HP 50g, 50g Graphing Calculator Owner's manual

HP 50g, 50g Graphing Calculator Owner's manual
HP 50g graphing calculator
user’s manual
H
Edition 1
HP part number F2229AA-90001
FrontPageQS49_E.backup.fm Page 2
Friday, February 24, 2006
4:54 PM
Notice
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Printing History
Edition 1
April 2006
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Preface
You have in your hands a compact symbolic and numerical computer that
will facilitate calculation and mathematical analysis of problems in a
variety of disciplines, from elementary mathematics to advanced
engineering and science subjects.
This manual contains examples that illustrate the use of the basic calculator
functions and operations. The chapters in this user’s manual are organized
by subject in order of difficulty: from the setting of calculator modes, to real
and complex number calculations, operations with lists, vectors, and
matrices, graphics, calculus applications, vector analysis, differential
equations, probability and statistics.
For symbolic operations the calculator includes a powerful Computer
Algebraic System (CAS), which lets you select different modes of operation,
e.g., complex numbers vs. real numbers, or exact (symbolic) vs.
approximate (numerical) mode. The display can be adjusted to provide
textbook-type expressions, which can be useful when working with
matrices, vectors, fractions, summations, derivatives, and integrals. The
high-speed graphics of the calculator are very convenient for producing
complex figures in very little time.
Thanks to the infrared port, the USB port, and the RS232 port and cable
provided with your calculator, you can connect your calculator with other
calculators or computers. This allows for fast and efficient exchange of
programs and data with other calculators and computers.
We hope your calculator will become a faithful companion for your school
and professional applications.
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Table of Contents
Chapter 1 - Getting started
Basic Operations, 1-1
Batteries, 1-1
Turning the calculator on and off, 1-2
Adjusting the display contrast, 1-2
Contents of the calculator’s display, 1-3
Menus, 1-3
The TOOL menu, 1-3
Setting time and date, 1-4
Introducing the calculator’s keyboard, 1-4
Selecting calculator modes, 1-6
Operating Mode, 1-7
Number Format and decimal dot or comma, 1-10
Standard format, 1-10
Fixed format with decimals, 1-10
Scientific format, 1-11
Engineering format, 1-12
Decimal comma vs. decimal point, 1-13
Angle Measure, 1-14
Coordinate System, 1-14
Selecting CAS settings, 1-15
Explanation of CAS settings, 1-16
Selecting Display modes, 1-17
Selecting the display font, 1-18
Selecting properties of the line editor, 1-18
Selecting properties of the Stack, 1-19
Selecting properties of the equation writer (EQW), 1-20
References, 1-20
Chapter 2 - Introducing the calculator
Calculator objects, 2-1
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Editing expressions in the stack, 2-1
Creating arithmetic expressions, 2-1
Creating algebraic expressions, 2-4
Using the Equation Writer (EQW) to create expressions, 2-5
Creating arithmetic expressions, 2-5
Creating algebraic expressions, 2-7
Organizing data in the calculator, 2-8
The HOME directory, 2-8
Subdirectories, 2-9
Variables, 2-9
Typing variable names , 2-9
Creating variables, 2-10
Algebraic mode, 2-10
RPN mode, 2-11
Checking variables contents, 2-13
Algebraic mode, 2-13
RPN mode, 2-13
Using the right-shift key followed by soft menu key labels, 2-13
Listing the contents of all variables in the screen, 2-14
Deleting variables, 2-14
Using function PURGE in the stack in Algebraic mode, 2-14
Using function PURGE in the stack in RPN mode, 2-15
UNDO and CMD functions, 2-16
CHOOSE boxes vs. Soft MENU, 2-16
References, 2-18
Chapter 3 - Calculations with real numbers
Examples of real number calculations, 3-1
Using powers of 10 in entering data, 3-3
Real number functions in the MTH menu, 3-5
Using calculator menus, 3-5
Hyperbolic functions and their inverses, 3-5
Operations with units, 3-7
The UNITS menu, 3-7
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Available units, 3-9
Attaching units to numbers, 3-9
Unit prefixes, 3-10
Operations with units, 3-11
Unit conversions, 3-12
Physical constants in the calculator, 3-13
Defining and using functions, 3-15
Reference, 3-16
Chapter 4 - Calculations with complex numbers
Definitions, 4-1
Setting the calculator to COMPLEX mode, 4-1
Entering complex numbers, 4-2
Polar representation of a complex number, 4-3
Simple operations with complex numbers, 4-4
The CMPLX menus, 4-4
CMPLX menu through the MTH menu, 4-4
CMPLX menu in keyboard, 4-6
Functions applied to complex numbers, 4-6
Function DROITE: equation of a straight line, 4-7
Reference, 4-7
Chapter 5 - Algebraic and arithmetic operations
Entering algebraic objects, 5-1
Simple operations with algebraic objects, 5-2
Functions in the ALG menu , 5-3
Operations with transcendental functions, 5-5
Expansion and factoring using log-exp functions, 5-5
Expansion and factoring using trigonometric functions, 5-6
Functions in the ARITHMETIC menu, 5-7
Polynomials, 5-8
The HORNER function, 5-8
The variable VX, 5-8
The PCOEF function, 5-8
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The PROOT function, 5-9
The QUOT and REMAINDER functions, 5-9
The PEVAL function , 5-9
Fractions, 5-9
The SIMP2 function, 5-10
The PROPFRAC function, 5-10
The PARTFRAC function, 5-10
The FCOEF function, 5-10
The FROOTS function, 5-11
Step-by-step operations with polynomials and fractions, 5-11
Reference, 5-12
Chapter 6 - Solution to equations
Symbolic solution of algebraic equations, 6-1
Function ISOL, 6-1
Function SOLVE, 6-2
Function SOLVEVX, 6-4
Function ZEROS, 6-4
Numerical solver menu, 6-5
Polynomial Equations, 6-6
Finding the solutions to a polynomial equation, 6-6
Generating polynomial coefficients given the polynomial's roots,
6-7
Generating an algebraic expression for the polynomial, 6-8
Financial calculations, 6-8
Solving equations with one unknown through NUM.SLV, 6-9
Function STEQ, 6-9
Solution to simultaneous equations with MSLV, 6-10
Reference, 6-11
Chapter 7 - Operations with lists
Creating and storing lists, 7-1
Operations with lists of numbers, 7-1
Changing sign , 7-1
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Addition, subtraction, multiplication, division, 7-2
Functions applied to lists, 7-4
Lists of complex numbers, 7-4
Lists of algebraic objects, 7-5
The MTH/LIST menu, 7-5
The SEQ function, 7-7
The MAP function, 7-7
Reference, 7-7
Chapter 8 - Vectors
Entering vectors , 8-1
Typing vectors in the stack, 8-1
Storing vectors into variables in the stack, 8-2
Using the Matrix Writer (MTRW) to enter vectors, 8-3
Simple operations with vectors, 8-5
Changing sign, 8-5
Addition, subtraction, 8-5
Multiplication by a scalar, and division by a scalar, 8-6
Absolute value function, 8-6
The MTH/VECTOR menu, 8-6
Magnitude, 8-7
Dot product , 8-7
Cross product, 8-7
Reference, 8-8
Chapter 9 - Matrices and linear algebra
Entering matrices in the stack, 9-1
Using the Matrix Writer, 9-1
Typing in the matrix directly into the stack, 9-2
Operations with matrices, 9-3
Addition and subtraction, 9-4
Multiplication, 9-4
Multiplication by a scalar, 9-4
Matrix-vector multiplication, 9-5
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Matrix multiplication, 9-5
Term-by-term multiplication, 9-6
Raising a matrix to a real power, 9-6
The identity matrix, 9-7
The inverse matrix, 9-7
Characterizing a matrix (The matrix NORM menu), 9-8
Function DET, 9-8
Function TRACE, 9-8
Solution of linear systems, 9-9
Using the numerical solver for linear systems, 9-9
Solution with the inverse matrix, 9-11
Solution by “division” of matrices, 9-11
References, 9-12
Chapter 10 - Graphics
Graphs options in the calculator, 10-1
Plotting an expression of the form y = f(x), 10-2
Generating a table of values for a function, 10-4
Fast 3D plots, 10-5
Reference, 10-7
Chapter 11 - Calculus Applications
The CALC (Calculus) menu, 11-1
Limits and derivatives, 11-1
Function lim, 11-1
Functions DERIV and DERVX, 11-3
Anti-derivatives and integrals, 11-3
Functions INT, INTVX, RISCH, SIGMA and SIGMAVX, 11-3
Definite integrals, 11-4
Infinite series, 11-5
Functions TAYLR, TAYLR0, and SERIES, 11-5
Reference, 11-6
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Chapter 12 - Multi-variate Calculus Applications
Partial derivatives, 12-1
Multiple integrals, 12-2
Reference, 12-2
Chapter 13 - Vector Analysis Applications
The del operator, 13-1
Gradient, 13-1
Divergence, 13-2
Curl, 13-2
Reference, 13-2
Chapter 14 - Differential Equations
The CALC/DIFF menu, 14-1
Solution to linear and non-linear equations, 14-1
Function LDEC, 14-1
Function DESOLVE, 14-3
The variable ODETYPE, 14-3
Laplace Transforms, 14-4
Laplace transform and inverses in the calculator, 14-4
Fourier series, 14-5
Function FOURIER, 14-5
Fourier series for a quadratic function, 14-6
Reference, 14-7
Chapter 15 - Probability Distributions
The MTH/PROBABILITY.. sub-menu - part 1, 15-1
Factorials, combinations, and permutations, 15-1
Random numbers, 15-2
The MTH/PROB menu - part 2, 15-3
The Normal distribution, 15-3
The Student-t distribution, 15-3
The Chi-square distribution, 15-4
The F distribution, 15-4
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Reference, 15-4
Chapter 16 - Statistical Applications
Entering data, 16-1
Calculating single-variable statistics, 16-2
Sample vs. population, 16-2
Obtaining frequency distributions, 16-3
Fitting data to a function y = f(x), 16-5
Obtaining additional summary statistics, 16-6
Confidence intervals, 16-7
Hypothesis testing, 16-9
Reference, 16-11
Chapter 17 - Numbers in Different Bases
The BASE menu, 17-1
Writing non-decimal numbers, 17-2
Reference, 17-2
Chapter 18 - Using SD cards
Inserting and removing an SD card, 18-1
Formatting an SD card, 18-1
Accessing objects on an SD card, 18-2
Storing objects on the SD card, 18-2
Recalling an object from the SD card, 18-3
Purging an object from the SD card, 18-3
Purging all objects on the SD card (by reformatting), 18-4
Specifying a directory on an SD card, 18-4
Chapter 19 - Equation Library
Reference, 19-4
Limited Warranty, W-1
Service, W-3
Regulatory information, W-5
Disposal of Waste Equipment by Users in Private Household in the European Union, W-7
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Chapter 1
Getting started
This chapter provides basic information about the operation of your
calculator. It is designed to familiarize you with the basic operations and
settings before you perform a calculation.
Basic Operations
Batteries
The calculator uses 4 AAA (LR03) batteries as main power and a CR2032
lithium battery for memory backup.
Before using the calculator, please install the batteries according to the
following procedure.
To install the main batteries
a. Make sure the calculator is OFF. Slide up the battery compartment
cover as illustrated.
b. Insert 4 new AAA (LR03) batteries into the main compartment. Make
sure each battery is inserted in the indicated direction.
To install the backup battery
a. Make sure the calculator is OFF. Press down the holder. Push the plate
to the shown direction and lift it.
Page 1-1
b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is
facing up.
c. Replace the plate and push it to the original place.
After installing the batteries, press $ to turn the power on.
Warning: When the low battery icon is displayed, you need to replace the
batteries as soon as possible. However, avoid removing the backup battery
and main batteries at the same time to avoid data lost.
Turning the calculator on and off
The $ key is located at the lower left corner of the keyboard. Press it
once to turn your calculator on. To turn the calculator off, press the rightshift key @ (first key in the second row from the bottom of the keyboard),
followed by the $ key. Notice that the $ key has a OFF label printed
in the upper right corner as a reminder of the OFF command.
Adjusting the display contrast
You can adjust the display contrast by holding the $ key while pressing
the + or - keys.
The $(hold) + key combination produces a darker display
The $(hold) - key combination produces a lighter display
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Contents of the calculator’s display
Turn your calculator on once more. At the top of the display you will have
two lines of information that describe the settings of the calculator. The first
line shows the characters:
RAD XYZ HEX R= 'X'
For details on the meaning of these symbols see Chapter 2 in the
calculator’s user’s guide.
The second line shows the characters
{ HOME }
indicating that the HOME directory is the current file directory in the
calculator’s memory.
At the bottom of the display you will find a number of labels, namely,
@EDIT @VIEW @@[email protected]@ @@[email protected] !PURGE !CLEAR
associated with the six soft menu keys, F1 through F6:
ABCDEF
The six labels displayed in the lower part of the screen will change
depending on which menu is displayed. But A will always be
associated with the first displayed label, B with the second displayed
label, and so on.
Menus
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
and Chapter 2 and Appendix L in the user’s guide for more
information on editing)
@VIEW B VIEW the contents of a variable
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@@[email protected] C ReCaLl the contents of a variable
@@[email protected] D STOre the contents of a variable
!PURGE E PURGE a variable
@CLEAR F 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 A CASCMD: CAS CoMmanD, used to launch a command from
the CAS (Computer Algebraic System) by selecting from a list
@HELP B 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 calculator’s user’s guide to learn how to set time and
date.
Introducing the calculator’s keyboard
The figure on the next page shows a diagram of the calculator’s 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 left-shift key, key (8,1), the right-shift key, key
(9,1), and the ALPHA key, key (7,1), can be combined with some of the
other keys to activate the alternative functions shown in the keyboard.
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For example, the P key, key(4,4), has the following six functions
associated with it:
P
Main function, to activate the SYMBolic menu
„´
Left-shift function, to activate the MTH (Math) menu
…N
Right-shift function, to activate the CATalog function
~p
ALPHA function, to enter the upper-case letter P
~„p
ALPHA-Left-Shift function, to enter the lower-case letter p
Page 1-5
~…p
ALPHA-Right-Shift function, to enter the symbol π
Of the six functions associated with a key only the first four are shown in
the keyboard itself. The figure in next page shows these four labels for the
P key. Notice that the color and the position of the labels in the key,
namely, SYMB, MTH, CAT and P, indicate which is the main function
(SYMB), and which of the other three functions is associated with the leftshift „(MTH), right-shift …(CAT ), and ~ (P) keys.
For detailed information on the calculator keyboard operation refer to
Appendix B in the calculator’s user’s guide.
Selecting calculator modes
This section assumes that you are now at least partially familiar with the
use of choose and dialog boxes (if you are not, please refer to appendix A
in the user’s guide).
Press the H button (second key from the left on the second row of keys
from the top) to show the following CALCULATOR MODES input form:
Press the [email protected]@OK#@ soft menu key to return to normal display. Examples of
selecting different calculator modes are shown next.
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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. If using the latter approach, use up and
down arrow keys, — ˜, to select the mode, and press the [email protected]@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:
⎛
⎝
3.0 ⋅ ⎜ 5.0 −
⎞
⎟
3.0 ⋅ 3.0 ⎠
2.5
+e
23.0
1
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:
[email protected]#*+-/R
Q¸Ü‚Oš™˜—`
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.*!Ü5.1./3.*3.
—————
/23.Q3™™+!¸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 [email protected]@OK#@):
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You could also type the expression directly into the display without using
the equation writer, as follows:
R!Ü3.*!Ü5.1/3.*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#@ soft menu key to complete the
operation. The display, for the RPN mode looks as follows:
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.
What RPN means is that, instead of writing an operation such as 3 + 2 by
pressing
3+2`
we write the operands first, in the proper order, and then the operator, i.e.,
3`2+
As you enter the operands, they occupy different stack levels. Entering
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, +, to the objects occupying
levels 1 and 2. The result, 5, is then placed in level 1.
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Let's try some other simple operations before trying the more complicated
expression used earlier for the algebraic operating mode:
123/32
123`32/
42
3
4`2Q
√(√27)
[email protected]»
Note the position of the y and 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
5`3+ Calculates (5 +3) first.
2X
Completes the calculation.
Let's try now the expression proposed earlier:
⎛
⎝
3 ⋅ ⎜5 −
23
⎞
⎟
3⋅3⎠
2.5
+e
1
3
3`
Enter 3 in level 1
5`
Enter 5 in level 1, 3 moves to level 2
3`
Enter 3 in level 1, 5 moves to level 2, 3 to level 3
3*
Place 3 and multiply, 9 appears in level 1
Y
1/(3×3), last value in lev. 1; 5 in level 2; 3 in level 3
-
5 - 1/(3×3) , occupies level 1 now; 3 in level 2
*
3 × (5 - 1/(3×3)), occupies level 1 now.
