The HOME directory

The HOME directory, as pointed out earlier, is the base directory for memory operation for the calculator. To get to the HOME directory, you can press the UPDIR function () -- repeat as needed -- until the {HOME} spec is shown in the second line of the display header. Alternatively, you can use (hold) , press ` if in the algebraic mode. For this example, the HOME directory contains nothing but the CASDIR. Pressing J will show the variables in the soft menu keys:
Subdirectories To store your data in a well organized directory tree you may want to create subdirectories under the HOME directory, and more subdirectories within

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

The CASDIR sub-directory

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

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GNAME means a global name, and REAL means a real (or floating-point) numeric variable. The fourth and last column represents the size, in bytes, of the variable truncated, without decimals (i.e., nibbles). Thus, for example, variable PERIOD takes 12.5 bytes, while variable REALASSUME takes 27.5 bytes (1 byte = 8 bits, 1 bit is the smallest unit of memory in computers and calculators).

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Use the down arrow key () to select the option 2. MEMORY , or just press 2. Then, press @@OK@@. This will produce the following pull-down menu:
Use the down arrow key () to select the 5. DIRECTORY option, or just press 5. Then, press @@OK@@. This will produce the following pull-down menu:
Use the down arrow key () to select the 5. CRDIR option, and press @@OK@@. Command CRDIR in Algebraic mode Once you have selected the CRDIR through one of the means shown above, the command will be available in your stack as follows: At this point, you need to type a directory name, say chap1 : ~~~chap1~` The name of the new directory will be shown in the soft menu keys, e.g.,

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Command CRDIR in RPN mode To use the CRDIR in RPN mode you need to have the name of the directory already available in the stack before accessing the command. For example: ~~~chap2~` Then access the CRDIR command by either of the means shown above, e.g., through the N key:
Press the @@OK@ soft menu key to activate the command, to create the subdirectory:
Moving among subdirectories
To move down the directory tree, you need to press the soft menu key corresponding to the sub-directory you want to move to. The list of variables in a sub-directory can be produced by pressing the J (VARiables) key. To move up in the directory tree, use the function UPDIR, i.e., enter. Alternatively, you can use the FILES menu, i.e., press. Use the up and down arrow keys () to select the sub-directory you want to move to, and then press the !CHDIR (CHange DIRectory) or A soft menu key. This will show the contents of the sub-directory you moved to in the soft menu key labels.

Deleting subdirectories

To delete a sub-directory, use one of the following procedures: Using the FILES menu Press the key to trigger the FILES menu. Select the directory containing the sub-directory you want to delete, and press the !CHDIR if needed. This will close the FILES menu and display the contents of the directory you selected. In this case you will need to press `. Press the @@OK@@ soft menu

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key to list the contents of the directory in the screen. Select the sub-directory (or variable) that you want to delete. Press L@PURGE. A screen similar to the following will be shown:
The S2 string in this form is the name of the sub-directory that is being deleted. The soft menu keys provide the following options: @YES@ (A) Proceed with deleting the sub-directory (or variable) @ALL@ (B) Proceed with deleting all sub-directories (or variables) !ABORT (E) Do not delete sub-directory (or variable) from a list @@NO@@ (F) Do not delete sub-directory (or variable) After selecting one of these four commands, you will be returned to the screen listing the contents of the sub-directory. The !ABORT command, however, will show an error message:

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UNDO and CMD functions
Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made. These functions are associated with the HIST key: UNDO results from the keystroke sequence , while CMD results from the keystroke sequence. To illustrate the use of UNDO, try the following exercise in algebraic (ALG) mode: 5*4/3`. The UNDO command () will simply erase the result. The same exercise in RPN mode, will follow these keystrokes: 5`4`*3`/. Using at this point will undo the most recent operation (20/3), leaving the original terms back in the stack:
To illustrate the use of CMD, lets enter the following entries in ALG mode. Press ` after each entry.
Next, use the CMD function () to show the four most recent commands entered by the user, i.e.,
You can use the up and down arrow keys () to navigate through these commands and highlight any of them that you want to entry anew. Once you have selected the command to enter, press @@@OK@@@. The CMD function operates in the same fashion when the calculator is in RPN mode, except that the list of commands only shows numbers or algebraics. It does not show functions entered. For example, try the following exercise in RPN mode: 5`2`3/*S S5*2`.

