, and the value last highlighted in the input form is provided for you. Suppose that you want to halve this value. The following screen follows in ALG mode after entering 1136.22/2:

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(In RPN mode, we would have used 1136.22 ` 2 `/). Press @@OK@@ to enter this new value. The input form will now look like this:
Press !TYPES to see the type of data in the PMT field (the highlighted field). As a result, you get the following specification:
This indicates that the value in the PMT field must be a real number. Press @@OK@@ to return to the input form, and press L to recover the first menu. Next, press the ` key or the $ key to return to the stack. In this instance, the following values will be shown:

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Grades: There are 400 grades (400 g) in a complete circumference.
The angle measure affects the trig functions like SIN, COS, TAN and associated functions. To change the angle measure mode, use the following procedure: Press the H button. Next, use the down arrow key, , twice. Select the Angle Measure 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 preferred mode, and press the !!@@OK#@ (F) soft menu key to complete the operation. For example, in the following screen, the Radians mode is selected:

Coordinate System

The coordinate system selection affects the way vectors and complex numbers are displayed and entered. To learn more about complex numbers and vectors, see Chapters 4 and 8, respectively, in this guide. There are three coordinate systems available in the calculator: Rectangular (RECT), Cylindrical (CYLIN), and Spherical (SPHERE). To change coordinate system: Press the H button. Next, use the down arrow key, , three times. Select the Coord System 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 preferred mode, and press the !!@@OK#@ ( F) soft menu key to complete the operation. For example, in the following screen, the Polar coordinate mode is selected:

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Selecting CAS settings
CAS stands for Computer Algebraic System. This is the mathematical core of the calculator where the symbolic mathematical operations and functions are programmed. The CAS offers a number of settings can be adjusted according to the type of operation of interest. To see the optional CAS settings use the following: Press the H button to activate the CALCULATOR MODES input form.
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

Selecting the display font
First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key

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(D) to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0:system 8 is selected. This is the default value of the display font. Pressing the @CHOOS soft menu key (B), will provide a list of available system fonts, as shown below:
The options available are three standard System Fonts (sizes 8, 7, and 6) and a Browse. option. The latter will let you browse the calculator memory for additional fonts that you may have created or downloaded into the calculator. Practice changing the display fonts to sizes 7 and 6. Press the OK soft menu key to effect the selection. When done with a font selection, press the @@@OK@@@ soft menu key to go back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more and see how the stack display change to accommodate the different font.
Selecting properties of the line editor
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. Press the down arrow key, , once, to get to the Edit line. This line shows three properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small _Full page _Indent Changes font size to small Allows to place the cursor after the end of the line Auto indent cursor when entering a carriage return
Instructions on the use of the line editor are presented in Chapter 2 in the users guide.

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Selecting properties of the Stack
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. Press the down arrow key, , twice, to get to the Stack line. This line shows two properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small. This maximizes the amount of information displayed on the screen. Note, this selection overrides the font selection for the stack display. Displays mathematical expressions in graphical mathematical notation

_Textbook

To illustrate these settings, either in algebraic or RPN mode, use the equation writer to type the following definite integral: O0\xx` In Algebraic mode, the following screen shows the result of these keystrokes with neither _Small nor _Textbook are selected:

With the _Small option selected only, the display looks as shown below:
With the _Textbook option selected (default value), regardless of whether the _Small option is selected or not, the display shows the following result:

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Selecting properties of the equation writer (EQW)
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. Press the down arrow key, , three times, to get to the EQW (Equation Writer) line. This line shows two properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small _Small Stack Disp Changes font size to small while using the equation editor Shows small font in the stack after using the equation editor
Detailed instructions on the use of the equation editor (EQW) are presented elsewhere in this manual. For the example of the integral
e X dX , presented above, selecting the
_Small Stack Disp in the EQW line of the DISPLAY MODES input form produces the following display:

References

Additional references on the subjects covered in this Chapter can be found in Chapter 1 and Appendix C of the calculators users guide.

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Chapter 2 Introducing the calculator
In this chapter we present a number of basic operations of the calculator including the use of the Equation Writer and the manipulation of data objects in the calculator. Study the examples in this chapter to get a good grasp of the capabilities of the calculator for future applications.

Calculator objects

Some of the most commonly used objects are: reals (real numbers, written with a decimal point, e.g., -0.0023, 3.56), integers (integer numbers, written without a decimal point, e.g., 1232, -123212123), complex numbers (written as an ordered pair, e.g., (3,-2)), lists, etc. Calculator objects are described in Chapters 2 and 24 in the calculators user guide.
Editing expressions in the stack
In this section we present examples of expression editing directly into the calculator display or stack.