23`Enter 23 in level 1, 14.66666 moves to level 2.
3Q
Enter 3, calculate 233 into level 1. 14.666 in lev. 2.
/
(3 × (5-1/(3×3)))/233 into level 1
2.5Enter 2.5 level 1
!¸
Page 1-9
e2.5, goes into level 1, level 2 shows previous value.
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+
(3 × (5 - 1/(3 × 3)))/233 + e2.5 = 12.18369, into lev. 1.
R
√((3 × (5 - 1/(3×3)))/233 + e2.5) = 3.4905156, into 1.
To select between the ALG vs. RPN operating mode, you can also set/
clear system flag 95 through the following keystroke sequence:
H @FLAGS! 9˜˜˜˜
`
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 (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 [email protected]@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, and select the
option Fixed with the arrow down key ˜.
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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.
Press the [email protected]@OK#@ soft menu key to complete the selection:
Press the [email protected]@OK#@ soft menu key return to the calculator display. The 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, and select the option Scientific with the arrow down key ˜.
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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).
Press the [email protected]@OK#@ soft menu key return to the calculator display. The number
now is shown as:
This result, 1.23E2, is the calculator’s 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, and select the option
Engineering with the arrow down key ˜. Keep the 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).
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Press the [email protected]@OK#@ soft menu key return to the calculator display. The number
now is shown as:
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:
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
soft menu key. The input form will look as follows:
Press the [email protected]@OK#@ soft menu key return to the calculator display. The number
123.4567890123456, entered earlier, now is shown as:
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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 (360°) in a complete circumference.
• Radians: There are 2π radians (2π r) in a complete circumference.
• 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. If using the latter approach, use up and down
arrow keys, —˜, to select the preferred mode, and press the [email protected]@OK#@
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. If using the latter approach, use up and down
arrow keys, —˜, to select the preferred mode, and press the [email protected]@OK#@
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soft menu key to complete the operation. For example, in the following
screen, the Polar coordinate mode is selected:
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 @@[email protected]@ 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: š™˜—.
• To select or deselect any of the settings shown above, select the
underline before the option of interest, and toggle the
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
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Non-Rational options above). Unselected options will show no check
mark in the underline preceding the option of interest (e.g., the
_Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow
options above).
• After having selected and unselected all the options that you want in
the CAS MODES input form, press the @@@[email protected]@@ soft menu key. This will
take you back to the CALCULATOR MODES input form. To return to
normal calculator display at this point, press the @@@[email protected]@@ soft menu key
once more.
Explanation of CAS settings
• Indep var: The independent variable for CAS applications. Typically,
VX = ‘X’.
• Modulo: For operations in modular arithmetic this variable holds the
modulus or modulo of the arithmetic ring (see Chapter 5 in the
calculator’s user’s guide).
• Numeric: If set, the calculator produces a numeric, or floating-point
result, in calculations. Note that constants will always be evaluated
numerically.
• Approx: If set, Approximate mode uses numerical results in calculations.
If unchecked, the CAS is in Exact mode, which produces symbolic
results in algebraic calculations.
• Complex: If set, complex number operations are active. If unchecked
the CAS is in Real mode, i.e., real number calculations are the default.
See Chapter 4 for operations with complex numbers.
• Verbose: If set, provides detailed information in certain CAS
operations.
• Step/Step: If set, provides step-by-step results for certain CAS
operations. Useful to see intermediate steps in summations, derivatives,
integrals, polynomial operations (e.g., synthetic division), and matrix
operations.
• Incr Pow: Increasing Power, means that, if set, polynomial terms are
shown in increasing order of the powers of the independent variable.
• Rigorous: If set, calculator does not simplify the absolute value function
|X| to X.
• Simp Non-Rational: If set, the calculator will try to simplify non-rational
expressions as much as possible.
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Selecting Display modes
The calculator display can be customized to your preference by selecting
different display modes. To see the optional display settings use the
following:
• First, press the H button to activate the CALCULATOR MODES input
form. Within the CALCULATOR MODES input form, press the @@[email protected]
soft menu key to display the DISPLAY MODES input form.
• To navigate through the many options in the DISPLAY MODES input
form, use the arrow keys: š™˜—.
• To select or deselect any of the settings shown above, that require a
check mark, select the underline before the option of interest, and
toggle the
soft menu key until the right setting is achieved.
When an option is selected, a check mark will be shown in the
underline (e.g., the Textbook option in the Stack: line above).
Unselected options will show no check mark in the underline preceding
the option of interest (e.g., the _Small, _Full page, and _Indent options
in the Edit: line above).
• To select the Font for the display, highlight the field in front of the Font:
option in the DISPLAY MODES input form, and use the @CHOOS soft menu.
• After having selected and unselected all the options that you want in
the DISPLAY MODES input form, press the @@@[email protected]@@ soft menu key. This will
take you back to the CALCULATOR MODES input form. To return to
normal calculator display at this point, press the @@@[email protected]@@ soft menu key
once more.
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Selecting the display font
First, press the H button to activate the CALCULATOR MODES input
form. Within the CALCULATOR MODES input form, press the @@[email protected] soft
menu key 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 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 @@@[email protected]@@ soft menu key to go back to the CALCULATOR MODES input
form. To return to normal calculator display at this point, press the @@@[email protected]@@
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 @@[email protected] soft
menu key 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
Changes font size to small
_Full page
Allows to place the cursor after the end of the line
_Indent
Auto indent cursor when entering a carriage return
Instructions on the use of the line editor are presented in Chapter 2 in the
user’s 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 @@[email protected] 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.
_Textbook
Displays mathematical expressions in graphical
mathematical notation
To illustrate these settings, either in algebraic or RPN mode, use the
equation writer to type the following definite integral:
‚O…Á0™„虄¸\x™x`
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 @@[email protected] soft
menu key 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
Changes font size to small while using the equation
editor
_Small Stack Disp 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
∫
∞
0
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 calculator’s user’s 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 calculator’s 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 − 2.0 3
1.0 +
To enter this expression use the following keystrokes:
5.*„Ü1.+1/7.5™/
„ÜR3.-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 user’s guide)
and you enter your expression using integer numbers for integer values,
the result is a symbolic quantity, e.g.,
5*„Ü1+1/7.5™/
„ÜR3-2Q3
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 preceded by a tickmark, however, the
calculator will reproduce the expression as entered. For example:
³5*„Ü1+1/7.5™/
„ÜR3-2Q3`
The result will be shown as follows:
To evaluate the expression we can use the EVAL function, as follows:
µ„î`
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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 …ï.
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.,
³5*„Ü1+1/7.5™/
„ÜR3-2Q3`
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 by pressing:
µ!î` or @ï`
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., ™…ï.
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: -µ. The result is zero (0.).
For additional information on editing arithmetic expressions in the display
or stack, see Chapter 2 in the calculator’s user’s guide.
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Creating algebraic expressions
Algebraic expressions include not only numbers, but also variable names.
As an example, we will enter the following algebraic expression:
x
R +2L
R+ y
b
2L 1 +
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:
³2*~l*R„Ü1+~„x/
~r™/„Ü~r+~„y™+2*~l/
~„b
Press ` to get the following result:
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
calculator’s display or stack see Chapter 2 in the calculator’s user’s guide.
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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 calculator’s
user’s guide.
Creating arithmetic expressions
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:
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:
*„Ü5+1/3
The edited expression looks as follows:
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Suppose that you want to replace the quantity between parentheses in the
denominator (i.e., 5+1/3) with (5+π2/2). First, we use the delete key
(ƒ) delete the current 1/3 expression, and then we replace that fraction
with π2/2, as follows:
ƒƒƒ„ìQ2
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 π2 expression. We do this by pressing the right arrow key (™)
once. At that point, we enter the following keystrokes:
/2
The expression now looks as follows:
Suppose that now you want to add the fraction 1/3 to this entire
expression, i.e., you want to enter the expression:
5
5 + 2 ⋅ (5 +
π
2
2
+
)
1
3
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First, we need to highlight the entire first term by using either the right
arrow (™) or the upper arrow (—) keys, repeatedly, until the entire
expression is highlighted, i.e., seven times, producing:
NOTE: Alternatively, from the original position of the cursor (to the
right of the 2 in the denominator of π2/2), we can use the keystroke
combination ‚—, interpreted as (‚ ‘ ).
Once the expression is highlighted as shown above, type +1/
3 to add the fraction 1/3. Resulting in:
Creating algebraic expressions
An algebraic expression is very similar to an arithmetic expression, except
that English and Greek letters may be included. The process of creating
an algebraic expression, therefore, follows the same idea as that of
creating an arithmetic expression, except that use of the alphabetic
keyboard is included.
To illustrate the use of the Equation Writer to enter an algebraic equation
we will use the following example. Suppose that we want to enter the
expression:
2
3
⎛ x + 2 µ ⋅ ∆y ⎞
⎟
1/ 3
⎝ θ
⎠
λ + e − µ ⋅ LN ⎜
Use the following keystrokes:
2/R3™™*~‚n+„¸\~‚m
™™*‚¹~„x+2*~‚m*~‚c
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~„y———/~‚tQ1/3
This results in the output:
In this example we used several lower-case English letters, e.g., x
(~„x), several Greek letters, e.g., λ(~‚n), and even a
∆y
combination
of
Greek
and
English
letters,
namely,
(~‚c~„y). Keep in mind that to enter a lower-case English
letter, you need to use the combination: ~„ followed by the letter you
want to enter. Also, you can always copy special characters by using the
CHARS menu (…±) if you don’t want to memorize the keystroke
combination that produces it. A listing of commonly used ~‚
keystroke combinations is listed in Appendix D of the user’s guide.
For additional information on editing, evaluating, factoring, and
simplifying algebraic expressions see Chapter 2 of the calculator’s user’s
guide.
Organizing data in the calculator
You can organize data in your calculator by storing variables in a
directory tree. The basis of the calculator’s directory tree is the HOME
directory described next.
The HOME directory
To get to the HOME directory, press the UPDIR function („§) -- repeat
as needed -- until the {HOME} spec is shown in the second line of the
display header. Alternatively, use „ (hold) §. For this example, the
HOME directory contains nothing but the CASDIR. Pressing J will show
the variables in the soft menu keys:
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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 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
off. For details on manipulation of directories see Chapter 2 in the
calculator’s user’s guide.
Variables
Variables are similar to files on a computer hard drive. One variable can
store one object (numerical values, algebraic expressions, lists, vectors,
matrices, programs, etc). Variables are referred to by their names, which
can be any combination of alphabetic and numerical characters, starting
with a letter (either English or Greek). Some non-alphabetic characters,
such as the arrow (→) can be used in a variable name, if combined with
an alphabetical character. Thus, ‘→A’ is a valid variable name, but ‘→’ is
not. Valid examples of variable names are: ‘A’, ‘B’, ‘a’, ‘b’, ‘α’, ‘β’, ‘A1’,
‘AB12’, ‘A12’, ’Vel’, ’Z0’, ’z1’, etc.
A variable can not have the same name as a function of the calculator.
Some of the reserved calculator variable names are the following:
ALRMDAT, CST, EQ, EXPR, IERR, IOPAR, MAXR, MINR, PICT, PPAR,
PRTPAR, VPAR, ZPAR, der_, e, i, n1,n2, …, s1, s2, …, ΣDAT, ΣPAR, π, ∞.
Variables can be organized into sub-directories (see Chapter 2 in the
calculator’s user’s guide).
Typing variable names
To name variables, you will have to type strings of letters at once, which
may or may not be combined with numbers. To type strings of characters
you can lock the alphabetic keyboard as follows:
~~ locks the alphabetic keyboard in upper case. When locked in
this fashion, pressing the „ before a letter key produces a lower case
letter, while pressing the ‚ key before a letter key produces a special
character. If the alphabetic keyboard is already locked in upper case, to
lock it in lower case, type, „~.
~~„~ locks the alphabetic keyboard in lower case. When
locked in this fashion, pressing the „ before a letter key produces an
upper case letter. To unlock lower case, press „~.
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To unlock the upper-case locked keyboard, press ~.
Try the following exercises:
~~math`
~~m„a„t„h`
~~m„~at„h`
The calculator display will show the following (left-hand side is Algebraic
mode, right-hand side is RPN mode):
Creating variables
The simplest way to create a variable is by using the K. The following
examples are used to store the variables listed in the following table (Press
J if needed to see variables menu):
Name
α
Contents
-0.25
Type
real
A12
3×105
‘r/(m+r)'
[3,2,1]
3+5i
<<→ r 'π*r^2' >>
real
Q
R
z1
p1
algebraic
vector
complex
program
Algebraic mode
To store the value of –0.25 into variable α: 0.25\K
~‚a. AT this point, the screen will look as follows:
Press ` to create the variable. The variable is now shown in the soft
menu key labels when you press J:
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The following are the keystrokes for entering the remaining variables:
A12: 3V5K~a12`
Q: ~„r/„Ü
~„m+~„r™™K~q`
R: „Ô3‚í2‚í1™K~r`
z1: 3+5*„¥K~„z1` (Accept change to
Complex mode if asked).
p1: å‚é~„r³„ì*
~„rQ2™™™K~„p1`.
The screen, at this point, will look as follows:
You will see six of the seven variables listed at the bottom of the screen:
p1, z1, R, Q, A12, a.
RPN mode
(Use H\@@[email protected]@ to change to RPN mode). Use the following keystrokes
to store the value of –0.25 into variable α: .25\`³
~‚a`. At this point, the screen will look as follows:
With –0.25 on the level 2 of the stack and 'α' on the level 1 of the stack,
you can use the K key to create the variable. The variable is now shown
in the soft menu key labels when you press J:
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To enter the value 3×105 into A12, we can use a shorter version of the
procedure: 3V5³~a12`K
Here is a way to enter the contents of Q:
Q: ~„r/„Ü
~„m+~„r™™³~q`K
To enter the value of R, we can use an even shorter version of the
procedure:
R: „Ô3#2#1™ ³~rK
Notice that to separate the elements of a vector in RPN mode we can use
the space key (#), rather than the comma (‚í) used above in
Algebraic mode.
z1: ³3+5*„¥³~„z1K
p1: ‚å‚é~„r³„ì*
~„rQ2™™™³~„p1™`K.
The screen, at this point, will look as follows:
You will see six of the seven variables listed at the bottom of the screen:
p1, z1, R, Q, A12, α.
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Checking variables contents
The simplest way to check a variable content is by pressing the soft menu
key label for the variable. For example, for the variables listed above,
press the following keys to see the contents of the variables:
Algebraic mode
Type these keystrokes: [email protected]@[email protected]@ ` @@@[email protected]@ `@@@[email protected]@@ `. At this point, the
screen looks as follows:
RPN mode
In RPN mode, you only need to press the corresponding soft menu key
label to get the contents of a numerical or algebraic variable. For the case
under consideration, we can try peeking into the variables z1, R, Q, A12,
α, created above, as follows: [email protected]@[email protected]@ @@@[email protected]@ @@@[email protected]@ @@[email protected]@ @@»@@
At this point, the screen looks like this:
Using the right-shift key followed by soft menu key labels
In Algebraic mode, you can display the content of a variable by pressing
[email protected] and then the corresponding soft menu key. Try the following
examples:
J‚@@[email protected]@ ‚ @@[email protected]@ ‚ @@@[email protected]@ ‚@@@[email protected]@ ‚ @@[email protected]@
NOTE: In RPN mode, you don’t need to press @ (just J and then
the corresponding soft menu key.)
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This produces the following screen (Algebraic mode in the left, RPN in the
right)
Notice that this time the contents of program p1 are listed in the screen. To
see the remaining variables in this directory, press L.
Listing the contents of all variables in the screen
Use the keystroke combination ‚˜ to list the contents of all variables
in the screen. For example:
Press $ to return to normal calculator display.
Deleting variables
The simplest way of deleting variables is by using function PURGE. This
function can be accessed directly by using the TOOLS menu (I), or by
using the FILES menu „¡@@[email protected]@ .
Using function PURGE in the stack in Algebraic mode
Our variable list contains variables p1, z1, Q, R, and α. We will use
command PURGE to delete variable p1. Press I @[email protected] J @@[email protected]@ `.