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Pressing produces the following selection box:
As you can see, the numbers 3, 2, and 5, used in the first calculation above, are listed in the selection box, as well as the algebraic SIN(5x2), but not the SIN function entered previous to the algebraic.
A flag is a Boolean value, that can be set or cleared (true or false), that specifies a given setting of the calculator or an option in a program. Flags in the calculator are identified by numbers. There are 256 flags, numbered from -128 to 128. Positive flags are called user flags and are available for programming purposes by the user. Flags represented by negative numbers are called system flags and affect the way the calculator operates. To see the current system flag setting press the H button, and then the @FLAGS! soft menu key (i.e., F1). You will get a screen labeled SYSTEM FLAGS listing flag numbers and the corresponding setting.
(Note: In this screen, as only system flags are present, only the absolute value of the flag number Is displayed). A flag is said to be set if you see a check mark ( ) in front of the flag number. Otherwise, the flag is not set or cleared. To change the status of a system flag press the @ CHK@ ! soft menu key while the flag you want to change is highlighted, or use the \ key. You can use the up and down arrow keys () to move about the list of system flags. Although there are 128 system flags, not all of them are used, and some of them are used for internal system control. System flags that are not accessible to the user are not visible in this screen. A complete list of flags is presented in Chapter 24.

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

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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

Page 5-11

MODULO menu
ADDTMOD DIVMOD DIV2MOD Add two expressions modulo current modulus Divides 2 polynomials modulo current modulus Euclidean division of 2 polynomials with modular coefficients EXPANDMOD Expands/simplify polynomial modulo current modulus FACTORMOD Factorize a polynomial modulo current modulus GCDMOD GCD of 2 polynomials modulo current modulus INVMOD inverse of integer modulo current modulus MOD (not entry available in the help facility) MODSTO Changes modulo setting to specified value MULTMOD Multiplication of two polynomials modulo current modulus POWMOD Raises polynomial to a power modulo current modulus SUBTMOD Subtraction of 2 polynomials modulo current modulus

Page 6-4

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

Numerical solver menu

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

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

Polynomial Equations

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

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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):
To see all the solutions, press the down-arrow key () to trigger the line editor:

Statistics of grouped data
Grouped data is typically given by a table showing the frequency (w) of data in data classes or bins. Each class or bin is represented by a class mark (s), typically the midpoint of the class. An example of grouped data is shown next: Class Frequency Class mark count boundaries 0-2 2-4 4-6 6--10 sk 9 wk 3
The class mark data can be stored in variable S, while the frequency count can be stored in variable W, as follows:

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Given the list of class marks S = {s1, s2, , sn }, and the list of frequency counts W = {w1, w2, , wn }, the weighted average of the data in S with weights W represents the mean value of the grouped data, that we call s, in this context:

wk s k

where N =
represents the total frequency count.
The mean value for the data in lists S and W, therefore, can be calculated using the procedure outlined above for the weighted average, i.e.,
Well store this value into a variable called XBAR:
The variance of this grouped data is defined as

wk ( s k s ) 2

(sk s ) 2 N

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To calculate this last result, we can use the following:
The standard deviation of the grouped data is the square root of the variance:

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Chapter 9 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.
From a mathematical point of view, a vector is an array of 2 or more elements arranged into a row or a column. These will be referred to as row and column vectors. Examples are shown below:
1 v = 3 , u = [1, 3, 5, 2] 6
Physical vectors have two or three components and can be used to represent physical quantities such as position, velocity, acceleration, forces, moments, linear and angular momentum, angular velocity and acceleration, etc. Referring to a Cartesian coordinate system (x,y,z), there exists unit vectors i, j, k associated with each coordinate direction, such that a physical vector A can be written in terms of its components Ax, Ay, Az, as A = Axi + Ayj + Azk. Alternative notation for this vector are: A = [Ax, Ay, Az], A = (Ax, Ay, Az), or A = < Ax, Ay, Az >. A two dimensional version of this vector will be written as A = Axi + Ayj, A = [Ax, Ay], A = (Ax, Ay), or A = < Ax, Ay >. Since in the calculator vectors are written between brackets [ ], we will choose the notation A = [Ax, Ay, Az] or A = [Ax, Ay, Az], to refer to two- and three-dimensional vectors from now on. The magnitude of a vector A is defined as |A| =
Ax + Ay + Az2. A unit vector in the direction of vector A, is defined as eA =

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If using RPN mode, we assume that the matrix is in the stack and use {6} ` RDM. Note: Function RDM provides a more direct and efficient way to transform lists to arrays and vice versa, than that provided at the end of Chapter 9.