Creating arithmetic expressions
For this example, we select the Algebraic operating mode and select a Fix format with 3 decimals for the display. We are going to enter the arithmetic expression:

1.0 7.5 5.0 3.0 2.1.0 +

To enter this expression use the following keystrokes: 5.*1.+1/7.5/ R3.-2.Q3 The resulting expression is: 5*(1+1/7.5)/( 3-2^3). Press ` to get the expression in the display as follows:

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Notice that, if your CAS is set to EXACT (see Appendix C in users guide) and you enter your expression using integer numbers for integer values, the result is a symbolic quantity, e.g., 5*1+1/7.5/ R3-2Q3 Before producing a result, you will be asked to change to Approximate mode. Accept the change to get the following result (shown in Fix decimal mode with three decimal places see Chapter 1):
In this case, when the expression is entered directly into the stack, as soon as you press `, the calculator will attempt to calculate a value for the expression. If the expression is entered between apostrophes, however, the calculator will reproduce the expression as entered. For example: 5*1+1/7.5/ R3-2Q3` The result will be shown as follows:

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To evaluate the expression we can use the EVAL function, as follows: ` If the CAS is set to Exact, you will be asked to approve changing the CAS setting to Approx. Once this is done, you will get the same result as before. An alternative way to evaluate the expression entered earlier between quotes is by using the option. We will now enter the expression used above when the calculator is set to the RPN operating mode. We also set the CAS to Exact, the display to Textbook, and the number format to Standard. The keystrokes to enter the expression between quotes are the same used earlier, i.e., 5*1+1/7.5/ R3-2Q3` Resulting in the output
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

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The cursor is shown as a left-facing key. The cursor indicates the current edition location. For example, for the cursor in the location indicated above, type now: *5+1/3 The edited expression looks as follows:
Suppose that you want to replace the quantity between parentheses in the denominator (i.e., 5+1/3) with (5+2/2). First, we use the delete key () delete the current 1/3 expression, and then we replace that fraction with 2/2, as follows: Q2 When hit this point the screen looks as follows:
In order to insert the denominator 2 in the expression, we need to highlight the entire 2 expression. We do this by pressing the right arrow key () once. At that point, we enter the following keystrokes: /2

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

@@OK@@

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

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

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The ORDER command is not shown in this screen. To find it we use the L key to find it:
To activate the ORDER command we press the C(@ORDER) soft menu key.
For additional information on entering and manipulating expressions in the display or in the Equation Writer see Chapter 2 of the calculators users guide. For CAS (Computer Algebraic System) settings, see Appendix C in the calculators users guide. For information on Flags see, Chapter 24 in the calculators users guide.

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

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6.3` 8.5 4.2` 2.5 * 2.3` 4.5 / Alternatively, in RPN mode, you can separate the operands with a space (#) before pressing the operator key. Examples: 3.7#5.2 6.3#8.5 4.2#2.5 2.3#4.5 + * /

x 2 + y 2 is the modulus of the complex number z, and = Arg(z) =
arctan(y/x) is the argument of the complex number z. The complex conjugate of a complex number z = x + iy = re i, isz = x iy = re -i. The negative of z, z = -x-iy = - re i, can be thought of as the reflection of z about the origin.
Setting the calculator to COMPLEX mode
To work with complex numbers select the CAS complex mode: H)@@CAS@ @ @CHK The COMPLEX mode will be selected if the CAS MODES screen shows the option _Complex checked off, i.e.,
Press @@OK@@ , twice, to return to the stack.

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Entering complex numbers
Complex numbers in the calculator can be entered in either of the two Cartesian representations, namely, x+iy, or (x,y). The results in the calculator will be shown in the ordered-pair format, i.e., (x,y). For example, with the calculator in ALG mode, the complex number (3.5,-1.2), is entered as: 3.5\1.2` A complex number can also be entered in the form x+iy. For example, in ALG mode, 3.5-1.2i is entered as (accept mode changes): 3.5 -1.2*` In RPN mode, these numbers could be entered using the following keystrokes: 3.51.2\` (Notice that the change-sign keystroke is entered after the number 1.2 has been entered, in the opposite order as the ALG mode exercise), and 3.5 -1.2*` (Notice the need to enter an apostrophe before typing the number 3.5-1.2i in RPN mode). To enter the unit imaginary number alone type : (the I key).
Polar representation of a complex number
The polar representation of the complex number 3.5-1.2i, entered above, is obtained by changing the coordinate system to cylindrical or polar (using function CYLIN). You can find this function in the catalog (N). You can also change the coordinate to POLAR using H. Changing to polar coordinate with the angular measure in radians, produces the result:

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The result shown above represents a magnitude, 3.7, and an angle 0.33029. The angle symbol () is shown in front of the angle measure. Return to Cartesian or rectangular coordinates by using function RECT (available in the catalog, N). A complex number in polar representation is written as z = rei. You can enter this complex number into the calculator by using an ordered pair of the form (r, ). The angle symbol () can be entered as ~6. For example, the complex number z = 5.2e1.5i, can be entered as follows (the figures show the RPN stack, before and after entering the number):