The screen will now show variable p1 removed:
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You can use the PURGE command to erase more than one variable by
placing their names in a list in the argument of PURGE. For example, if
now we wanted to purge variables R and Q, simultaneously, we can try
the following exercise. Press :
I @[email protected] „ä³J @@@[email protected]@ ™‚í³J @@@[email protected]@
At this point, the screen will show the following command ready to be
executed:
To finish deleting the variables, press `. The screen will now show the
remaining variables:
Using function PURGE in the stack in RPN mode
Assuming that our variable list contains the variables p1, z1, Q, R, and α.
We will use command PURGE to delete variable p1. Press ³ @@[email protected]@ `
I @[email protected] The screen will now show variable p1 removed:
To delete two variables simultaneously, say variables R and Q, first create
a list (in RPN mode, the elements of the list need not be separated by
commas as in Algebraic mode):
J „ä³ @@@[email protected]@ ™³ @@@[email protected]@ `
Then, press [email protected]@ use to purge the variables.
Additional information on variable manipulation is available in Chapter 2
of the calculator’s user’s guide.
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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 „®.
CHOOSE boxes vs. Soft MENU
In some of the exercises presented in this chapter we have seen menu lists
of commands displayed in the screen. These menu lists are referred to as
CHOOSE boxes. Herein we indicate the way to change from CHOOSE
boxes to Soft MENUs, and vice versa, through an exercise.
Although not applied to a specific example, the present exercise shows the
two options for menus in the calculator (CHOOSE boxes and soft MENUs).
In this exercise, we use the ORDER command to reorder variables in a
directory. The steps are shown for Algebraic mode.
„°˜
Show PROG menu list and select MEMORY
@@[email protected]@ ˜˜˜˜ Show the MEMORY menu list and select
DIRECTORY
@@[email protected]@ ——
Show the DIRECTORY menu list and select ORDER
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@@[email protected]@
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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 calculator’s user’s guide). To set this flag try the following:
[email protected]! ———————
The screen shows flag 117 not set (CHOOSE boxes), as shown here:
Press the
soft menu key to set flag 117 to soft MENU. The screen
will reflect that change:
Press @@[email protected]@ twice to return to normal calculator display.
Now, we’ll 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.,
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Press B to select the MEMORY soft menu ()@@[email protected]@). The display now
shows:
Press E to select the DIRECTORY soft menu ()@@[email protected]@)
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.
NOTE: most of the examples in this user manual assume that the
current setting of flag 117 is its default setting (that is, not set). If you
have set the flag but want to strictly follow the examples in this manual,
you should clear the flag before continuing.
References
For additional information on entering and manipulating expressions in the
display or in the Equation Writer see Chapter 2 of the calculator’s user’s
guide. For CAS (Computer Algebraic System) settings, see Appendix C in
the calculator’s user’s guide. For information on Flags see, Chapter 24 in
the calculator’s user’s 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 calculator’s 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 4Y.
• For addition, subtraction, multiplication, division, use the proper
operation key, namely, +-*/.
Examples in ALG mode:
3.7+5.2`
6.3-8.5`
4.2*2.5`
2.3/4.5`
Examples in RPN mode:
3.7` 5.2+
6.3` 8.54.2` 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+
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6.3#8.54.2#2.5*
2.3#4.5/
• Parentheses („Ü) can be used to group operations, as well as to
enclose arguments of functions.
In ALG mode:
„Ü5+3.2™/„Ü72.2`
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:
³„Ü5+3.2™/
„Ü7-2.2`µ
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
————@[email protected] or, ‚—@[email protected]
• The absolute value function, ABS, is available through „Ê.
Example in ALG mode:
„Ê\2.32`
Example in RPN mode:
2.32\„Ê
• The square function, SQ, is available through „º.
Example in ALG mode:
„º\2.3`
Example in RPN mode:
2.3\„º
The square root function, √, is available through the R key. When
calculating in the stack in ALG mode, enter the function before the
argument, e.g.,
R123.4`
In RPN mode, enter the number first, then the function, e.g.,
123.4R
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• The power function, ^, is available through the Q key. When
calculating in the stack in ALG mode, enter the base (y) followed by the
Q key, and then the exponent (x), e.g.,
5.2Q1.25`
In RPN mode, enter the number first, then the function, e.g.,
5.2`1.25Q
• The root function, XROOT(y,x), is available through the keystroke
combination ‚». When calculating in the stack in ALG mode,
enter the function XROOT followed by the arguments (y,x), separated
by commas, e.g.,
‚»3‚í27`
In RPN mode, enter the argument y, first, then, x, and finally the
function call, e.g.,
27`3‚»
• Logarithms of base 10 are calculated by the keystroke combination
‚Ã (function LOG) while its inverse function (ALOG, or
antilogarithm) is calculated by using „Â. In ALG mode, the
function is entered before the argument:
‚Ã2.45`
„Â\2.3`
In RPN mode, the argument is entered before the function
2.45‚Ã
2.3\„Â
Using powers of 10 in entering data
Powers of ten, i.e., numbers of the form -4.5 ×10 -2, etc., are entered by
using the V key. For example, in ALG mode:
\4.5V\2`
Or, in RPN mode:
4.5\V2\`
• Natural logarithms are calculated by using ‚¹ (function LN) while
the exponential function (EXP) is calculated by using „¸. In ALG
mode, the function is entered before the argument:
‚¹2.45`
„¸\2.3`
In RPN mode, the argument is entered before the function
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2.45`‚¹
2.3\`„¸
• Three trigonometric functions are readily available in the keyboard: sine
(S), cosine (T), and tangent (U). Arguments of these functions
are angles in either degrees, radians, grades. The following examples
use angles in degrees (DEG):
In ALG mode:
S30`
T45`
U135`
In RPN mode:
30S
45T
135U
• The inverse trigonometric functions available in the keyboard are the
arcsine („¼), arccosine („¾), and arctangent („À).
The answer from these functions will be given in the selected angular
measure (DEG, RAD, GRD). Some examples are shown next:
In ALG mode:
„¼0.25`
„¾0.85`
„À1.35`
In RPN mode:
0.25„¼
0.85„¾
1.35„À
All the functions described above, namely, ABS, SQ, √, ^, XROOT, LOG,
ALOG, LN, EXP, SIN, COS, TAN, ASIN, ACOS, ATAN, can be combined
with the fundamental operations (+-*/) to form more complex
expressions. The Equation Writer, whose operations is described in
Chapter 2, is ideal for building such expressions, regardless of the
calculator operation mode.
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Real number functions in the MTH menu
The MTH („´) menu include a number of mathematical functions
mostly applicable to real numbers. With the default setting of CHOOSE
boxes for system flag 117 (see Chapter 2), the MTH menu shows the
following functions:
The functions are grouped by th type of argument (1. vectors, 2. matrices,
3. lists, 7. probability, 9. complex) or by the type of function (4. hyperbolic,
5. real, 6. base, 8. fft). It also contains an entry for the mathematical
constants available in the calculator, entry 10.
In general, be aware of the number and order of the arguments required
for each function, and keep in mind that, in ALG mode you should select
first the function and then enter the argument, while in RPN mode, you
should enter the argument in the stack first, and then select the function.
Using calculator menus
1. We will describe in detail the use of the 4. HYPERBOLIC.. menu in this
section with the intention of describing the general operation of
calculator menus. Pay close attention to the process for selecting
different options.
2. To quickly select one of the numbered options in a menu list (or
CHOOSE box), simply press the number for the option in the keyboard.
For example, to select option 4. HYPERBOLIC.. in the MTH menu,
simply press 4.
Hyperbolic functions and their inverses
Selecting Option 4. HYPERBOLIC.. , in the MTH menu, and pressing @@[email protected]@,
produces the hyperbolic function menu:
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For example, in ALG mode, the keystroke sequence to calculate, say,
tanh(2.5), is the following:
„´[email protected]@[email protected]@ [email protected]@[email protected]@ 2.5`
In the RPN mode, the keystrokes to perform this calculation are the
following:
2.5`„´[email protected]@[email protected]@ [email protected]@[email protected]@
The operations shown above assume that you are using the default setting
for system flag 117 (CHOOSE boxes). If you have changed the setting of
this flag (see Chapter 2) to SOFT menu, the MTH menu will show as
follows (left-hand side in ALG mode, right – hand side in RPN mode):
Pressing L shows the remaining options:
Thus, to select, for example, the hyperbolic functions menu, with this menu
format press )@@[email protected] , to produce:
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Finally, in order to select, for example, the hyperbolic tangent (tanh)
function, simply press @@[email protected]
NOTE: To see additional options in these soft menus, press the L
key or the „«keystroke sequence.
For example, to calculate tanh(2.5), in the ALG mode, when using SOFT
menus over CHOOSE boxes, follow this procedure:
„´@@[email protected] @@[email protected] 2.5`
In RPN mode, the same value is calculated using:
2.5`„´)@@[email protected] @@[email protected]
As an exercise of applications of hyperbolic functions, verify the following
values:
SINH (2.5) = 6.05020..
ASINH(2.0) = 1.4436…
COSH (2.5) = 6.13228..
ACOSH (2.0) = 1.3169…
TANH(2.5) = 0.98661..
ATANH(0.2) = 0.2027…
EXPM(2.0) = 6.38905….
LNP1(1.0) = 0.69314….
Operations with units
Numbers in the calculator can have units associated with them. Thus, it is
possible to calculate results involving a consistent system of units and
produce a result with the appropriate combination of units.
The UNITS menu
The units menu is launched by the keystroke combination
‚Û(associated with the 6 key). With system flag 117 set to
CHOOSE boxes, the result is the following menu:
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Option 1. Tools.. contains functions used to operate on units (discussed
later). Options 2. Length.. through 17.Viscosity.. contain menus with a
number of units for each of the quantities described. For example,
selecting option 8. Force.. shows the following units menu:
The user will recognize most of these units (some, e.g., dyne, are not used
very often nowadays) from his or her physics classes: N = newtons, dyn =
dynes, gf = grams – force (to distinguish from gram-mass, or plainly gram,
a unit of mass), kip = kilo-poundal (1000 pounds), lbf = pound-force (to
distinguish from pound-mass), pdl = poundal.
To attach a unit object to a number, the number must be followed by an
underscore. Thus, a force of 5 N will be entered as 5_N.
For extensive operations with units SOFT menus provide a more convenient
way of attaching units. Change system flag 117 to SOFT menus (see
Chapter 2), and use the keystroke combination ‚Û to get the
following menus. Press L to move to the next menu page.
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Pressing on the appropriate soft menu key will open the sub-menu of units
for that particular selection. For example, for the @)SPEED sub-menu, the
following units are available:
Pressing the soft menu key @)UNITS will take you back to the UNITS menu.
Recall that you can always list the full menu labels in the screen by using
‚˜, e.g., for the @)ENRG set of units the following labels will be listed:
NOTE: Use the L key or the „«keystroke sequence to
navigate through the menus.
Available units
For a complete list of available units see Chapter 3 in the calculator’s
user’s guide.
Attaching units to numbers
To attach a unit object to a number, the number must be followed by an
underscore (‚Ý, key(8,5)). Thus, a force of 5 N will be entered as
5_N.
Here is the sequence of steps to enter this number in ALG mode, system
flag 117 set to CHOOSE boxes:
5‚Ý‚Û[email protected]@[email protected]@ @@[email protected]@ `
NOTE: If you forget the underscore, the result is the expression 5*N,
where N here represents a possible variable name and not Newtons.
To enter this same quantity, with the calculator in RPN mode, use the
following keystrokes:
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5‚Û[email protected]@[email protected]@ @@[email protected]@
Notice that the underscore is entered automatically when the RPN mode is
active.
The keystroke sequences to enter units when the SOFT menu option is
selected, in both ALG and RPN modes, are illustrated next. For example,
in ALG mode, to enter the quantity 5_N use:
5‚Ý‚ÛL @)@FORCE @@@[email protected]@ `
The same quantity, entered in RPN mode uses the following keystrokes:
5‚ÛL @)@FORCE @@@[email protected]@
NOTE: You can enter a quantity with units by typing the underline and
units with the ~keyboard, e.g., 5‚Ý~n will produce the
entry: 5_N
Unit prefixes
You can enter prefixes for units according to the following table of prefixes
from the SI system. The prefix abbreviation is shown first, followed by its
name, and by the exponent x in the factor 10x corresponding to each
prefix:
Prefix
Name
x
Prefix
Name
x
Y
yotta
+24
d
deci
-1
Z
zetta
+21
c
centi
-2
E
exa
+18
m
milli
-3
P
peta
+15
µ
micro
-6
T
tera
+12
n
nano
-9
G
giga
+9
p
pico
-12
M
mega
+6
f
femto
-15
k,K
kilo
+3
a
atto
-18
h,H
hecto
+2
z
zepto
-21
D(*)
deka
+1
y
yocto
-24
(*) In the SI system, this prefix is da rather than D. Use D for deka in the
calculator, however.
To enter these prefixes, simply type the prefix using the ~ keyboard. For
example, to enter 123 pm (picometer), use:
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123‚Ý~„p~„m
Using UBASE (type the name) to convert to the default unit (1 m) results in:
Operations with units
Here are some calculation examples using the ALG operating mode. Be
warned that, when multiplying or dividing quantities with units, you must
enclosed each quantity with its units between parentheses. Thus, to enter,
for example, the product 12.5m × 5.2 yd, type it to read
(12.5_m)*(5.2_yd) `:
which shows as 65_(m⋅yd). To convert to units of the SI system, use
function UBASE (find it using the command catalog, ‚N):
NOTE: Recall that the ANS(1) variable is available through the
keystroke combination „î(associated with the ` key).
To calculate a division, say, 3250 mi / 50 h, enter it as
(3250_mi)/(50_h) `
which transformed to SI units, with function UBASE, produces:
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Addition and subtraction can be performed, in ALG mode, without using
parentheses, e.g., 5 m + 3200 mm, can be entered simply as
5_m + 3200_mm `.
More complicated expression require the use of parentheses, e.g.,
(12_mm)*(1_cm^2)/(2_s) `:
Stack calculations in the RPN mode do not require you to enclose the
different terms in parentheses, e.g.,
12 @@@[email protected]@@ `1.5 @@[email protected]@ `*
3250 @@[email protected]@ `50 @@@[email protected]@@ `/
These operations produce the following output:
Unit conversions
The UNITS menu contains a TOOLS sub-menu, which provides the
following functions:
CONVERT(x,y) convert unit object x to units of object y
UBASE(x)
convert unit object x to SI units
UVAL(x)
extract the value from unit object x
UFACT(x,y)
factors a unit y from unit object x
UNIT(x,y)
combines value of x with units of y
Examples of function CONVERT are shown below. Examples of the other
UNIT/TOOLS functions are available in Chapter 3 of the calculator’s user’s
guide.
For example, to convert 33 watts to btu’s use either of the following entries:
CONVERT(33_W,1_hp) `
CONVERT(33_W,11_hp) `
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Physical constants in the calculator
The calculator’s physical constants are contained in a constants library
activated with the command CONLIB. To launch this command you could
simply type it in the stack: ~~conlib`, or, you can select
the command CONLIB from the command catalog, as follows: First,
launch the catalog by using: ‚N~c. Next, use the up and down
arrow keys —˜ to select CONLIB. Finally, press @@[email protected]@. Press `, if
needed. Use the up and down arrow keys (—˜) to navigate through
the list of constants in your calculator.
The soft menu keys corresponding to this CONSTANTS LIBRARY screen
include the following functions:
SI
when selected, constants values are shown in SI units (*)
ENGL
when selected, constants values are shown in English units
(*)
UNIT
when selected, constants are shown with units attached (*)
VALUE
when selected, constants are shown without units
STK
copies value (with or without units) to the stack
QUIT
exit constants library
(*) Activated only if the VALUE option is selected.
This is the way the top of the CONSTANTS LIBRARY screen looks when the
option VALUE is selected (units in the SI system):
To see the values of the constants in the English (or Imperial) system, press
the @ENGL option:
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If we de-select the UNITS option (press @UNITS ) only the values are shown
(English units selected in this case):
To copy the value of Vm to the stack, select the variable name, and press
@²STK, then, press @[email protected] For the calculator set to the ALG, the screen will
look like this:
The display shows what is called a tagged value, Vm:359.0394. In
here, Vm, is the tag of this result. Any arithmetic operation with this
number will ignore the tag. Try, for example:
‚¹2*„î`
which produces:
The same operation in RPN mode will require the following keystrokes
(after the value of Vm was extracted from the constants library):
2`*‚¹
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Defining and using functions
Users can define their own functions by using the DEFINE command
available thought the keystroke sequence „à (associated with the
2 key). The function must be entered in the following format:
Function_name(arguments) = expression_containing_arguments
For example, we could define a simple function
H(x) = ln(x+1) + exp(-x)
Suppose that you have a need to evaluate this function for a number of
discrete values and, therefore, you want to be able to press a single button
and get the result you want without having to type the expression in the
right-hand side for each separate value. In the following example, we
assume you have set your calculator to ALG mode. Enter the following
sequence of keystrokes:
„à³~h„Ü~„x™‚Å
‚¹~„x+1™+„¸~„x`
The screen will look like this:
Press the J key, and you will notice that there is a new variable in your
soft menu key (@@@[email protected]@). To see the contents of this variable press ‚@@@[email protected]@.