Function RANM

Function RANM (RANdom Matrix) will generate a matrix with random integer elements given a list with the number of rows and columns (i.e., the dimensions of the matrix). For example, in ALG mode, two different 23 matrices with random elements are produced by using the same command, namely, RANM({2,3}) :
In RPN mode, use {2,3} ` RANM. Obviously, the results you will get in your calculator will most certainly be different than those shown above. The random numbers generated are integer numbers uniformly distributed in the range [-10,10], i.e., each one of those 21 numbers has the same probability of being selected. Function RANM is useful for generating matrices of any size to illustrate matrix operations, or the application of matrix functions.

Function SUB

Function SUB extracts a sub-matrix from an existing matrix, provided you indicate the initial and final position of the sub-matrix. For example, if we

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want to extract elements a12, a13, a22, and a23 from the last result, as a 22 sub-matrix, in ALG mode, use:
In RPN mode, assuming that the original 23 matrix is already in the stack, use {1,2} ` {2,3} ` SUB.

Function REPL

Function REPL replaces or inserts a sub-matrix into a larger one. The input for this function is the matrix where the replacement will take place, the location where the replacement begins, and the matrix to be inserted. For example, keeping the matrix that we inherited from the previous example, enter the matrix: [[1,2,3],[4,5,6],[7,8,9]]. In ALG mode, the following screen shot to the left shows the new matrix before pressing `. The screen shot to the right shows the application of function RPL to replace the matrix in ANS(2), the 22 matrix, into the 33 matrix currently located in ANS(1), starting at position {2,2}:
If working in the RPN mode, assuming that the 22 matrix was originally in the stack, we proceed as follows: [[1,2,3],[4,5,6],[7,8,9]]` (this last key swaps the contents of stack levels 1 and 2) {1,2} ` (another swapping of levels 1 and 2) REPL.

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Function DIAG
Function DIAG takes the main diagonal of a square matrix of dimensions nn, and creates a vector of dimension n containing the elements of the main diagonal. For example, for the matrix remaining from the previous exercise, we can extract its main diagonal by using:

These programs can be useful for statistical applications, specifically to create the statistical matrix DAT. Examples of the use of these program are shown in a latter chapters.

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Manipulating matrices by columns
The calculator provides a menu with functions for manipulating matrices by operating in their columns. This menu is available through the MTH/MATRIX/COL. sequence: () shown in the figure below with system flag 117 set to CHOOSE boxes:
or through the MATRICES/CREATE/COLUMN sub-menu:
Both approaches will show the same functions:
When system flag 117 is set to SOFT menus, the COL menu is accessible through !)MATRX !)@MAKE@ !)@@COL@ , or through !)@CREAT@ !)@@COL@. Both approaches will show the same set of functions:
The operation of these functions is presented below.

Function COL

Function COL takes as argument a matrix and decomposes it into vectors corresponding to its columns. An application of function COL in ALG mode is shown below. The matrix used has been stored earlier in variable A. The matrix is shown in the figure to the left. The figure to the right shows the matrix

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decomposed in columns. To see the full result, use the line editor (triggered by pressing ).
In RPN mode, you need to list the matrix in the stack, and the activate function COL, i.e., @@@A@@@ COL. The figure below shows the RPN stack before and after the application of function COL.
In this result, the first column occupies the highest stack level after decomposition, and stack level 1 is occupied by the number of columns of the original matrix. The matrix does not survive decomposition, i.e., it is no longer available in the stack.
Function COL has the opposite effect of Function COL, i.e., given n vectors of the same length, and the number n, function COL builds a matrix by placing the input vectors as columns of the resulting matrix. Here is an example in ALG mode. The command used was: COL ([1,2,3],[4,5,6],[7,8,9],3)

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The latter result can be defined as a function, FW(X), as follows (cutting and pasting the last result into the command):
We can now plot the real part of this function. Change the decimal mode to Standard, and use the following:
The solution is shown below:

Fourier Transforms

Before presenting the concept of Fourier transforms, well discuss the general definition of an integral transform. In general, an integral transform is a transformation that relates a function f(t) to a new function F(s) by an

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integration of the form F ( s ) =

(s, t ) f (t ) dt.