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

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Press ` to return to stack. The stack will show the following results in ALG mode (the same result would be shown in RPN mode):
All the solutions are complex numbers: (0.432,-0.389), (0.432,0.389), (0.766, 0.632), (-0.766, -0.632). Generating polynomial coefficients given the polynomial's roots Suppose you want to generate the polynomial whose roots are the numbers [1, 5, -2, 4]. To use the calculator for this purpose, follow these steps: @@OK@@ 15 2\ 4@@OK@@ @SOLVE@ Select Solve poly Enter vector of roots Solve for coefficients
Press ` to return to stack, the coefficients will be shown in the stack.
Press to trigger the line editor to see all the coefficients.

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Generating an algebraic expression for the polynomial You can use the calculator to generate an algebraic expression for a polynomial given the coefficients or the roots of the polynomial. The resulting expression will be given in terms of the default CAS variable X. To generate the algebraic expression using the coefficients, try the following example. Assume that the polynomial coefficients are [1,5,-2,4]. Use the following keystrokes: @@OK@@ 15 2\ 4@@OK@@ @SYMB@ ` Select Solve poly Enter vector of coefficients Generate symbolic expression Return to stack.

The operation of the MTH/LIST menu is as follows: LIST : Calculate increment among consecutive elements in list LIST : Calculate summation of elements in the list LIST : Calculate product of elements in the list SORT : Sorts elements in increasing order REVLIST : Reverses order of list ADD : Operator for term-by-term addition of two lists of the same length (examples of this operator were shown above) Examples of application of these functions in ALG mode are shown next:
SORT and REVLIST can be combined to sort a list in decreasing order:

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

The MAP function

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

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

Entering vectors

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

@STK@@

key will place the contents of the selected cell on the stack.
The @GOTO@ key, when pressed, will request that the user indicate the number of the row and column where he or she wants to position the cursor. Pressing L once more produces the last menu, which contains only one function @@DEL@ (delete). The function @@DEL@ will delete the contents of the selected cell and replace it with a zero. To see these keys in action try the following exercise: (1) Activate the Matrix Writer by using. Make sure the @VEC @GO keys are selected. (2) Enter the following: 1`2`3` L @GOTO@ 2@@OK@@ 1 @@OK@@ @@OK@@ 4`5`6` 7`8`9` and

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(3) Move the cursor up two positions by using. Then press @-ROW. The second row will disappear. (4) Press @+ROW@. A row of three zeroes appears in the second row. (5) Press @-COL@. The first column will disappear. (6) Press @+COL@. A column of two zeroes appears in the first column. (7) Press @GOTO@ 3@@OK@@ 3@@OK@@ @@OK@@ to move to position (3,3). (8) Press @STK@@. This will place the contents of cell (3,3) on the stack, although you will not be able to see it yet. Press ` to return to normal display. The number 9, element (3,3), and the full matrix entered will be available in the stack.
Simple operations with vectors
To illustrate operations with vectors we will use the vectors u2, u3, v2, and v3, stored in an earlier exercise. Also, store vector A=[-1,-2,-3,-4,-5] to be used in the following exercises.
To change the sign of a vector use the key \, e.g.,

Addition, subtraction

Addition and subtraction of vectors require that the two vector operands have the same length:

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Attempting to add or subtract vectors of different length produces an error message:

The Chi-square (2) distribution has one parameter , known as the degrees of freedom. The calculator provides for values of the upper-tail (cumulative) distribution function for the 2-distribution using [UTPC] given the value of x and the parameter. The definition of this function is, therefore, UTPC(,x) = P(X>x) = 1 - P(X<x). For example, UTPC(5, 2.5) = 0.776495
The Chi-square distribution
The F distribution has two parameters N = numerator degrees of freedom, and D = denominator degrees of freedom. The calculator provides for values of the upper-tail (cumulative) distribution function for the F distribution, function UTPF, given the parameters N and D, and the value of F. The definition of this function is, therefore, UTPF(N,D,F) = P( >F) = 1 - P( <F). For example, to calculate UTPF(10,5, 2.5) = 0.1618347

The F distribution

For additional probability distributions and probability applications, refer to Chapter 17 in the calculators users guide.