The screen will show now:
Thus, the variable H contains a program defined by:
<< x ‘LN(x+1) + EXP(x)’ >>
This is a simple program in the default programming language of the
calculator. This programming language is called UserRPL (See Chapters
20 and 21 in the calculator’s user’s guide). The program shown above is
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relatively simple and consists of two parts, contained between the program
containers
• Input:
x
x
• Process:
‘LN(x+1) + EXP(x) ‘
This is to be interpreted as saying: enter a value that is temporarily
assigned to the name x (referred to as a local variable), evaluate the
expression between quotes that contain that local variable, and show the
evaluated expression.
To activate the function in ALG mode, type the name of the function
followed by the argument between parentheses, e.g., @@@[email protected]@@
„Ü2`. Some examples are shown below:
In the RPN mode, to activate the function enter the argument first, then
press the soft menu key corresponding to the variable name @@@[email protected]@@ . For
example, you could try: [email protected]@@[email protected]@@ . The other examples shown above can
be entered by using: [email protected]@@[email protected]@@ , 2`3/@@@[email protected]@@.
Reference
Additional information on operations with real numbers with the calculator
is contained in Chapter 3 of the user’s guide.
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Chapter 4
Calculations with complex numbers
This chapter shows examples of calculations and application of functions to
complex numbers.
Definitions
A complex number z is a number z = x + iy, where x and y are real
numbers, and i is the imaginary unit defined by i² = –1. The complex
number x + iy has a real part, x = Re(z), and an imaginary part, y = Im(z).
The complex number z = zx + iy is often used to represent a point P(x,y) in
the x–y plane, with the x-axis referred to as the real axis, and the y-axis
referred to as the imaginary axis.
A complex number in the form x + iy is said to be in a rectangular
representation. An alternative representation is the ordered pair z = (x,y).
A complex number can also be represented in polar coordinates (polar
representation) as z = reiθ = r·cosθ + i r·sinθ, where r = |z| =
x2 + y2
is the magnitude of the complex number z, and θ = Arg(z) = arctan(y/x) is
the argument of the complex number z.
The relationship between the Cartesian and polar representation of
complex numbers is given by the Euler formula: ei iθ = cosθ + i sinθ. The
complex conjugate of a complex number (z = x + iy = re iθ) is z = x – iy
= re –iθ . The complex conjugate of i can be thought of as the reflection of
z about the real (x) axis. Similarly, the negative of z, –z = –x –iy = –re iθ,
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)@@[email protected] ˜˜™
The COMPLEX mode will be selected if the CAS MODES screen shows the
option _Complex checked, i.e.,
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Press @@[email protected]@ , twice, to return to the stack.
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:
„Ü3.5‚í\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*„¥`
NOTE: to enter the unit imaginary number alone type „¥, the I
key.
In RPN mode, these numbers could be entered using the following
keystrokes:
„Ü3.5‚í1.2\`
(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).
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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 standard notation and the angular
measure in radians, produces the result in RPN mode:
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 = r⋅eiθ. You can enter this complex number
into the calculator by using an ordered pair of the form (r, ∠θ). 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):
Because the coordinate system is set to rectangular (or Cartesian), the
calculator automatically converts the number entered to Cartesian
coordinates, i.e., x = r cos θ, y = r sin θ, resulting, for this case, in
(0.3678…, 5.18…).
On the other hand, if the coordinate system is set to cylindrical coordinates
(use CYLIN), entering a complex number (x,y), where x and y are real
numbers, will produce a polar representation. For example, in cylindrical
coordinates, enter the number (3.,2.). The figure below shows the RPN
stack, before and after entering this number:
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Simple operations with complex numbers
Complex numbers can be combined using the four fundamental operations
(+-*/). The results follow the rules of algebra with the caveat
that i2= -1. Operations with complex numbers are similar to those with
real numbers. For example, with the calculator in ALG mode and the CAS
set to Complex, try the following operations:
(3+5i) + (6-3i) = (9,2);
(5-2i) - (3+4i) = (2,-6)
(3-i)·(2-4i) = (2,-14);
(5-2i)/(3+4i) = (0.28,-1.04)
1/(3+4i) = (0.12, -0.16) ;
-(5-3i) = -5 + 3i
The CMPLX menus
There are two CMPLX (CoMPLeX numbers) menus available in the
calculator. One is available through the MTH menu (introduced in
Chapter 3) and one directly into the keyboard (‚ß). The two CMPLX
menus are presented next.
CMPLX menu through the MTH menu
Assuming that system flag 117 is set to CHOOSE boxes (see Chapter 2),
the CMPLX sub-menu within the MTH menu is accessed by using:
„´[email protected]@[email protected]@ . The functions available are the following:
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The first menu (options 1 through 6) shows the following functions:
Examples of applications of these functions are shown next in RECT
RE(z)
Real part of a complex number
IM(z)
Imaginary part of a complex number
C→R(z)
Separates a complex number into its real and imaginary
parts
R→C(x,y)
Forms the complex number (x,y) out of real numbers x and
y
ABS(z)
Calculates the magnitude of a complex number.
ARG(z)
Calculates the argument of a complex number.
SIGN(z)
Calculates a complex number of unit magnitude as z/
|z|.
NEG(z)
Changes the sign of z
CONJ(z)
Produces the complex conjugate of z
coordinates. Recall that, for ALG mode, the function must precede the
argument, while in RPN mode, you enter the argument first, and then select
the function. Also, recall that you can get these functions as soft menu
labels by changing the setting of system flag 117 (See Chapter 2).
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CMPLX menu in keyboard
A second CMPLX menu is accessible by using the right-shift option
associated with the 1 key, i.e., ‚ß. With system flag 117 set to
CHOOSE boxes, the keyboard CMPLX menu shows up as the following
screens:
The resulting menu include some of the functions already introduced in the
previous section, namely, ARG, ABS, CONJ, IM, NEG, RE, and SIGN. It
also includes function i which serves the same purpose as the keystroke
combination „¥.
Functions applied to complex numbers
Many of the keyboard-based functions and MTH menu functions defined in
Chapter 3 for real numbers (e.g., SQ, LN, ex, etc.), can be applied to
complex numbers. The result is another complex number, as illustrated in
the following examples.
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NOTE: When using trigonometric functions and their inverses with
complex numbers the arguments are no longer angles. Therefore, the
angular measure selected for the calculator has no bearing in the
calculation of these functions with complex arguments.
Function DROITE: equation of a straight line
Function DROITE takes as argument two complex numbers, say, x1 + iy1
and x2+iy2, and returns the equation of the straight line, say, y = a + bx,
that contains the points (x1, y1) and (x2, y2). For example, the line
between points A(5, -3) and B(6, 2) can be found as follows (example in
Algebraic mode):
Function DROITE is found in the command catalog (‚N).
calculator is in APPROX mode, the result will be Y = 5.*(X-5.)-3.
If the
Reference
Additional information on complex number operations is presented in
Chapter 4 of the calculator’s user’s guide.
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Chapter 5
Algebraic and arithmetic operations
An algebraic object, or simply, algebraic, is any number, variable name or
algebraic expression that can be operated upon, manipulated, and
combined according to the rules of algebra. Examples of algebraic objects
are the following:
12.3, 15.2_m, ‘π’, ‘e’, ‘i’
• A number:
• A variable name: ‘a’, ‘ux’, ‘width’, etc.
• An expression:
‘p*D^2/4’, ’f*(L/D)*(V^2/(2*g))’,
• An equation:
‘p*V = n*R*T’, ‘Q=(Cu/n)*A(y)*R(y)^(2/3)*√So’
Entering algebraic objects
Algebraic objects can be created by typing the object between single
quotes directly into stack level 1 or by using the equation writer (EQW).
For example, to enter the algebraic object ‘π*D^2/4’ directly into stack
level 1 use:
³„ì*~dQ2/4`
An algebraic object can also be built in the Equation Writer and then sent
to the stack, or operated upon in the Equation Writer itself. The operation
of the Equation Writer was described in Chapter 2. As an exercise, build
the following algebraic object in the Equation Writer:
After building the object, press ` to show it in the stack (ALG and RPN
modes shown below):
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Simple operations with algebraic objects
Algebraic objects can be added, subtracted, multiplied, divided (except by
zero), raised to a power, used as arguments for a variety of standard
functions (exponential, logarithmic, trigonometry, hyperbolic, etc.), as you
would any real or complex number. To demonstrate basic operations with
algebraic objects, let’s create a couple of objects, say ‘π*R^2’ and
‘g*t^2/4’, and store them in variables A1 and A2 (See Chapter 2 to learn
how to create variables and store values in them). Here are the keystrokes
for storing variables A1 in ALG mode:
³„ì*~rQ2™K~a1`
resulting in:
The keystrokes corresponding to RPN mode are:
„ì~r`2Q*~a1K
After storing the variable A2 and pressing the key, the screen will show the
variables as follows:
In ALG mode, the following keystrokes will show a number of operations
with the algebraics contained in variables @@[email protected]@ and @@[email protected]@ (press J to
recover variable menu):
@@[email protected]@ + @@[email protected]@ `
@@[email protected]@ - @@[email protected]@ `
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@@[email protected]@ *@@[email protected]@ `
‚¹@@[email protected]@
1:31 PM
@@[email protected]@ / @@[email protected]@ `
„¸@@[email protected]@
The same results are obtained in RPN mode if using the following
keystrokes:
@@[email protected]@ @@[email protected]@ +µ
@@[email protected]@ @@[email protected]@ *µ
@@[email protected]@ ‚ ¹µ
@@[email protected]@ @@[email protected]@ -µ
@@[email protected]@ @@[email protected]@ /µ
@@[email protected]@ „ ¸µ
Functions in the ALG menu
The ALG (Algebraic) menu is available by using the keystroke sequence
‚× (associated with the 4 key). With system flag 117 set to
CHOOSE boxes, the ALG menu shows the following functions:
Rather than listing the description of each function in this manual, the user
is invited to look up the description using the calculator’s help facility:
[email protected])[email protected]`. 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.
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To complete the operation press @@[email protected]@. 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:
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:
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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:
³„¸+~x+~y`
At this point, select function TEXPAND from menu ALG (or directly from
the catalog ‚N), to complete the operation.
Operations with transcendental functions
The calculator offers a number of functions that can be used to replace
expressions containing logarithmic and exponential functions („Ð),
as well as trigonometric functions (‚Ñ).
Expansion and factoring using log-exp functions
The „Ð produces the following menu:
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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 ‚Ñ, shows the following
functions:
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 calculator’s help facility ([email protected]). The user is
invited to explore this facility to find information on the commands in the
TRIG menu.
Page 5-6
Functions in the ARITHMETIC menu
The ARITHMETIC menu is triggered through the keystroke combination
„Þ (associated with the 1 key). With system flag 117 set to
CHOOSE boxes, „Þ 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 sub-menus of functions that apply to specific mathematical
objects. When system flag 117 is set to SOFT menus, the ARITHMETIC
menu („Þ) produces:
Following, we present the help facility entries for functions FACTORS
and SIMP2 in the ARITHMETIC menu([email protected]):
FACTORS:
SIMP2:
The functions associated with the ARITHMETIC submenus: INTEGER,
POLYNOMIAL, MODULO, and PERMUTATION, are presented in detail in
Chapter 5 in the calculator’s user’s guide. The following sections show
some applications to polynomials and fractions.
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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 („Þ, POLYNOMIAL, HORNER) produces the
Horner division, or synthetic division, of a polynomial P(X) by the factor (Xa), 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+5 2
11}
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
i.e.,
X6 -1
=
-5 15624}
(X5-5*X4+25X3-125X2+625X-3125)(X+5)+15624.
The variable VX
Most polynomial examples above were written using variable X. This is
because a variable called VX exists in the calculator’s {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 calculator’s user’s 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.
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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.].
The QUOT and REMAINDER functions
The functions QUOT and REMAINDER provide, respectively, the quotient
Q(X) and the remainder R(X), resulting from dividing two polynomials,
P1(X) and P2(X). In other words, they provide the values of Q(X) and R(X)
from P1(X)/P2(X) = Q(X) + R(X)/P2(X). For example,
QUOT(‘X^3-2*X+2’, ‘X-1’) = ‘X^2+X-1’
REMAINDER(‘X^3-2*X+2’, ‘X-1’) = 1.
Thus, we can write: (X3-2X+2)/(X-1) = X2+X-1 + 1/(X-1).
NOTE: you could get the latter result by using PARTFRAC:
PARTFRAC(‘(X^3-2*X+2)/(X-1)’) = ‘X^2+X-1 + 1/(X-1)’.
The PEVAL function
The function PEVAL (Polynomial EVALuation) can be used to evaluate a
polynomial
p(x) = an⋅xn+an-1⋅x n-1+ …+ a2⋅x2+a1⋅x+ a0,
given an array of coefficients [an, an-1, … a2, a1, a0] and a value of x0.
The result is the evaluation p(x0). Function PEVAL is not available in the
ARITHMETIC menu, instead use the CALC/DERIV&INTEG Menu. Example:
PEVAL([1,5,6,1],5) = 281.
Additional applications of polynomial functions are presented in Chapter 5
in the calculator’s user’s guide.
Fractions
Fractions can be expanded and factored by using functions EXPAND and
FACTOR, from the ALG menu (‚×). For example:
EXPAND(‘(1+X)^3/((X-1)*(X+3))’)=‘(X^3+3*X^2+3*X+1)/(X^2+2*X-3)’
EXPAND(‘(X^2)*(X+Y)/(2*X-X^2)^2)’)=‘(X+Y)/(X^2-4*X+4)’
FACTOR(‘(3*X^3-2*X^2)/(X^2-5*X+6)’)=‘X^2*(3*X-2)/((X-2)*(X-3))’
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FACTOR(‘(X^3-9*X)/(X^2-5*X+6)’ )=‘X*(X+3)/(X-2)’
The SIMP2 function
Function SIMP2, in the ARITHMETIC menu, takes as arguments two
numbers or polynomials, representing the numerator and denominator of a
rational fraction, and returns the simplified numerator and denominator.
For example:
SIMP2(‘X^3-1’,’X^2-4*X+3’) = {‘X^2+X+1’,‘X-3’}
The PROPFRAC function
The function PROPFRAC converts a rational fraction into a “proper”
fraction, i.e., an integer part added to a fractional part, if such
decomposition is possible. For example:
PROPFRAC(‘5/4’) = ‘1+1/4’
PROPFRAC(‘(x^2+1)/x^2’) = ‘1+1/x^2’
The PARTFRAC function
The function PARTFRAC decomposes a rational fraction into the partial
fractions that produce the original fraction. For example:
PARTFRAC(‘(2*X^6-14*X^5+29*X^4-37*X^3+41*X^2-16*X+5)/(X^57*X^4+11*X^3-7*X^2+10*X)’) =
‘2*X+(1/2/(X-2)+5/(X-5)+1/2/X+X/(X^2+1))’
The FCOEF function
The function FCOEF, available through the ARITHMETIC/POLYNOMIAL
menu, is used to obtain a rational fraction, given the roots and poles of the
fraction.
NOTE: If a rational fraction is given as F(X) = N(X)/D(X), the roots of
the fraction result from solving the equation N(X) = 0, while the poles
result from solving the equation D(X) = 0.