The function (s,t) is
known as the kernel of the transformation. The use of an integral transform allows us to resolve a function into a given spectrum of components. To understand the concept of a spectrum, consider the Fourier series
f (t ) = a0 + (an cos n x + bn sin n x ),
representing a periodic function with a period T. This Fourier series can be re-written as f ( x ) = a0 +

cos( n x + n ), where

b An = a n + bn , n = tan 1 n , a n
for n =1,2, The amplitudes An will be referred to as the spectrum of the function and will be a measure of the magnitude of the component of f(x) with frequency fn = n/T. The basic or fundamental frequency in the Fourier series is f0 = 1/T, thus, all other frequencies are multiples of this basic frequency, i.e., fn = nf0. Also, we can define an angular frequency, n = 2n/T = 2fn = 2 nf0 = n0, where 0 is the basic or fundamental angular frequency of the Fourier series. Using the angular frequency notation, the Fourier series expansion is written as
f ( x) = a 0 + An cos( n x + n ).
= a 0 + (a n cos n x + bn sin n x )

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A plot of the values An vs. n is the typical representation of a discrete spectrum for a function. The discrete spectrum will show that the function has components at angular frequencies n which are integer multiples of the fundamental angular frequency 0. Suppose that we are faced with the need to expand a non-periodic function into sine and cosine components. A non-periodic function can be thought of as having an infinitely large period. Thus, for a very large value of T, the fundamental angular frequency, 0 = 2/T, becomes a very small quantity, say. Also, the angular frequencies corresponding to n = n0 = n, (n = 1, 2, , ), now take values closer and closer to each other, suggesting the need for a continuous spectrum of values. The non-periodic function can be written, therefore, as
f ( x) = [C ( ) cos( x) + S ( ) sin( x)]d ,

C ( ) =

f ( x) cos( x) dx,

S ( ) =

1 f ( x) sin( x) dx 2
The continuous spectrum is given by
A( ) = [C ( )]2 + [ S ( )] 2
The functions C(), S(), and A() are continuous functions of a variable , which becomes the transform variable for the Fourier transforms defined below. Example 1 Determine the coefficients C(), S(), and the continuous spectrum A(), for the function f(x) = exp(-x), for x > 0, and f(x) = 0, x < 0.

For the analysis of a single set of data (a sample) we can use applications number 1, 2, and 4 from the list above. All of these applications 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,. This operation may become tedious for large number of data points. Instead, you may want to enter the data as a list (see Chapter 8) and convert the list into a column vector by using program CRMC (see Chapter 10). Alternatively, you can enter the following program to convert a list into a column vector. Type the program in RPN mode: OBJ 12 LIST ARRY

Page 18-1

Store the program in a variable called LXC. After storing this program in RPN mode you can also use it in ALG mode. To store a column vector into variable DAT use function STO, available through the catalog (N), e.g., STO (ANS(1)) in ALG mode. Example 1 Using the program LXC, defined above, create a column vector using the following data: 2.1 1.2 3.1 4.5 2.3 1.1 2.3 1.5 1.6 2.2 1.2 2.5. In RPG mode, type in the data in a list: {2.1 1.2 3.1 4.5 2.3 1.1 2.3 1.5 1.6 2.2 1.2 2.5 } `@LXC Use function STO to store the data into DAT.
Calculating single-variable statistics
Assuming that the single data set was stored as a column vector in variable DAT. To access the different STAT programs, press. Press @@@OK@@ to select 1. Single-var. There will be available to you an input form labeled SINGLE-VARIABLE STATISTICS, with the data currently in your DAT variable listed in the form as a vector. Since you only have one column, the field Col: should have the value 1 in front of it. The Type field determines whether you are working with a sample or a population, the default setting is Sample. Move the cursor to the horizontal line preceding the fields Mean, Std Dev, Variance, Total, Maximum, Minimum, pressing the @ CHK@ soft menu key to select those measures 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. Example 1 -- For the data stored in the previous example, the single-variable statistics results are the following: Mean: 2.133, Std Dev: 0.964, Variance: 0.929 Total: 25.6, Maximum: 4.5, Minimum: 1.1

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Definitions The definitions used for these quantities are the following: Suppose that you have a number data points x1, x2, x3, , representing different measurements of the same discrete or continuous variable x. The set of all possible values of the quantity x is referred to as the population of x. A finite population will have only a fixed number of elements xi. If the quantity x represents the measurement of a continuous quantity, and since, in theory, such a quantity can take an infinite number of values, the population of x in this case is infinite. If you select a sub-set of a population, represented by the n data values {x1, x2, , xn}, we say you have selected a sample of values of x. Samples are characterized by a number of measures or statistics. There are measures of central tendency, such as the mean, median, and mode, and measures of spreading, such as the range, variance, and standard deviation. Measures of central tendency The mean (or arithmetic mean) of the sample, x, is defined as the average value of the sample elements,

Additional notes on linear regression
In this section we elaborate the ideas of linear regression presented earlier in the chapter and present a procedure for hypothesis testing of regression parameters.
The method of least squares
Let x = independent, non-random variable, and Y = dependent, random variable. The regression curve of Y on x is defined as the relationship between x and the mean of the corresponding distribution of the Ys. Assume that the regression curve of Y on x is linear, i.e., mean distribution of Ys is given by + x. Y differs from the mean ( + x) by a value , thus Y = + x + , where is a random variable. To visually check whether the data follows a linear trend, draw a scattergram or scatter plot.