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Chapter 16 Statistical Applications
The calculator provides the following pre-programmed statistical features accessible through the keystroke combination (the 5 key):

Entering data

Applications number 1, 2, and 4 from the list above require that the data be available as columns of the matrix DAT. This can be accomplished by entering the data in columns using the Matrix Writer, , and then using functions STO to store the matrix into DAT. For example, enter the following data using the Matrix Writer (see Chapters 8 or 9 in this guide), and store the data into DAT: 2.1 1.2 3.1 4.5 2.3 1.1 2.3 1.5 1.6 2.2 1.2 2.5. The screen may look like this:
Notice the variable @DAT listed in the soft menu keys.
Calculating single-variable statistics
After entering the column vector into DAT, press @@@OK@@ to select Single-var. The following input form will be provided:

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The form lists the data in DAT, shows that column 1 is selected (there is only one column in the current DAT). Move about the form with the arrow keys, and press the @ CHK@ soft menu key to select those measures (Mean, Standard Deviation, Variance, Total number of data points, Maximum and Minimum values) that you want as output of this program. When ready, press @@@OK@@. The selected values will be listed, appropriately labeled, in the screen of your calculator. For example:
Sample vs. population The pre-programmed functions for single-variable statistics used above can be applied to a finite population by selecting the Type: Population in the SINGLE-VARIABLE STATISTICS screen. The main difference is in the values of the variance and standard deviation which are calculated using n in the denominator of the variance, rather than (n-1). For the example above, use now the @CHOOS soft menu key to select population as Type: and re-calculate measures:

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Obtaining frequency distributions
The application 2. Frequencies. in the STAT menu can be used to obtain frequency distributions for a set of data. The data must be present in the form of a column vector stored in variable DAT. To get started, press @@@OK@@@. The resulting input form contains the following fields:
DAT: Col: X-Min: Bin Count: Bin Width:
the matrix containing the data of interest. the column of DAT that is under scrutiny. the minimum class boundary to be used in the frequency distribution (default = -6.5). the number of classes used in the frequency distribution (default = 13). the uniform width of each class in the frequency distribution (default = 1).
Given a set of n data values: {x1, x2, , xn} listed in no particular order, one can group the data into a number of classes, or bins by counting the frequency or number of values corresponding to each class. The application 2. Frequencies. in the STAT menu will perform this frequency count, and will keep track of those values that may be below the minimum and above the maximum class boundaries (i.e., the outliers). As an example, generate a relatively large data set, say 200 points, by using the command RANM({200,1}), and storing the result into variable DAT, by using function STO (see example above). Next, obtain single-variable information using: @@@OK@@@. The results are:

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This information indicates that our data ranges from -9 to 9. To produce a frequency distribution we will use the interval (-8,8) dividing it into 8 bins of width 2 each. Select the program 2. Frequencies. by using @@@OK@@@. The data is already loaded in DAT, and the option Col should hold the value 1 since we have only one column in DAT. Change X-Min to -8, Bin Count to 8, and Bin Width to 2, then press @@@OK@@@.
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.

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.
5. Warranty does not apply to defects resulting from (a) improper or inadequate maintenance or calibration, (b) software, interfacing, parts or supplies not supplied by HP, (c) unauthorized modification or misuse, (d) operation outside of the published environmental specifications for the product, or (e) improper site preparation or maintenance. 6. HP MAKES NO OTHER EXPRESS WARRANTY OR CONDITION WHETHER WRITTEN OR ORAL. TO THE EXTENT ALLOWED BY LOCAL LAW, ANY IMPLIED WARRANTY OR CONDITION OF MERCHANTABILITY, SATISFACTORY QUALITY, OR FITNESS FOR A PARTICULAR PURPOSE IS LIMITED TO THE DURATION OF THE EXPRESS WARRANTY SET FORTH ABOVE. Some countries, states or provinces do not allow limitations on the duration of an implied warranty, so the above limitation or exclusion might not apply to you. This warranty gives you specific legal rights and you might also have other rights that vary from country to country, state to state, or province to province.

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7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE REMEDIES. EXCEPT AS INDICATED ABOVE, IN NO EVENT WILL HP OR ITS SUPPLIERS BE LIABLE FOR LOSS OF DATA OR FOR DIRECT, SPECIAL, INCIDENTAL, CONSEQUENTIAL (INCLUDING LOST PROFIT OR DATA), OR OTHER DAMAGE, WHETHER BASED IN CONTRACT, TORT, OR OTHERWISE. Some countries, States or provinces do not allow the exclusion or limitation of incidental or consequential damages, so the above limitation or exclusion may not apply to you. 8. The only warranties for HP products and services are set forth in the express warranty statements accompanying such products and services. HP shall not be liable for technical or editorial errors or omissions contained herein. FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND NEW ZEALAND: THE WARRANTY TERMS CONTAINED IN THIS STATEMENT, EXCEPT TO THE EXTENT LAWFULLY PERMITTED, DO NOT EXCLUDE, RESTRICT OR MODIFY AND ARE IN ADDITION TO THE MANDATORY STATUTORY RIGHTS APPLICABLE TO THE SALE OF THIS PRODUCT TO YOU.