The input for the function is a vector listing the roots followed by their
multiplicity (i.e., how many times a given root is repeated), and the poles
followed by their multiplicity represented as a negative number. For
example, if we want to create a fraction having roots 2 with multiplicity 1,
0 with multiplicity 3, and -5 with multiplicity 2, and poles 1 with multiplicity
2 and –3 with multiplicity 5, use:
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FCOEF([2,1,0,3,–5,2,1,–2,–3,–5])=‘(X--5)^2*X^3*(X-2)/(X-+3)^5*(X-1)^2’
If you press µ„î` (or, simply µ, in RPN mode) you will get:
‘(X^6+8*X^5+5*X^4-50*X^3)/(X^7+13*X^6+61*X^5+105*X^445*X^3-297*X62-81*X+243)’
The FROOTS function
The function FROOTS, in the ARITHMETIC/POLYNOMIAL menu, obtains
the roots and poles of a fraction. As an example, applying function
FROOTS to the result produced above, will result in: [1 –2. –3 –5. 0 3. 2
1. –5 2.]. The result shows poles followed by their multiplicity as a
negative number, and roots followed by their multiplicity as a positive
number. In this case, the poles are (1, -3) with multiplicities (2,5)
respectively, and the roots are (0, 2, -5) with multiplicities (3, 1, 2),
respectively.
Another example is: FROOTS(‘(X^2-5*X+6)/(X^5-X^2)’) = [0 –2. 1 –1. 3
1. 2 1.], i.e., poles = 0 (2), 1(1), and roots = 3(1), 2(1). If you have had
Complex mode selected, then the results would be:
[0 –2. 1 –1. – ((1+i*√3)/2) –1. – ((1–i*√3)/2) –1. 3 1. 2 1.].
Step-by-step operations with polynomials and
fractions
By setting the CAS modes to Step/step the calculator will show
simplifications of fractions or operations with polynomials in a step-by-step
fashion. This is very useful to see the steps of a synthetic division. The
example of dividing
X 3 − 5X 2 + 3X − 2
X −2
is shown in detail in Appendix C of the calculator’s user’s guide. The
following example shows a lengthier synthetic division (DIV2 is available in
the ARITH/POLYNOMIAL menu):
X 9 −1
X 2 −1
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Reference
Additional information, definitions, and examples of algebraic and
arithmetic operations are presented in Chapter 5 of the calculator’s user’s
guide.
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Chapter 6
Solution to equations
Associated with the 7 key there are two menus of equation-solving
functions, the Symbolic SOLVer („Î), and the NUMerical SoLVer
(‚Ï). Following, we present some of the functions contained in
these menus.
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 „Î. With
system flag 117 set to CHOOSE boxes, the following menu lists will be
available:
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.
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:
Using the RPN mode, the solution is accomplished by entering the
equation in the stack, followed by the variable, before entering function
ISOL. Right before the execution of ISOL, the RPN stack should look as in
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the figure to the left. After applying ISOL, the result is shown in the figure
to the right:
The first argument in ISOL can be an expression, as shown above, or an
equation. For example, in ALG mode, try:
NOTE: To type the equal sign (=) in an equation, use ‚Å
(associated with the \ key).
The same problem can be solved in RPN mode as illustrated below (figures
show the RPN stack before and after the application of function ISOL):
Function SOLVE
Function SOLVE has the same syntax as function ISOL, except that SOLVE
can also be used to solve a set of polynomial equations. The help-facility
entry for function SOLVE, with the solution to equation X^4 – 1 = 3 , is
shown next:
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The following examples show the use of function SOLVE in ALG and RPN
modes (Use Complex mode in the CAS):
The screen shot shown above displays two solutions. In the first one, β4 -5β
=125, SOLVE produces no solutions { }. In the second one, β4 - 5β = 6,
SOLVE produces four solutions, shown in the last output line. The very last
solution is not visible because the result occupies more characters than the
width of the calculator’s screen. However, you can still see all the solutions
by using the down arrow key (˜), which triggers the line editor (this
operation can be used to access any output line that is wider than the
calculator’s screen):
The corresponding RPN screens for these two examples, before and after
the application of function SOLVE, are shown next:
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Function SOLVEVX
The function SOLVEVX solves an equation for the default CAS variable
contained in the reserved variable name VX. By default, this variable is set
to ‘X’. Examples, using the ALG mode with VX = ‘X’, are shown below:
In the first case SOLVEVX could not find a solution. In the second case,
SOLVEVX found a single solution, X = 2.
The following screens show the RPN stack for solving the two examples
shown above (before and after application of SOLVEVX):
Function ZEROS
The function ZEROS finds the solutions of a polynomial equation, without
showing their multiplicity. The function requires having as input the
expression for the equation and the name of the variable to solve for.
Examples in ALG mode are shown next:
To use function ZEROS in RPN mode, enter first the polynomial expression,
then the variable to solve for, and then function ZEROS. The following
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screen shots show the RPN stack before and after the application of
ZEROS to the two examples above (Use Complex mode in the CAS):
The Symbolic Solver functions presented above produce solutions to
rational equations (mainly, polynomial equations). If the equation to be
solved for has all numerical coefficients, a numerical solution is possible
through the use of the Numerical Solver features of the calculator.
Numerical solver menu
The calculator provides a very powerful environment for the solution of
single algebraic or transcendental equations. To access this environment
we start the numerical solver (NUM.SLV) by using ‚Ï. This
produces a drop-down menu that includes the following options:
Following, we present applications of items 3. Solve poly.., 5. Solve
finance, and 1. Solve equation.., in that order. Appendix 1-A, in the
calculator’s user’s guide, contains instructions on how to use input forms
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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 calculator’s 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:
‚Ϙ˜@@[email protected]@
Select Solve poly…
„Ô3‚í2‚í0
Enter vector of coefficients
‚í1\‚í[email protected]@[email protected]@ @[email protected] 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):
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:
‚Ϙ˜@@[email protected]@
Select Solve poly…
˜„Ô1‚í5
Enter vector of roots
‚í2\‚í[email protected]@[email protected]@ @[email protected] 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:
‚Ϙ˜
Select Solve poly…
„Ô1‚í5
Enter vector of coefficients
‚í2\‚í[email protected]@[email protected]@—@[email protected] Generate symbolic expression
Return to stack.
`
The expression thus generated is shown in the stack as: 'X^3+5*X^2+2*X+4'
To generate the algebraic expression using the roots, try the following
example. Assume that the polynomial roots are [1, 3, -2, 1]. Use the
following keystrokes:
‚Ϙ˜@@[email protected]@
Select Solve poly…
˜„Ô1‚í3
Enter vector of roots
‚í2\‚í[email protected]@[email protected]@˜@[email protected] symbolic
expression
Return to stack.
`
The expression thus generated is shown in the stack as:
'(X-1)*(X-3)*(X+2)*(X-1)'.
To expand the products, you can use the EXPAND command.
The resulting expression is: 'X^4+-3*X^3+ -3*X^2+11*X-6'.
Financial calculations
The calculations in item 5. Solve finance.. in the Numerical Solver
(NUM.SLV) are used for calculations of time value of money of interest in
the discipline of engineering economics and other financial applications.
This application can also be started by using the keystroke combination
„Ò (associated with the 9 key). Detailed explanations of these
types of calculations are presented in Chapter 6 of the calculator’s user’s
guide.
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Solving equations with one unknown through
NUM.SLV
The calculator's NUM.SLV menu provides item 1. Solve equation.. solve
different types of equations in a single variable, including non-linear
algebraic and transcendental equations. For example, let's solve the
equation: ex-sin(πx/3) = 0.
Simply enter the expression as an algebraic object and store it into
variable EQ. The required keystrokes in ALG mode are the following:
³„¸~„x™-S„ì
*~„x/3™‚Å0™
K~e~q`
Function STEQ
Function STEQ will store its argument into variable EQ, e.g., in ALG mode:
In RPN mode, enter the equation between apostrophes and activate
command STEQ. Thus, function STEQ can be used as a shortcut to store
an expression into variable EQ.
Press J to see the newly created EQ variable:
Then, enter the SOLVE environment and select Solve equation…, by using:
‚Ï@@[email protected]@. The corresponding screen will be shown as:
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The equation we stored in variable EQ is already loaded in the Eq field in
the SOLVE EQUATION input form. Also, a field labeled x is provided. To
solve the equation all you need to do is highlight the field in front of X: by
using ˜, and press @[email protected] The solution shown is X: 4.5006E-2:
This, however, is not the only possible solution for this equation. To obtain
a negative solution, for example, enter a negative number in the X: field
before solving the equation. Try 3\ @@@[email protected]@ ˜ @[email protected] The solution is
now X: -3.045.
Solution to simultaneous equations with MSLV
Function MSLV is available in the ‚Ï menu. The help-facility entry for
function MSLV is shown next:
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.
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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.
Reference
Additional information on solving single and multiple equations is provided
in Chapters 6 and 7 of the calculator’s user’s guide.
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Chapter 7
Operations with lists
Lists are a type of calculator’s object that can be useful for data processing.
This chapter presents examples of operations with lists. To get started with
the examples in this Chapter, we use the Approximate mode (See Chapter
1).
Creating and storing lists
To create a list in ALG mode, first enter the braces key „ä , then type
or enter the elements of the list, separating them with commas (‚í).
The following keystrokes will enter the list {1.,2.,3.,4.} and store it into
variable L1.
„ä1‚í2‚í3‚í4
™K~l1`
Entering the same list in RPN mode requires the following keystrokes:
„ä1#2#3#4`
³~l1`K
Operations with lists of numbers
To demonstrate operations with lists of numbers enter and store the
following lists in the corresponding variables.
L2 = {-3.,2.,1.,5.} L3 = {-6.,5.,3.,1.,0.,3.,-4.} L4 = {3.,-2.,1.,5.,3.,2.,1.}
Changing sign
The sign-change key (\), when applied to a list of numbers, will change
the sign of all elements in the list. For example:
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Addition, subtraction, multiplication, division
Multiplication and division of a list by a single number is distributed across
the list, for example:
Subtraction of a single number from a list will subtract the same number
from each element in the list, for example:
Addition of a single number to a list produces a list augmented by the
number, and not an addition of the single number to each element in the
list. For example:
Subtraction, multiplication, and division of lists of numbers of the same
length produce a list of the same length with term-by-term operations.
Examples:
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The division L4/L3 will produce an infinity entry because one of the
elements in L3 is zero, and an error message is returned.
NOTE: If we had entered the elements in lists L4 and L3 as integers,
the infinite symbol would be shown whenever a division by zero
occurs. To produce the following result you need to re-enter the lists as
integer (remove decimal points) using Exact mode:
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:
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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.,
ABS
INVERSE (1/x)
Lists of complex numbers
You can create a complex number list, say L1 ADD i*L2. In RPN mode, you
could enter this as L1 i L2 ADD *. The result is:
Functions such as LN, EXP, SQ, etc., can also be applied to a list of
complex numbers, e.g.,
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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:
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
ΣLIST:
Calculate summation of elements in the list
ΠLIST:
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)
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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:
If you are working in RPN mode, enter the list onto the stack and then
select the operation you want. For example, to calculate the increment
between consecutive elements in list L3, press:
l3`!´˜˜#OK# #OK#
This places L3 onto the stack and then selects the ∆LIST operation from the
MTH menu.
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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 that function to the list of numbers. For
example, the following call to function MAP applies the function SIN(X) to
the list {1,2,3}:
In ALG mode, the syntax is:
~~map~!Ü!ä[email protected]í[email protected]í3™
@íS~X`
In RPN mode, the syntax is:
!ä[email protected]í[email protected]í3`³S~X`~
~map`
In both cases, you can either type out the MAP command (as in the
examples above) or select the command from the CAT menu.
Reference
For additional references, examples, and applications of lists see Chapter
8 in the calculator’s user’s 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
„Ô, 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 („Ô) and typing the components or
elements of the vector separated by commas (‚í). 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 calculator’s 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 (‚í) or spaces (#). Notice that after pressing `, in
either mode, the calculator shows the vector elements separated by
spaces.
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Storing vectors into variables in the stack
Vectors can be stored into variables. The screen shots below show the
vectors
u2 = [1, 2], u3 = [-3, 2, -2], v2 = [3,-1], v3 = [1, -5, 2]
Stored into variables @@@[email protected]@, @@@[email protected]@, @@@[email protected]@, and @@@[email protected]@, respectively. First, in
ALG mode:
Then, in RPN mode (before pressing K, repeatedly):
NOTE: The apostrophes (‘) are not needed ordinarily in entering the
names u2, v2, etc. in RPN mode. In this case, they are used to
overwrite the existing variables created earlier in ALG mode. Thus, the
apostrophes must be used if the existing variables are not purged
previously.
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Using the Matrix Writer (MTRW) to enter vectors
Vectors can also be entered by using the Matrix Writer „² (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 Chapter 9). 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!
The @EDIT key is used to edit the contents of a selected cell in the Matrix
Writer.
The
key, when selected, will produce a vector, as opposed to a
matrix of one row and many columns.
The
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
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
key, when selected, automatically selects the next cell to the
right of the current cell when you press `. This option is selected by
default. This option, if desired, needs to be selected before entering
elements.
The
key, when selected, automatically selects the next cell below
the current cell when you press `. This option, if desired, needs to
be selected before entering elements.
Moving to the right vs. moving down in the Matrix Writer
Activate the Matrix Writer and enter 3`5`2``
with the
key selected (default). Next, enter the same sequence
of numbers with the
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.
Activate the Matrix Writer again by using „², and press L to
check out the second soft key menu at the bottom of the display. It will
show the keys:
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@[email protected] @-ROW @[email protected] @[email protected]
@[email protected]
The @[email protected] 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 @[email protected] key will add a column full of zeros at the location of the
selected cell of the spreadsheet.
The @[email protected] key will delete the column corresponding to the selected cell
of the spreadsheet.
The
key will place the contents of the selected cell on the stack.
The @[email protected] 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 @@[email protected] (delete).
The function @@[email protected] 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 „². Make sure the
and
keys are selected.
(2) Enter the following:
1`2`3`
L @[email protected] 2 @@[email protected]@ 1 @@[email protected]@ @@[email protected]@
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 @[email protected] A row of three zeroes appears in the second row.
(5) Press @[email protected] The first column will disappear.
(6) Press @[email protected] A column of two zeroes appears in the first column.
(7) Press @[email protected] 3 @@[email protected]@ 3 @@[email protected]@ @@[email protected]@ to move to position (3,3).
(8) Press
. 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.
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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.
Changing sign
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:
Attempting to add or subtract vectors of different length produces an error
message:
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Multiplication by a scalar, and division by a scalar
Multiplication by a scalar or division by a scalar is straightforward:
Absolute value function
The absolute value function (ABS), when applied to a vector, produces the
magnitude of the vector. For example: ABS([1,-2,6]), ABS(A),
ABS(u3), will show in the screen as follows:
The MTH/VECTOR menu
The MTH menu („´) contains a menu of functions that specifically to
vector objects:
The VECTOR menu contains the following functions (system flag 117 set to
CHOOSE boxes):
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Magnitude
The magnitude of a vector, as discussed earlier, can be found with function
ABS. This function is also available from the keyboard („Ê).
Examples of application of function ABS were shown above.
Dot product
Function DOT (option 2 in CHOOSE box above) is used to calculate the
dot product of two vectors of the same length. Some examples of
application of function DOT, using the vectors A, u2, u3, v2, and v3,
stored earlier, are shown next in ALG mode. Attempts to calculate the dot
product of two vectors of different length produce an error message:
Cross product
Function CROSS (option 3 in the MTH/VECTOR menu) is used to calculate
the cross product of two 2-D vectors, of two 3-D vectors, or of one 2-D and
one 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:
Reference
Additional information on operations with vectors, including applications
in the physical sciences, is presented in Chapter 9 of the calculator’s user’s
guide.
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Chapter 9
Matrices and linear algebra
This chapter shows examples of creating matrices and operations with
matrices, including linear algebra applications.
Entering matrices in the stack
In this section we present two different methods to enter matrices in the
calculator stack: (1) using the Matrix Writer, and (2) typing the matrix
directly into the stack.
Using the Matrix Writer
As with the case of vectors, discussed in Chapter 8, matrices can be
entered into the stack by using the Matrix Writer. For example, to enter the
matrix:
⎡− 2.5 4.2 2.0⎤
1.9 2.8⎥ ,
⎢ 0.3
− 0.1 0.5⎦⎥
⎣⎢ 2
first, start the Matrix Writer by using „². Make sure that the option
is selected. Then use the following keystrokes:
2.5\`4.2`2`˜ššš
.3`1.9`2.8`
2`.1\`.5`
At this point, the Matrix Writer screen looks like this:
Press ` once more to place the matrix on the stack. The ALG mode
stack is shown next, before and after pressing `, once more:
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If you have selected the textbook display option (using [email protected])DISP! and
checking off Textbook), the matrix will look like the one shown above.
Otherwise, the display will show:
The display in RPN mode will look very similar to these.