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Suppose that we have n paired observations (xi, yi); we predict y by means of y = a + bx, where a and b are constant. Define the prediction error as, ei = yi - yi = yi - (a + bxi). The method of least squares requires us to choose a, b so as to minimize the sum of squared errors (SSE)
SSE = ei2 = [ y i (a + bxi )]2

i =1 i =1

the conditions

( SSE ) = 0 a

We get the, so-called, normal equations:

( SSE ) = 0 b

i =1 n i =1

= a n + b xi

i =1 n n

xi yi = a xi + b xi2

This is a system of linear equations with a and b as the unknowns, which can be solved using the linear equation features of the calculator. There is, however, no need to bother with these calculations because you can use the 3. Fit Data option in the menu as presented earlier. ____________________________________________________________________ Notes: a,b are unbiased estimators of ,. The Gauss-Markov theorem of probability indicates that among all unbiased estimators for and , the least-square estimators (a,b) are the most efficient. ____________________________________________________________________

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Additional equations for linear regression
The summary statistics such as x, x2, etc., can be used to define the following quantities:
n n 1 n S xx = ( xi x ) 2 = (n 1) s x = xi xi n i =1 i =1 i =1
n n 1 n S y = ( yi y ) 2 = (n 1) s y = y i yi n i =1 i =1 i =1 n n 1 n n = ( xi x )( y i y ) 2 = (n 1) s xy = xi y i xi y i n i =1 i =1 i =1 i =1
From which it follows that the standard deviations of x and y, and the covariance of x,y are given, respectively, by

The POLYNOMIAL sub-menu
The POLYNOMIAL sub-menu includes functions for generating and manipulating polynomials. These functions are presented in Chapter 5:

The TESTS sub-menu

The TESTS sub-menu includes relational operators (e.g., ==, <, etc.), logical operators (e.g., AND, OR, etc.), the IFTE function, and the ASSUME and UNASSUME commands.
Relational and logical operators are presented in Chapter 21 in the context of programming the calculator in User RPL language. The IFTE function is introduced in Chapter 3. Functions ASSUME and UNASSUME are presented next, using their CAS help facility entries (see Appendix C).

ASSUME

UNASSUME

Page J-3

Appendix K The MAIN menu
The MAIN menu is available in the command catalog. This menu include the following sub-menus:

The CASCFG command

This is the first entry in the MAIN menu. This command configures the CAS. For CAS configuration information see Appendix C.

The ALGB sub-menu

The ALGB sub-menu includes the following commands:
These functions, except for 0.MAIN MENU and 11.UNASSIGN are available in the ALG keyboard menu (). Detailed explanation of these functions can be found in Chapter 5. Function UNASSIGN is described in the following entry from the CAS menu:

Page K-1

The DIFF sub-menu
The DIFF sub-menu contains the following functions:
These functions are also available through the CALC/DIFF sub-menu (start with ). These functions are described in Chapters 13, 14, and 15, except for function TRUNC, which is described next using its CAS help facility entry:

The MATHS sub-menu

The MATHS menu is described in detail in Appendix J.

The TRIGO sub-menu

The TRIGO menu contains the following functions:
These functions are also available in the TRIG menu (). Description of these functions is included in Chapter 5.

Page K-2

The SOLVER sub-menu
The SOLVER menu includes the following functions:
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.