Typing in the matrix directly into the stack
The same result as above can be achieved by entering the following
directly into the stack:
„Ô
„Ô2.5\‚í4.2‚í2™
‚í
„Ô.3‚í1.9‚í2.8™
‚í
„Ô2‚í.1\‚í.5`
Thus, to enter a matrix directly into the stack open a set of brackets
(„Ô) and enclose each row of the matrix with an additional set of
brackets („Ô).
Commas (‚í.) should separate the
elements of each row, as well as the brackets between rows.
For future exercises, let’s save this matrix under the name A. In ALG mode
use K~a. In RPN mode, use ³~aK.
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Operations with matrices
Matrices, like other mathematical objects, can be added and subtracted.
They can be multiplied by a scalar, or among themselves, and raised to a
real power. An important operation for linear algebra applications is the
inverse of a matrix. Details of these operations are presented next.
To illustrate the operations we will create a number of matrices that we will
store in the following variables. Here are the matrices A22, B22, A23,
B23, A33 and B33 (The random matrices in your calculator may be
different):
In RPN mode, the steps to follow are:
{2,2}` RANM 'A22'`K {2,2}` RANM 'B22'`K
{2,3}` RANM 'A23'`K {2,3}` RANM 'B23'`K
{3,2}` RANM 'A32'`K {3,2}` RANM 'B32'`K
{3,3}` RANM 'A33'`K {3,3}` RANM 'B33'`K
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Addition and subtraction
Four examples are shown below using the matrices stored above (ALG
mode).
In RPN mode, try the following eight examples:
A22 ` B22`+
A22 ` B22`-
A23 ` B23`+
A23 ` B23`-
A32 ` B32`+
A32 ` B32`-
A33 ` B33`+
A33 ` B33`-
Multiplication
There are a number of multiplication operations that involve matrices.
These are described next. The examples are shown in algebraic mode.
Multiplication by a scalar
Some examples of multiplication of a matrix by a scalar are shown below.
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Matrix-vector multiplication
Matrix-vector multiplication is possible only if the number of columns of the
matrix is equal to the length of the vector. A couple of examples of matrixvector multiplication follow:
Vector-matrix multiplication, on the other hand, is not defined. This
multiplication can be performed, however, as a special case of matrix
multiplication as defined next.
Matrix multiplication
Matrix multiplication is defined by Cm×n = Am×p⋅Bp×n. Notice that matrix
multiplication is only possible if the number of columns in the first operand
is equal to the number of rows of the second operand. The general term in
the product, cij, is defined as
p
cij = ∑ aik ⋅ bkj , for i = 1,2,K, m; j = 1,2,K, n.
k =1
Matrix multiplication is not commutative, i.e., in general, A⋅B ≠ B⋅A.
Furthermore, one of the multiplications may not even exist. The following
screen shots show the results of multiplications of the matrices that we
stored earlier:
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Term-by-term multiplication
Term-by-term multiplication of two matrices of the same dimensions is
possible through the use of function HADAMARD. The result is, of course,
another matrix of the same dimensions. This function is available through
Function catalog (‚N), or through the MATRICES/OPERATIONS submenu („Ø). Applications of function HADAMARD are presented
next:
Raising a matrix to a real power
You can raise a matrix to any power as long as the power is either an
integer or a real number with no fractional part. The following example
shows the result of raising matrix B22, created earlier, to the power of 5:
You can also raise a matrix to a power without first storing it as a variable:
In algebraic mode, the keystrokes are: [enter or select the matrix] Q
[enter the power] `.
In RPN mode, the keystrokes are: [enter or select the matrix] † [enter the
power] Q`.
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The identity matrix
The identity matrix has the property that A⋅I = I⋅A = A. To verify this
property we present the following examples using the matrices stored
earlier on. Use function IDN (find it in the MTH/MATRIX/MAKE menu) to
generate the identity matrix as shown here:
The inverse matrix
The inverse of a square matrix A is the matrix A-1 such that A⋅A-1 = A-1⋅A
= I, where I is the identity matrix of the same dimensions as A. The inverse
of a matrix is obtained in the calculator by using the inverse function, INV
(i.e., the Y key). Examples of the inverse of some of the matrices stored
earlier are presented next:
To verify the properties of the inverse matrix, we present the following
multiplications:
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Characterizing a matrix (The matrix NORM
menu)
The matrix NORM (NORMALIZE) menu is accessed through the keystroke
sequence „´. This menu is described in detail in Chapter 10 of the
calculator’s user’s guide. Some of these functions are described next.
Function DET
Function DET calculates the determinant of a square matrix. For example,
Function TRACE
Function TRACE calculates the trace of square matrix, defined as the sum
of the elements in its main diagonal, or
n
tr (A) = ∑ aii
i =1
.
Examples:
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Solution of linear systems
A system of n linear equations in m variables can be written as
a11⋅x1 + a12⋅x2 + a13⋅x3 + …+ a1,m-1⋅x
m-1
+ a1,m⋅x
m
= b 1,
a21⋅x1 + a22⋅x2 + a23⋅x3 + …+ a2,m-1⋅x
m-1
+ a2,m⋅x
m
= b2,
a31⋅x1 + a32⋅x2 + a33⋅x3 + …+ a3,m-1⋅x
.
.
.
…
m-1
.
+ a3,m⋅x
m
.
= b 3,
.
an-1,1⋅x1 + an-1,2⋅x2 + an-1,3⋅x3 + …+ an-1,m-1⋅x m-1 + an-1,m⋅x m = bn-1,
an1⋅x1 + an2⋅x2 + an3⋅x3 + …+ an,m-1⋅x
m-1
+ an,m⋅x
m
= bn.
This system of linear equations can be written as a matrix equation,
An×m⋅xm×1 = bn×1, if we define the following matrix and vectors:
⎡ a11
⎢a
A = ⎢ 21
⎢ M
⎢
⎣an1
a12 L a1m ⎤
⎡ x1 ⎤
⎡ b1 ⎤
⎥
⎥
⎢
⎢b ⎥
a22 L a2 m ⎥
x
x = ⎢ 2⎥
b = ⎢ 2⎥
⎢M⎥
⎢M⎥
M O M ⎥
⎥
⎢ ⎥
⎢ ⎥
an 2 L anm ⎦ n×m
⎣ xm ⎦ m×1
⎣bn ⎦ n×1
,
,
Using the numerical solver for linear systems
There are many ways to solve a system of linear equations with the
calculator. One possibility is through the numerical solver ‚Ï. From
the numerical solver screen, shown below (left), select the option 4. Solve
lin sys.., and press @@@[email protected]@@. The following input form will be provide (right):
To solve the linear system A⋅x = b, enter the matrix A, in the format
[[ a11, a12, … ], … [….]] in the A: field. Also, enter the vector b in the B:
field. When the X: field is highlighted, press @SOLVE. If a solution is
available, the solution vector x will be shown in the X: field. The solution is
also copied to stack level 1. Some examples follow.
The system of linear equations
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2x1 + 3x2 –5x3 = 13,
x1 – 3x2 + 8x3 = -13,
2x1 – 2x2 + 4x3 = -6,
can be written as the matrix equation A⋅x = b, if
⎡ x1 ⎤
⎡ 2 3 − 5⎤
⎥
⎢
A = ⎢1 − 3 8 ⎥, x = ⎢⎢ x 2 ⎥⎥, and
⎢⎣ x3 ⎥⎦
⎢⎣ 2 − 2 4 ⎥⎦
⎡ 13 ⎤
b = ⎢⎢− 13⎥⎥.
⎢⎣ − 6 ⎥⎦
This system has the same number of equations as of unknowns, and will be
referred to as a square system. In general, there should be a unique
solution to the system. The solution will be the point of intersection of the
three planes in the coordinate system (x1, x2, x3) represented by the three
equations.
To enter matrix A you can activate the Matrix Writer while the A: field is
selected. The following screen shows the Matrix Writer used for entering
matrix A, as well as the input form for the numerical solver after entering
matrix A (press ` in the Matrix Writer):
Press ˜ to select the B: field. The vector b can be entered as a row
vector with a single set of brackets, i.e., [13,-13,-6] @@@[email protected]@@ .
After entering matrix A and vector b, and with the X: field highlighted, we
can press @SOLVE! to attempt a solution to this system of equations:
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A solution was found as shown next.
Solution with the inverse matrix
The solution to the system A⋅x = b, where A is a square matrix is
x = A-1⋅ b. For the example used earlier, we can find the solution in the
calculator as follows (First enter matrix A and vector b once more):
Solution by “division” of matrices
While the operation of division is not defined for matrices, we can use the
calculator’s / key to “divide” vector b by matrix A to solve for x in the
matrix equation A⋅x = b. The procedure for the case of “dividing” b by A
is illustrated below for the example above.
The procedure is shown in the following screen shots (type in matrices A
and vector b once more):
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References
Additional information on creating matrices, matrix operations, and matrix
applications in linear algebra is presented in Chapters 10 and 11 of the
calculator’s user’s guide.
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Chapter 10
Graphics
In this chapter we introduce some of the graphics capabilities of the
calculator. We will present graphics of functions in Cartesian coordinates
and polar coordinates, parametric plots, graphics of conics, bar plots,
scatterplots, and fast 3D plots.
Graphs options in the calculator
To access the list of graphic formats available in the calculator, use the
keystroke sequence „ô(D) Please notice that if you are using the
RPN mode these two keys must be pressed simultaneously to activate any
of the graph functions. After activating the 2D/3D function, the calculator
will produce the PLOT SETUP window, which includes the TYPE field as
illustrated below.
Right in front of the TYPE field you will, most likely, see the option Function
highlighted. This is the default type of graph for the calculator. To see the
list of available graph types, press the soft menu key labeled @CHOOS. This
will produce a drop down menu with the following options (use the up- and
down-arrow keys to see all the options):
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Plotting an expression of the form y = f(x)
As an example, let's plot the function,
f ( x) =
1
2π
exp(−
x2
)
2
• First, enter the PLOT SETUP environment by pressing, „ô. Make
sure that the option Function is selected as the TYPE, and that ‘X’ is
selected as the independent variable (INDEP). Press [email protected]@@[email protected]@@ to
return to normal calculator display. The PLOT SET UP window should
look similar to this:
• Enter the PLOT environment by pressing „ñ(press them
simultaneously if in RPN mode). Press @ADD to get you into the equation
writer. You will be prompted to fill the right-hand side of an equation
Y1(x) = . Type the function to be plotted so that the Equation Writer
shows the following:
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• Press ` to return to the PLOT - FUNCTION window. The expression
‘Y1(X) = EXP(-X^2/2)/√(2*π)’ will be highlighted. Press [email protected]@@[email protected]@@ to
return to normal calculator display.
• Enter the PLOT WINDOW environment by entering „ò (press
them simultaneously if in RPN mode). Use a range of –4 to 4 for HVIEW, then press @AUTO to generate the V-VIEW automatically. The PLOT
WINDOW screen looks as follows:
• Plot the graph: @ERASE @DRAW (wait till the calculator finishes the graphs)
• To see labels: @EDIT L @LABEL @MENU
• To recover the first graphics menu: [email protected])PICT
• To trace the curve: @TRACE @@X,[email protected]@ . Then use the right- and left-arrow keys
(š™) to move about the curve. The coordinates of the points you
trace will be shown at the bottom of the screen. Check that for x =
1.05 , y = 0.0231. Also, check that for x = -1.48 , y = 0.134. Here is
picture of the graph in tracing mode:
• To recover the menu, and return to the PLOT WINDOW environment,
press [email protected] Press [email protected]@[email protected]@ to return to normal display.
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Generating a table of values for a function
The combinations „õ(E) and „ö(F), pressed
simultaneously if in RPN mode, let’s the user produce a table of values of
functions. For example, we will produce a table of the function Y(X) = X/
(X+10), in the range -5 < X < 5 following these instructions:
• We will generate values of the function f(x), defined above, for values
of x from –5 to 5, in increments of 0.5. First, we need to ensure that the
graph type is set to FUNCTION in the PLOT SETUP screen („ô,
press them simultaneously, if in RPN mode). The field in front of the
Type option will be highlighted. If this field is not already set to
FUNCTION, press the soft key @CHOOS and select the FUNCTION
option, then press @@@[email protected]@@.
• Next, press ˜ to highlight the field in front of the option EQ, and type
the function expression: ‘X/(X+10)’. Press `.
• To accept the changes made to the PLOT SETUP screen press [email protected]@@[email protected]@@.
You will be returned to normal calculator display.
• The next step is to access the Table Set-up screen by using the keystroke
combination „õ (i.e., soft key E) – simultaneously if in RPN
mode. This will produce a screen where you can select the starting
value (Start) and the increment (Step). Enter the following: 5\
@@@[email protected]@@ 0.5 @@@[email protected]@@0.5 @@@[email protected]@@ (i.e., Zoom factor = 0.5).
Toggle the
soft menu key until a check mark appears in front of
the option Small Font if you so desire. Then press @@@[email protected]@@. This will return
you to normal calculator display.
• To see the table, press „ö(i.e., soft menu key F) –
simultaneously if in RPN mode. This will produce a table of values of x
= -5, -4.5, …, and the corresponding values of f(x), listed as Y1 by
default. You can use the up and down arrow keys to move about in the
table. You will notice that we did not have to indicate an ending value
for the independent variable x. Thus, the table continues beyond the
maximum value for x suggested early, namely x = 5.
Some options available while the table is visible are @ZOOM, @@[email protected], and @DEFN:
• The @DEFN, when selected, shows the definition of the independent
variable.
• The @@[email protected] key simply changes the font in the table from small to big, and
vice versa. Try it.
• The @ZOOM key, when pressed, produces a menu with the options: In,
Out, Decimal, Integer, and Trig. Try the following exercises:
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• With the option In highlighted, press @@@[email protected]@@. The table is expanded
so that the x-increment is now 0.25 rather than 0.5. Simply, what
the calculator does is to multiply the original increment, 0.5, by the
zoom factor, 0.5, to produce the new increment of 0.25. Thus, the
zoom in option is useful when you want more resolution for the
values of x in your table.
• To increase the resolution by an additional factor of 0.5 press @ZOOM,
select In once more, and press @@@[email protected]@@. The x-increment is now
0.0125.
• To recover the previous x-increment, press @ZOOM —@@@[email protected]@@ to select
the option Un-zoom. The x-increment is increased to 0.25.
• To recover the original x-increment of 0.5 you can do an un-zoom
again, or use the option zoom out by pressing @ZOOM ˜@@@[email protected]@@.
• The option Decimal in @ZOOM produces x-increments of 0.10.
• The option Integer in @ZOOM produces x-increments of 1.
• The option Trig in produces increments related to fractions of π, thus
being useful when producing tables of trigonometric functions.
• To return to normal calculator display press `.
Fast 3D plots
Fast 3D plots are used to visualize three-dimensional surfaces represented
by equations of the form z = f(x,y). For example, if you want to visualize z
= f(x,y) = x2+y2, we can use the following:
• Press „ô, simultaneously if in RPN mode, to access to the PLOT
SETUP window.
• Change TYPE to Fast3D. ( @CHOOS!, find Fast3D, @@[email protected]@).
• Press ˜ and type ‘X^2+Y^2’ @@@[email protected]@@.
• Make sure that ‘X’ is selected as the Indep: and ‘Y’ as the Depnd:
variables.
• Press [email protected]@@[email protected]@@ to return to normal calculator display.
• Press „ò, simultaneously if in RPN mode, to access the PLOT
WINDOW screen.
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• Keep the default plot window ranges to read:
X-Left:-1
X-Right:1
Y-Near:-1
Y-Far: 1
Z-Low: -1
Z-High: 1
Step Indep: 10
Depnd: 8
NOTE: The Step Indep: and Depnd: values represent the number of
gridlines to be used in the plot. The larger these number, the slower it
is to produce the graph, although, the times utilized for graphic
generation are relatively fast. For the time being we’ll keep the default
values of 10 and 8 for the Step data.
• Press @ERASE @DRAW to draw the three-dimensional surface. The result is a
wireframe picture of the surface with the reference coordinate system
shown at the lower left corner of the screen. By using the arrow keys
(š™—˜) you can change the orientation of the surface. The
orientation of the reference coordinate system will change accordingly.
Try changing the surface orientation on your own. The following figures
show a couple of views of the graph:
• When done, press @EXIT.
• Press @CANCL to return to the PLOT WINDOW environment.
• Change the Step data to read: Step Indep: 20 Depnd: 16
• Press @ERASE @DRAW to see the surface plot. Sample views:
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• When done, press @EXIT.
• Press @CANCL to return to PLOT WINDOW.
• Press $, or [email protected]@@[email protected]@@, to return to normal calculator display.