HADAMARD, 11-5 HALT, L-2 Harmonic mean, 8-15 HEAD, 8-11 Header size, 1-29 Heaviside's step function, 16-15 HELP, 2-26 HERMITE, 5-11, 5-20 Hermite polynomials, 16-60 HESS, 15-2 Hessian matrix, 15-2 HEX, 19-2 HEX, 3-1 Hexadecimal numbers, 19-7
Graphs Graphs Graphs Graphs
truth plots, 12-29 bar plots, 12-30 histograms, 12-30 scatterplots, 12-30

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Higher-order derivatives, 13-13 Higher-order partial derivatives, 14-3 HILBERT, 10-14 Histograms, 12-30 HMS-, 25-3 HMS+, 25-3 HMS-->, 25-3 HORNER, 5-11, 5-20 H-VIEW, 12-20 Hyperbolic functions graphs, 1216 Hypothesis testing, 18-34 Hypothesis testing errors, 18-35 Hypothesis testing in linear regression, 18-52 Hypothesis testing in the calculator, 18-42 Hypothesis testing on variances, 18-46 HZIN, 12-50 HZOUT, 12-50
i, 3-16 I/O functions menu, F-2 IR, 5-28 IABCUV, 5-11 IBERNOULLI, 5-11 ICHINREM, 5-11 Identity matrix, 10-1 Identity matrix, 11-6 IDIV2, 5-11 IDN, 10-9 IEGCD, 5-11 IF.THEN.ELSE.END, 21-48
IF.THEN.END, 21-47 IFERR sub-menu, 21-65 IFTE, 3-35 ILAP, 16-11 Illumination units, 3-20 IM, 4-6 IMAGE, 11-54 Imaginary part, 4-1 Implicit derivatives, 13-7 Improper integrals, 13-21 Increasing-power CAS mode, C-9 INDEP, 22-6, Independent variable in CAS, C-2, Infinite series, 13-21, 13-23 INFO, 22-4 INPUT, 21-22 , Input forms programming, 21-21 Input forms use of A-1 Input string prompt programming, 21-21 Input-output functions menu, F-2 INS , L-1 INT, 13-14 Integer numbers, C-5 Integers, 2-1 Integrals, 13-14 Integrals definite, 13-15 Integrals step-by-step, 13-17 Integrals improper, 13-21 Integrals double, 14-6 Integrals multiple, 14-6
Integration change of variable, 13-19
Integration substitution, 13-18 Integration techniques, 13-18
Integration by partial fractions, 13-20
Integration by parts, 13-19

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Interactive drawing, 12-43
Interactive input programming, 21-19
Interactive plots with PLOT menu, 22-15 Interactive self-test, G-3 INTVX, 13-14 INV, 4-4 INV, L-4 Inverse cdfs, 17-14 Inverse cumulative distribution functions, 17-13 Inverse function graph, 12-12 Inverse Laplace transforms, 16-10 Inverse matrix, 11-6 INVMOD, 5-12 IP, 3-14 IQUOT, 5-11 IREMAINDER, 5-11 Irrotational fields, 15-5 ISECT in plots, 12-7 ISOL, 6-1 ISOM, 11-54 ISPRIME? , 5-11 ITALI, L-4
Keyboard alternate key functions B-4 Keyboard left-shift functions B-5 Keyboard right-shift functions B-8 Keyboard ALPHA characters B-9 Keyboard ALPHA-left-shift characters B-10 Keyboard ALPHA-right-shift characters, B-12 Keyboard ALPHA function, 1-12 Keyboard left-shift function, 1-12 Keyboard main function, 1-12 Keyboard right-shift function, 1-12 Kronecker's delta, 10-1

The six labels associated with the keys A through F form part of a menu of functions. Since the calculator has only six soft menu keys, it only display 6 labels at any point in time. However, a menu can have more than six entries. Each group of 6 entries is called a Menu page. To move to the next menu page (if available), press the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard.

The TOOL menu

The soft menu keys for the default menu, known as the TOOL menu, are associated with operations related to manipulation of variables (see section on variables in this Chapter): @EDIT A EDIT the contents of a variable (see Chapter 2 in this guide

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@VIEW @@ RCL @@ @@STO@ ! PURGE CLEAR

B C D E F

and Chapter 2 and Appendix L in the users guide for more information on editing) VIEW the contents of a variable ReCaLl the contents of a variable STOre the contents of a variable PURGE a variable CLEAR the display or stack
These six functions form the first page of the TOOL menu. This menu has actually eight entries arranged in two pages. The second page is available by pressing the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. In this case, only the first two soft menu keys have commands associated with them. These commands are: @CASCM @HELP A B CASCMD: CAS CoMmanD, used to launch a command from the CAS (Computer Algebraic System) by selecting from a list HELP facility describing the commands available in the calculator
Pressing the L key will show the original TOOL menu. Another way to recover the TOOL menu is to press the I key (third key from the left in the second row of keys from the top of the keyboard).

Setting time and date

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

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

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

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

Because this number has three figures in the integer part, it is shown with four significative figures and a zero power of ten, while using the Engineering format. For example, the number 0.00256, will be shown as:

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

Angle Measure

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

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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:

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

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

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

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

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

Rather than listing the description of each function in this manual, the user is invited to look up the description using the calculators help facility: I L @)HELP@ `. To locate a particular function, type the first letter of the function. For example, for function COLLECT, we type ~c, then use the up and down arrow keys, , to locate COLLECT within the help window. To complete the operation press @@OK@@. Here is the help screen for function COLLECT:
We notice that, at the bottom of the screen, the line See: EXPAND FACTOR suggests links to other help facility entries, the functions EXPAND and FACTOR. To move directly to those entries, press the soft menu key @SEE1! for EXPAND, and @SEE2! for FACTOR. Pressing @SEE1!, for example, shows the following information for EXPAND, while @SEE2! shows information for FACTOR:

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

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

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These functions allow to simplify expressions by replacing some category of trigonometric functions for another one. For example, the function ACOS2S allows to replace the function arccosine (acos(x)) with its expression in terms of arcsine (asin(x)). Description of these commands and examples of their applications are available in the calculators help facility (IL@HELP). The user is invited to explore this facility to find information on the commands in the TRIG menu.

Functions in the ARITHMETIC menu
The ARITHMETIC menu is triggered through the keystroke combination (associated with the 1 key). With system flag 117 set to CHOOSE boxes, shows the following menu:
Out of this menu list, options 5 through 9 (DIVIS, FACTORS, LGCD, PROPFRAC, SIMP2) correspond to common functions that apply to integer numbers or to polynomials. The remaining options (1. INTEGER, 2. POLYNOMIAL, 3. MODULO, and 4. PERMUTATION) are actually submenus of functions that apply to specific mathematical objects. When system flag 117 is set to SOFT menus, the ARITHMETIC menu () produces:
Following, we present the help facility entries for functions FACTORS and SIMP2 in the ARITHMETIC menu:

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FACTORS:

SIMP2:

The functions associated with the ARITHMETIC submenus: INTEGER, POLYNOMIAL, MODULO, and PERMUTATION, are presented in detail in Chapter 5 in the calculators users guide. The following sections show some applications to polynomials and fractions.

Polynomials

Polynomials are algebraic expressions consisting of one or more terms containing decreasing powers of a given variable. For example, X^3+2*X^2-3*X+2 is a third-order polynomial in X, while SIN(X)^2-2 is a second-order polynomial in SIN(X). Functions COLLECT and EXPAND, shown earlier, can be used on polynomials. Other applications of polynomial functions are presented next:

The HORNER function

The function HORNER (, POLYNOMIAL, HORNER) produces the Horner division, or synthetic division, of a polynomial P(X) by the factor (X-a), i.e., HORNER(P(X),a) = {Q(X), a, P(a)}, where P(X) = Q(X)(X-a)+P(a). For example, HORNER(X^3+2*X^2-3*X+1,2) = {X^2+4*X+i.e., X3+2X2-3X+1 = (X2+4X+5)(X-2)+11. Also, HORNER(X^6-1,-5)=
{ X^5-5*X^4+25*X^3-125*X^2+625*X-3125 -5 15624}
X6-1 = (X5-5*X4+25X3-125X2+625X-3125)(X+5)+15624.

Page 5-8

The variable VX
Most polynomial examples above were written using variable X. This is because a variable called VX exists in the calculators {HOME CASDIR} directory that takes, by default, the value of X. This is the name of the preferred independent variable for algebraic and calculus applications. Avoid using the variable VX in your programs or equations, so as to not get it confused with the CAS VX. For additional information on the CAS variable see Appendix C in the calculators users guide.

The PCOEF function

Given an array containing the roots of a polynomial, the function PCOEF generates an array containing the coefficients of the corresponding polynomial. The coefficients correspond to decreasing order of the independent variable. For example: PCOEF([-2, 1, 0 ,1, 1, 2]) = [1. 1. 5. 5. 4. 4. 0.], which represents the polynomial X6-X5-5X4+5X3+4X2-4X.

The PROOT function

Given an array containing the coefficients of a polynomial, in decreasing order, the function PROOT provides the roots of the polynomial. Example, from X2+5X+6 =0, PROOT([1, 5, 6]) = [2. 3.].