Try also a Fast 3D plot for the surface z = f(x,y) = sin (x2+y2)
• Press „ô, simultaneously if in RPN mode, to access the PLOT
SETUP window.
• Press ˜ and type ‘SIN(X^2+Y^2)’ @@@[email protected]@@.
• Press @ERASE @DRAW to draw the slope field plot. Press @EXIT @EDIT L
@)LABEL @MENU to see the plot unencumbered by the menu and with
identifying labels.
• Press [email protected])PICT to leave the EDIT environment.
• Press @CANCL to return to the PLOT WINDOW environment. Then, press
$, or [email protected]@@[email protected]@@, to return to normal calculator display.
Reference
Additional information on graphics is available in Chapters 12 and 22 in
the calculator’s user’s guide.
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Chapter 11
Calculus Applications
In this Chapter we discuss applications of the calculator’s functions to
operations related to Calculus, e.g., limits, derivatives, integrals, power
series, etc.
The CALC (Calculus) menu
Many of the functions presented in this Chapter are contained in the
calculator’s CALC menu, available through the keystroke sequence
„Ö (associated with the 4 key):
The first four options in this menu are actually sub-menus that apply to (1)
derivatives and integrals, (2) limits and power series, (3) differential
equations, and (4) graphics. The functions in entries (1) and (2) will be
presented in this Chapter. Functions DERVX and INTVX are discussed in
page 11-3, respectively.
Limits and derivatives
Differential calculus deals with derivatives, or rates of change, of functions
and their applications in mathematical analysis. The derivative of a
function is defined as a limit of the difference of a function as the increment
in the independent variable tends to zero. Limits are used also to check the
continuity of functions.
Function lim
The calculator provides function lim to calculate limits of functions. This
function uses as input an expression representing a function and the value
where the limit is to be calculated. Function lim is available through the
command catalog (‚N~„l) or through option 2. LIMITS &
SERIES… of the CALC menu (see above).
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Function lim is entered in ALG mode as lim(f(x),x=a) to calculate
f ( x) .
the limit lim
x→ a
In RPN mode, enter the function first, then the
expression ‘x=a’, and finally function lim. Examples in ALG mode are
shown next, including some limits to infinity, and one-sided limits. The
infinity symbol is associated with the 0 key, i.e.., „è.
To calculate one-sided limits, add +0 or -0 to the value to the variable. A
“+0” means limit from the right, while a “–0” means limit from the left. For
example, the limit of
x − 1 as x approaches 1 from the left can be
determined with the following keystrokes (ALG mode):
‚N~„l˜$OK$ R!ÜX1™@í[email protected]Å1+0`
The result is as follows:
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Functions DERIV and DERVX
The function DERIV is used to take derivatives in terms of any independent
variable, while the function DERVX takes derivatives with respect to the
CAS default variable VX (typically ‘X’). While function DERVX is available
directly in the CALC menu, both functions are available in the
DERIV.&INTEG sub-menu within the CALCL menu ( „Ö).
Function DERIV requires a function, say f(t), and an independent variable,
say, t, while function DERVX requires only a function of VX. Examples are
shown next in ALG mode. Recall that in RPN mode the arguments must be
entered before the function is applied.
Anti-derivatives and integrals
An anti-derivative of a function f(x) is a function F(x) such that f(x) = dF/dx.
One way to represent an anti-derivative is as a indefinite integral, i.e.,
∫ f ( x)dx = F ( x) + C
if and only if, f(x) = dF/dx, and C = constant.
Functions INT, INTVX, RISCH, SIGMA and SIGMAVX
The calculator provides functions INT, INTVX, RISCH, SIGMA and
SIGMAVX to calculate anti-derivatives of functions. Functions INT, RISCH,
and SIGMA work with functions of any variable, while functions INTVX,
and SIGMAVX utilize functions of the CAS variable VX (typically, ‘x’).
Functions INT and RISCH require, therefore, not only the expression for the
function being integrated, but also the independent variable name.
Function INT, requires also a value of x where the anti-derivative will be
evaluated. Functions INTVX and SIGMAVX require only the expression of
the function to integrate in terms of VX. Functions INTVX, RISCH, SIGMA
and SIGMAVX are available in the CALC/DERIV&INTEG menu, while INT
is available in the command catalog. Some examples are shown next in
ALG mode (type the function names to activate them):
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Please notice that functions SIGMAVX and SIGMA are designed for
integrands that involve some sort of integer function like the factorial (!)
function shown above. Their result is the so-called discrete derivative, i.e.,
one defined for integer numbers only.
Definite integrals
In a definite integral of a function, the resulting anti-derivative is evaluated
at the upper and lower limit of an interval (a,b) and the evaluated values
subtracted. Symbolically,
∫
b
a
f ( x)dx = F (b) − F (a ), where f(x) = dF/dx.
The PREVAL(f(x),a,b) function of the CAS can simplify such calculation by
returning f(b)-f(a) with x being the CAS variable VX.
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Infinite series
A function f(x) can be expanded into an infinite series around a point x=x0
by using a Taylor’s series, namely,
∞
f ( x) = ∑
n =0
f ( n) ( xo )
⋅ ( x − xo ) n
n!
,
where f(n)(x) represents the n-th derivative of f(x) with respect to x, f(0)(x) =
f(x).
If the value x0 = 0, the series is referred to as a Maclaurin’s series.
Functions TAYLR, TAYLR0, and SERIES
Functions TAYLR, TAYLR0, and SERIES are used to generate Taylor
polynomials, as well as Taylor series with residuals. These functions are
available in the CALC/LIMITS&SERIES menu described earlier in this
Chapter.
Function TAYLOR0 performs a Maclaurin series expansion, i.e., about X =
0, of an expression in the default independent variable, VX (typically ‘X’).
The expansion uses a 4-th order relative power, i.e., the difference
between the highest and lowest power in the expansion is 4. For example,
Function TAYLR produces a Taylor series expansion of a function of any
variable x about a point x = a for the order k specified by the user. Thus,
the function has the format TAYLR(f(x-a),x,k). For example,
Function SERIES produces a Taylor polynomial using as arguments the
function f(x) to be expanded, a variable name alone (for Maclaurin’s
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series) or an expression of the form ‘variable = value’ indicating the point
of expansion of a Taylor series, and the order of the series to be produced.
Function SERIES returns two output items: a list with four items, and an
expression for h = x - a, if the second argument in the function call is ‘x=a’,
i.e., an expression for the increment h. The list returned as the first output
object includes the following items:
1. Bi-directional limit of the function at point of expansion, i.e.,
lim f ( x)
x→ a
2. An equivalent value of the function near x = a
3. Expression for the Taylor polynomial
4. Order of the residual or remainder
Because of the relatively large amount of output, this function is easier to
handle in RPN mode. For example, the following screen shots show the
RPN stack before and after using the TAYLR function, as illustrated above:
The keystrokes that generate this particular example are:
~!s`!ì2/S~!s`6!Ö˜$OK$ ˜˜˜˜$OK$
Reference
Additional definitions and applications of calculus operations are
presented in Chapter 13 in the calculator’s user’s guide.
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Chapter 12
Multi-variate Calculus Applications
Multi-variate calculus refers to functions of two or more variables. In this
Chapter we discuss basic concepts of multi-variate calculus: partial
derivatives and multiple integrals.
Partial derivatives
To quickly calculate partial derivatives of multi-variate functions, use the
rules of ordinary derivatives with respect to the variable of interest, while
considering all other variables as constant. For example,
∂
(x cos( y ) ) = cos( y ), ∂ (x cos( y ) ) = − x sin( y )
∂x
∂y
,
You can use the derivative functions in the calculator: DERVX, DERIV, ∂,
described in detail in Chapter 11 of this manual, to calculate partial
derivatives (DERVX uses the CAS default variable VX, typically, ‘X’). Some
examples of first-order partial derivatives are shown next. The functions
used in the first two examples are f(x,y) = x cos(y), and g(x,y,z) =
(x2+y2)1/2sin(z).
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To define the functions f(x,y) and g(x,y,z), in ALG mode, use:
DEF(f(x,y)=x*COS(y)) `
DEF(g(x,y,z)=√(x^2+y^2)*SIN(z) `
To type the derivative symbol use ‚¿. The derivative
∂
( f ( x, y )) ,
∂x
for example, will be entered as ∂x(f(x,y)) ` in ALG mode in the screen.
Multiple integrals
A physical interpretation of the double integral of a function f(x,y) over a
region R on the x-y plane is the volume of the solid body contained under
the surface f(x,y) above the region R. The region R can be described as R
= {a<x<b, f(x)<y<g(x)} or as R = {c<y<d, r(y)<x<s(y)}. Thus, the double
integral can be written as
b g( x )
d
s( y )
∫∫ φ( x , y )dA = ∫ a ∫ f ( x ) φ( x , y )dydx = ∫ c ∫ r ( y ) φ( x , y )dydx
R
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 user’s 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.
Reference
For additional details of multi-variate calculus operations and their
applications see Chapter 14 in the calculator’s user’s 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
vector-based 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 φ(x,y,z) is a vector function defined by
gradφ = ∇φ . Function HESS can be used to obtain the gradient of a
function.. The function takes as input a function of n independent variables
φ(x1, x2, …,xn), and a vector of the functions [‘x1’ ‘x2’…’xn’]. The function
returns the Hessian matrix of the function, H = [hij] = [∂φ/∂xi∂xj], the
gradient of the function with respect to the n-variables, grad f = [ ∂φ/∂x1
∂φ/∂x2 … ∂φ/∂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 φ(X,Y,Z) = X2 + XY + XZ, we’ll apply function HESS to this scalar
field in the following example:
Thus, the gradient is [2X+Y+Z, X, X].
Alternatively, use function DERIV as follows:
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Divergence
The divergence of a vector function, F(x,y,z) = f(x,y,z)i + g(x,y,z)j
+h(x,y,z)k, is defined by taking a “dot-product” of the del operator with
the function, i.e., divF = ∇ • F . Function DIV can be used to calculate
the divergence of a vector field.
For example, for F(X,Y,Z) =
2
2
2
[XY,X +Y +Z ,YZ], the divergence is calculated, in ALG mode, as follows:
DIV([X*Y,X^2+Y^2+Z^2,Y*Z],[X,Y,Z])
Curl
The curl of a vector field F(x,y,z) = f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,is defined
by a “cross-product” of the del operator with the vector field, i.e.,
curlF = ∇ × F . The curl of vector field can be calculated with function
CURL. For example, for the function F(X,Y,Z) = [XY,X2+Y2+Z2,YZ], the curl
is calculated as follows: CURL([X*Y,X^2+Y^2+Z^2,Y*Z],[X,Y,Z])
Reference
For additional information on vector analysis applications see Chapter 15
in the calculator’s user’s guide.
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Chapter 14
Differential Equations
In this Chapter we present examples of solving ordinary differential
equations (ODE) using calculator functions. A differential equation is an
equation involving derivatives of the independent variable. In most cases,
we seek the dependent function that satisfies the differential equation.
The CALC/DIFF menu
The DIFFERENTIAL EQNS.. sub-menu within the CALC („Ö) menu
provides functions for the solution of differential equations. The menu is
listed below with system flag 117 set to CHOOSE boxes:
These functions are briefly described next. They will be described in more
detail in later parts of this Chapter.
DESOLVE:
Differential Equation SOLVEr, solves differential equations,
when possible
ILAP:
Inverse LAPlace transform, L-1[F(s)] = f(t)
LAP:
LAPlace transform, L[f(t)]=F(s)
LDEC:
Linear Differential Equation Command
Solution to linear and non-linear equations
An equation in which the dependent variable and all its pertinent
derivatives are of the first degree is referred to as a linear differential
equation. Otherwise, the equation is said to be non-linear.
Function LDEC
The calculator provides function LDEC (Linear Differential Equation
Command) to find the general solution to a linear ODE of any order with
constant coefficients, whether it is homogeneous or not. This function
requires you to provide two pieces of input:
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• the right-hand side of the ODE
• the characteristic equation of the ODE
Both of these inputs must be given in terms of the default independent
variable for the calculator’s CAS (typically X). The output from the function
is the general solution of the ODE. The examples below are shown in the
RPN mode:
Example 1 – To solve the homogeneous ODE
d3y/dx3-4⋅(d2y/dx2)-11⋅(dy/dx)+30⋅y = 0.
Enter:
0 ` 'X^3-4*X^2-11*X+30'` LDEC µ
The solution is (figure put together from EQW screenshots):
where cC0, cC1, and cC2 are constants of integration.
equivalent to
This result is
y = K1⋅e–3x + K2⋅e5x + K3⋅e2x.
Example 2 – Using the function LDEC, solve the non-homogeneous ODE:
d3y/dx3-4⋅(d2y/dx2)-11⋅(dy/dx)+30⋅y = x2.
Enter:
'X^2' ` 'X^3-4*X^2-11*X+30'` LDEC µ
The solution is:
which is equivalent to
y = K1⋅e–3x + K2⋅e5x + K3⋅e2x + (450⋅x2+330⋅x+241)/13500.
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Function DESOLVE
The calculator provides function DESOLVE (Differential Equation SOLVEr) to
solve certain types of differential equations. The function requires as input
the differential equation and the unknown function, and returns the solution
to the equation if available. You can also provide a vector containing the
differential equation and the initial conditions, instead of only a differential
equation, as input to DESOLVE. The function DESOLVE is available in the
CALC/DIFF menu. Examples of DESOLVE applications are shown below
using RPN mode.
Example 1 – Solve the first-order ODE:
dy/dx + x2⋅y(x) = 5.
In the calculator use:
'd1y(x)+x^2*y(x)=5' ` 'y(x)' ` DESOLVE
The solution provided is
{‘y(x) = (5*INT(EXP(xt^3/3),xt,x)+cC0)*1/EXP(x^3/3)}’ }
, which simplifies to
(
)
y (x ) = 5 ⋅ exp( − x 3 / 3) ⋅ ∫ exp( x 3 / 3) ⋅ dx + C 0 .
The variable ODETYPE
You will notice in the soft-menu key labels a new variable called @ODETY
(ODETYPE). This variable is produced with the call to the DESOL function
and holds a string showing the type of ODE used as input for DESOLVE.
Press @ODETY to obtain the string “1st order linear”.
Example 2 – Solving an equation with initial conditions. Solve
d2y/dt2 + 5y = 2 cos(t/2),
with initial conditions
y(0) = 1.2, y’(0) = -0.5.
In the calculator, use:
[‘d1d1y(t)+5*y(t) = 2*COS(t/2)’ ‘y(0) = 6/5’ ‘d1y(0) = -1/2’] `
‘y(t)’ `
DESOLVE
Notice that the initial conditions were changed to their Exact expressions,
‘y(0) = 6/5’, rather than ‘y(0)=1.2’, and ‘d1y(0) = -1/2’, rather than,
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‘d1y(0) = -0.5’.
solution.
1:31 PM
Changing to these Exact expressions facilitates the
NOTE: To obtain fractional expressions for decimal values use function
Q (See Chapter 5).
Press µµ to simplify the result. Use ˜ @EDIT to see this result:
i.e.,
‘y(t) = -((19*√5*SIN(√5*t)-(148*COS(√5*t)+80*COS(t/2)))/190)’.
Press ``[email protected] to get the string “Linear w/ cst coeff” for
the ODE type in this case.
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. Use of the Laplace transform converts the linear ODE involving f(t) into
an algebraic equation.
2. The unknown F(s) is solved for in the image domain through algebraic
manipulation.
3. 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)’`LAP in RPN mode, or LAP(F(X))in ALG mode.
The calculator returns the result (RPN, left; ALG, right):
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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 ,
∞
0
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: ‘X⋅e-X’, meaning that L -1{1/(s+1)2} =
x⋅e-x.
Fourier series
A complex Fourier series is defined by the following expression
f (t ) =
+∞
∑c
n = −∞
n
⋅ exp(
2inπt
),
T
where
cn =
1
T
∫
T
0
f (t ) ⋅ exp(
2 ⋅ i ⋅ n ⋅π
⋅ 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 („Ö).
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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.
Using the calculator in ALG mode, first we define functions f(t) and g(t):
Next, we move to the CASDIR sub-directory under HOME to change the
value of variable PERIOD, e.g.,
„(hold) §`[email protected])CASDI`[email protected] `
Return to the sub-directory where you defined functions f and g, and
calculate the coefficients. Set CAS to Complex mode (see chapter 2)
before trying the exercises. Function COLLECT is available in the ALG
menu (‚×).
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Thus,
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c0 = 1/3, c1 = (π⋅i+2)/π2, c2 = (π⋅i+1)/(2π2).