If the lists involved in the operation have different lengths, an error message (Invalid Dimensions) is produced. Try, for example, L1-L4. The plus sign (+), when applied to lists, acts a concatenation operator, putting together the two lists, rather than adding them term-by-term. For example:
In order to produce term-by-term addition of two lists of the same length, we need to use operator ADD. This operator can be loaded by using the function catalog (N). The screen below shows an application of ADD to add lists L1 and L2, term-by-term:
Functions applied to lists
Real number functions from the keyboard (ABS, ex, LN, 10x, LOG, SIN, x2, , COS, TAN, ASIN, ACOS, ATAN, yx) as well as those from the MTH/HYPERBOLIC menu (SINH, COSH, TANH, ASINH, ACOSH, ATANH), and MTH/REAL menu (%, etc.), can be applied to lists, e.g.,

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INVERSE (1/x)

Lists of complex numbers

You can create a complex number list, say, L5 = L1 ADD i*L2 (type the instruction as indicated before), as follows:
Functions such as LN, EXP, SQ, etc., can also be applied to a list of complex numbers, e.g.,
Lists of algebraic objects
The following are examples of lists of algebraic objects with the function SIN applied to them (select Exact mode for these examples -- See Chapter 1):

The MTH/LIST menu

The MTH menu provides a number of functions that exclusively to lists. With system flag 117 set to CHOOSE boxes, the MTH/LIST menu offers the following functions:

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With system flag 117 set to SOFT menus, the MTH/LIST menu shows the following functions:
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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Calculating a double integral in the calculator is straightforward. A double integral can be built in the Equation Writer (see example in Chapter 2 in the users guide), as shown below. This double integral is calculated directly in the Equation Writer by selecting the entire expression and using function @EVAL. The result is 3/2.
For additional details of multi-variate calculus operations and their applications see Chapter 14 in the calculators users guide.

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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.,

i.e., y(t) = -((19*5*SIN(5*t)-(148*COS(5*t)+80*COS(t/2)))/190). Press ``J @ODETY to get the string Linear w/ cst coeff for the ODE type in this case.

Page 14-4

Laplace Transforms
The Laplace transform of a function f(t) produces a function F(s) in the image domain that can be utilized to find the solution of a linear differential equation involving f(t) through algebraic methods. The steps involved in this application are three: 1. 2. 3. Use of the Laplace transform converts the linear ODE involving f(t) into an algebraic equation. The unknown F(s) is solved for in the image domain through algebraic manipulation. An inverse Laplace transform is used to convert the image function found in step 2 into the solution to the differential equation f(t).
Laplace transform and inverses in the calculator
The calculator provides the functions LAP and ILAP to calculate the Laplace transform and the inverse Laplace transform, respectively, of a function f(VX), where VX is the CAS default independent variable (typically X). The calculator returns the transform or inverse transform as a function of X. The functions LAP and ILAP are available under the CALC/DIFF menu. The examples are worked out in the RPN mode, but translating them to ALG mode is straightforward. Example 1 You can get the definition of the Laplace transform use the following: f(X) ` L P in RPN mode, or L P(F(X))in ALG mode. The calculator returns the result (RPN, left; ALG, right):
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):

Page 15-3

UTPT, given the parameter and the value of t, i.e., UTPT(,t) = P(T>t) = 1P(T<t). For example, UTPT(5,2.5) = 2.7245E-2.
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:

Hypothesis testing

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

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

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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.

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Chapter 17 Numbers in Different Bases

L.America

Telephone numbers 1800-HP INVENT (905) 206-4663 or 800- HP INVENT ROTC = Rest of the country Please logon to http://www.hp.com for the latest service and support information.

N.America

Country : U.S. Canada

Page W-3

Regulatory information
This section contains information that shows how the hp 49g+ graphing calculator complies with regulations in certain regions. Any modifications to the calculator not expressly approved by Hewlett-Packard could void the authority to operate the 49g+ in these regions.
This calculator generates, uses, and can radiate radio frequency energy and may interfere with radio and television reception. The calculator complies with the limits for a Class B digital device, pursuant to Part 15 of the FCC Rules. These limits are designed to provide reasonable protection against harmful interference in a residential installation. However, there is no guarantee that interference will not occur in a particular installation. In the unlikely event that there is interference to radio or television reception(which can be determined by turning the calculator off and on), the user is encouraged to try to correct the interference by one or more of the following measures: n Reorient or relocate the receiving antenna. n Relocate the calculator, with respect to the receiver. To maintain compliance with FCC rules and regulations, use only the cable accessories provided.
Connections to Peripheral Devices

Canada

This Class B digital apparatus complies with Canadian ICES-003. Cet appareil numerique de la classe B est conforme a la norme NMB-003 du Canada.

(VCCI)

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