The Fourier series with three elements will be written as
g(t) ≈ Re[(1/3) + (π⋅i+2)/π2⋅exp(i⋅π⋅t)+ (π⋅i+1)/(2π2)⋅exp(2⋅i⋅π⋅t)].
Reference
For additional definitions, applications, and exercises on solving
differential equations, using Laplace transform, and Fourier series and
transforms, as well as numerical and graphical methods, see Chapter 16
in the calculator’s user’s guide.
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Chapter 15
Probability Distributions
In this Chapter we provide examples of applications of the pre-defined
probability distributions in the calculator.
The MTH/PROBABILITY.. sub-menu - part 1
The MTH/PROBABILITY.. sub-menu is accessible through the keystroke
sequence „´. With system flag 117 set to CHOOSE boxes, the
following functions are available in the PROBABILITY.. menu:
In this section we discuss functions COMB, PERM, ! (factorial), and RAND.
Factorials, combinations, and permutations
The factorial of an integer n is defined as: n! = n⋅ (n-1) ⋅ (n-2)…3⋅2⋅1. 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
n
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!(n − r )!
⎝r⎠
We can calculate combinations, permutations, and factorials with functions
COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The
operation of those functions is described next:
• COMB(n,r): Calculates the number of combinations of n items taken r at
a time
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• PERM(n,r): Calculates the number of permutations of n items taken r at
a time
• n!: Factorial of a positive integer. For a non-integer, x! returns Γ(x+1),
where Γ(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. To generate
a random number, use function RAND from the MTH/PROBABILITY submenu. The following screen shows a number of random numbers
produced using RAND. (Note: The random numbers in your calculator will
differ from these).
Additional details on random numbers in the calculator are provided in
Chapter 17 of the user’s guide. Specifically, the use of function RDZ, to restart lists of random numbers is presented in detail in Chapter 17 of the
user’s guide.
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The MTH/PROB menu - part 2
In this section we discuss four continuous probability distributions that are
commonly used for problems related to statistical inference: the normal
distribution, the Student’s t distribution, the Chi-square (χ2) distribution, and
the F-distribution. The functions provided by the calculator to evaluate
probabilities for these distributions are NDIST, UTPN, UTPT, UTPC, and
UTPF. These functions are contained in the MTH/PROBABILITY menu
introduced earlier in this chapter. To see these functions activate the MTH
menu: „´ and select the PROBABILITY option:
The Normal distribution
Functions NDIST and UTPN relate to the Normal distribution with mean µ ,
and variance σ2.
To calculate the value of probability density function, or pdf, of the f(x) for
the normal distribution, use function NDIST(µ, σ2, x). For example, check
that for a normal distribution, NDIST(1.0, 0.5, 2.0) = 0.20755374. This
function is useful to plot the Normal distribution pdf.
The calculator also provides function UTPN that calculates the upper-tail
normal distribution, i.e., UTPN(µ, σ2, x) = P(X>x) = 1 - P(X<x), where P()
represents a probability. For example, check that for a normal distribution,
with µ = 1.0, σ2 = 0.5, UTPN(1.0, 0.5, 0.75) = 0.638163.
The Student-t distribution
The Student-t, or simply, the t-, distribution has one parameter ν, known as
the degrees of freedom of the distribution. The calculator provides for
values of the upper-tail (cumulative) distribution function for the tdistribution, function UTPT, given the parameter ν and the value of t, i.e.,
UTPT(ν,t) = P(T>t) = 1-P(T<t). For example, UTPT(5,2.5) = 2.7245…E-2.
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The Chi-square distribution
The Chi-square (χ2) distribution has one parameter ν, known as the
degrees of freedom. The calculator provides for values of the upper-tail
(cumulative) distribution function for the χ2-distribution using UTPC given
the value of x and the parameter ν. The definition of this function is,
therefore, UTPC(ν,x) = P(X>x) = 1 - P(X<x). For example, UTPC(5, 2.5) =
0.776495…
The F distribution
The F distribution has two parameters νN = numerator degrees of freedom,
and νD = 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 νN and νD, and the value
of F. The definition of this function is, therefore, UTPF(νN,νD,F) = P(ℑ>F) =
1 - P(ℑ<F). For example, to calculate UTPF(10,5, 2.5) = 0.1618347…
Reference
For additional probability distributions and probability applications, refer
to Chapter 17 in the calculator’s user’s guide.
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Chapter 16
Statistical Applications
The calculator provides the following pre-programmed statistical features
accessible through the keystroke combination ‚Ù (the 5 key):
Entering data
Applications numbered 1, 2, and 4 in the list above require that the data
be available as columns of the matrix ΣDAT. One way this can be
accomplished is by entering the data in columns using the Matrix Writer,
„², and then using function STOΣ to store the matrix into ΣDAT.
For example, enter the following data using the Matrix Writer (see
Chapters 8 or 9 in this guide), and store the data into ΣDAT:
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.
A simpler way to enter statistical data is to launch a statistics application
(such as Single-var, Frequencies or Summary stats, see
first screenshot above) and press #EDIT#. This launches the Matrix Writer.
Enter the data as before. In this case, when you exit the Matrix Writer, the
data you have entered is automatically saved in ΣDAT.
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Calculating single-variable statistics
After entering the column vector into ΣDAT, press ‚Ù @@@[email protected]@ to select
1. Single-var.. The following input form will be provided:
The form lists the data in ΣDAT, shows that column 1 is selected (there is
only one column in the current ΣDAT). Move about the form with the arrow
keys, and press the
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 @@@[email protected]@. 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 ΣDAT. To get started,
press ‚Ù˜@@@[email protected]@@. The resulting input form contains the following
fields:
ΣDAT:
the matrix containing the data of interest.
Col:
the column of ΣDAT that is under scrutiny.
X-Min:
the minimum class boundary to be used in the frequency
distribution (default = -6.5).
Bin Count:
the number of classes used in the frequency distribution
(default = 13).
Bin Width:
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
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ΣDAT, by using function STOΣ (see example above). Next, obtain singlevariable information using: ‚Ù @@@[email protected]@@. The results are:
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 ‚Ù˜ @@@[email protected]@@.
The data is already loaded in ΣDAT, and the option Col should hold the
value 1 since we have only one column in ΣDAT.
• Change X-Min to -8, Bin Count to 8, and Bin Width to 2, then press
@@@[email protected]@@.
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 ΣDAT vector, 14 values smaller than -8 and
8 larger than 8.
• Press ƒ 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.
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Fitting data to a function y = f(x)
The program 3. Fit data.., available as option number 3 in the STAT
menu, can be used to fit linear, logarithmic, exponential, and power
functions to data sets (x, y), stored in columns of the ΣDAT matrix. For this
application, you need to have at least two columns in your ΣDAT variable.
For example, to fit a linear relationship to the data shown in the table
below:
x
y
0
0.5
1
2.3
2
3.6
3
6.7
4
7.2
5
11
• First, enter the two columns of data into variable ΣDAT by using the
Matrix Writer, and function STOΣ.
• To access the program 3. Fit data.., use the following keystrokes:
‚Ù˜˜@@@[email protected]@@. The input form will show the current ΣDAT,
already loaded. If needed, change your set up screen to the following
parameters for a linear fitting:
• To obtain the data fitting press @@[email protected]@. The output from this program,
shown below for our particular data set, consists of the following three
lines in RPN mode:
3: '0.195238095238 + 2.00857242857*X'
2: Correlation: 0.983781424465
1: Covariance: 7.03
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Level 3 shows the form of the equation. Level 2 shows the sample
correlation coefficient, and level 1 shows the covariance of x-y. For
definitions of these parameters see Chapter 18 in the user’s guide.
For additional information on the data-fit feature of the calculator see
Chapter 18 in the user’s guide.
Obtaining additional summary statistics
The application 4. Summary stats.. in the STAT menu can be useful in
some calculations for sample statistics. To get started, press ‚Ù once
more, move to the fourth option using the down-arrow key ˜, and press
@@@[email protected]@@. The resulting input form contains the following fields:
ΣDAT:
the matrix containing the data of interest.
X-Col, Y-Col: these options apply only when you have more than two
columns in the matrix ΣDAT. By default, the x column is
column 1, and the y column is column 2. If you have only
one column, then the only setting that makes sense is to
have X-Col: 1.
_ΣX _ ΣY…:
summary statistics that you can choose as results of this
program by checking the appropriate field using
when that field is selected.
Many of these summary statistics are used to calculate statistics of two
variables (x, y) that may be related by a function y = f(x). Therefore, this
program can be thought off as a companion to program 3. Fit data..
As an example, for the x-y data currently in ΣDAT, obtain all the summary
statistics.
• To access the summary stats… option, use:
‚Ù˜˜˜@@@[email protected]@@
• Select the column numbers corresponding to the x- and y-data, i.e.,
X-Col: 1, and Y-Col: 2.
• Using the
etc.
key select all the options for outputs, i.e., _ΣX, _ΣY,
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• Press @@@[email protected]@@ to obtain the following results:
Confidence intervals
The application 6. Conf Interval can be accessed by using
‚Ù—@@@[email protected]@@. The application offers the following options:
These options are to be interpreted as follows:
1. Z-INT: 1 µ.: Single sample confidence interval for the population mean,
µ, with known population variance, or for large samples with unknown
population variance.
2. Z-INT: µ1−µ2.: Confidence interval for the difference of the population
means, µ1- µ2, with either known population variances, or for large
samples with unknown population variances.
3. Z-INT: 1 p.: Single sample confidence interval for the proportion, p, for
large samples with unknown population variance.
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4. Z-INT: p1− p2.: Confidence interval for the difference of two
proportions, p1-p2, for large samples with unknown population
variances.
5. T-INT: 1 µ.: Single sample confidence interval for the population mean,
µ, for small samples with unknown population variance.
6. T-INT: µ1−µ2.: Confidence interval for the difference of the population
means, µ1- µ2, for small samples with unknown population variances.
Example 1 – Determine the centered confidence interval for the mean of a
population if a sample of 60 elements indicate that the mean value of the
sample is ⎯x = 23.3, and its standard deviation is s = 5.2. Use
α = 0.05. The confidence level is C = 1-α = 0.95.
Select case 1 from the menu shown above by pressing @@@[email protected]@@. Enter the
values required in the input form as shown:
Press @HELP to obtain a screen explaining the meaning of the confidence
interval in terms of random numbers generated by a calculator. To scroll
down the resulting screen use the down-arrow key ˜. Press @@@[email protected]@@ when
done with the help screen. This will return you to the screen shown above.
To calculate the confidence interval, press @@@[email protected]@@. The result shown in the
calculator is:
Press @GRAPH to see a graphical display of the confidence interval
information:
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The graph shows the standard normal distribution pdf (probability density
function), the location of the critical points ±zα/2, the mean value (23.3)
and the corresponding interval limits (21.98424 and 24.61576). Press
@TEXT to return to the previous results screen, and/or press @@@[email protected]@@ to exit the
confidence interval environment. The results will be listed in the calculator’s
display.
Additional examples of confidence interval calculations are presented in
Chapter 18 in the calculator’s user’s guide.
Hypothesis testing
A hypothesis is a declaration made about a population (for instance, with
respect to its mean). Acceptance of the hypothesis is based on a statistical
test on a sample taken from the population. The consequent action and
decision-making are called hypothesis testing.
The calculator provides hypothesis testing procedures under application 5.
Hypoth. tests.. can be accessed by using ‚Ù——@@@[email protected]@@.
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:
1. Z-Test: 1 µ.: Single sample hypothesis testing for the population mean,
µ, with known population variance, or for large samples with unknown
population variance.
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1:31 PM
2. Z-Test: µ1−µ2.: Hypothesis testing for the difference of the population
means, µ1- µ2, 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 µ.: Single sample hypothesis testing for the population mean,
µ, for small samples with unknown population variance.
6. T-Test: µ1−µ2.: Hypothesis testing for the difference of the population
means, µ1- µ2, for small samples with unknown population variances.
Example 1 – For µ0 = 150, σ = 10, ⎯x = 158, n = 50, for α = 0.05, test
the hypothesis H0: µ = µ0, against the alternative hypothesis, H1: µ ≠ µ0.
Press ‚Ù——@@@[email protected]@@ to access the confidence interval feature in the
calculator. Press @@@[email protected]@@ to select option 1. Z-Test: 1 µ.
Enter the following data and press @@@[email protected]@@:
You are then asked to select the alternative hypothesis:
Select µ ≠ 150. Then, press @@@[email protected]@@. The result is:
Page 16-10
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1:31 PM
Then, we reject H0: µ = 150, against H1: µ ≠ 150. The test z value is
z0 = 5.656854. The P-value is 1.54×10 -8. The critical values of
±zα/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:
Reference
Additional materials on statistical analysis, including definitions of
concepts, and advanced statistical applications, are available in Chapter
18 in the user’s guide.
Page 16-11
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SG49A.book
Page 1
Friday, September 16, 2005
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Chapter 17
Numbers in Different Bases
Besides our decimal (base 10, digits = 0-9) number system, you can work
with a binary system (base 2, digits = 0,1), an octal system (base 8, digits
= 0-7), or a hexadecimal system (base 16, digits=0-9,A-F), among others.
The same way that the decimal integer 321 means 3x102+2x101+1x100,
the number 100110, in binary notation, means
1x25 + 0x24 + 0x23 + 1x22 + 1x21 + 0x20 = 32+0+0+4+2+0 = 38.
The BASE menu
The BASE menu is accessible through ‚ã(the 3 key). With system
flag 117 set to CHOOSE boxes (see Chapter 1 in this guide), the
following entries are available:
With system flag 117 set to SOFT menus, the BASE menu shows the
following:
This figure shows that the LOGIC, BIT, and BYTE entries within the BASE
menu are themselves sub-menus. These menus are discussed in detail in
Chapter 19 of the calculator’s user’s guide.
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1:31 PM
Writing non-decimal numbers
Numbers in non-decimal systems, referred to as binary integers, are written
preceded by the # symbol („â) in the calculator. To select the current
base to be used for binary integers, choose either HEX (adecimal), DEC
(imal), OCT (al), or BIN (ary) in the BASE menu. For example, if
is
selected, binary integers will be a hexadecimal numbers, e.g., #53,
#A5B, etc. As different systems are selected, the numbers will be
automatically converted to the new current base.
To write a number in a particular system, start the number with # and end
with either h (hexadecimal), d (decimal), o (octal), or b (binary), examples:
HEX
DEC
OCT
BIN
Reference
For additional details on numbers from different bases see Chapter 19 in
the calculator’s user’s guide.
Page 17-2
Ch18_Using SD cardQS.fm
Page 1
Friday, February 24, 2006
8:39 PM
Chapter 18
Using SD cards
The calculator has a memory card slot into which you can insert an SD
flash card for backing up calculator objects, or for downloading objects
from other sources. The SD card in the calculator will appear as port
number 3.
Inserting and removing an SD card
The SD slot is located on the bottom edge of the calculator, just below the
number keys. SD cards must be inserted facing down. Most cards have a
label on what would usually be considered the top of the card. If you are
holding the HP 50g with the keyboard facing up, then this side of the SD
card should face down or away from you when being inserted into the HP
50g. The card will go into the slot without resistance for most of its length
and then it will require slightly more force to fully insert it. A fully inserted
card is almost flush with the case, leaving only the top edge of the card
visible.
To remove an SD card, turn off the HP 50g, press gently on the exposed
edge of the card and push in. The card should spring out of the slot a small
distance, allowing it now to be easily removed from the calculator.
Formatting an SD card
Most SD cards will already be formatted, but they may be formatted with a
file system that is incompatible with the HP 50g. The HP 50g will only work
with cards in the FAT16 or FAT32 format.
You can format an SD card from a PC, or from the calculator. If you do it
from the calculator (using the method described below), make sure that
your calculator has fresh or fairly new batteries.
NOTE: formatting an SD card deletes all the data that is currently on
it.
1. Insert the SD card into the card slot (as explained in the previous
section).
2. Hold down the ‡ key and then press the D key. Release the D
key and then release the ‡ key. The system menu is displayed with
several choices.
3. Press 0 for FORMAT. The formatting process begins.
Page 18-1
Ch18_Using SD cardQS.fm
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Friday, February 24, 2006
8:39 PM
4. When the formatting is finished, the HP 50g displays the message
"FORMAT FINISHED. PRESS ANY KEY TO EXIT". To exit the system
menu, hold down the ‡ key, press and release the C key and then
release the ‡ key.
The SD card is now ready for use. It will have been formatted in FAT32
format.
Accessing objects on an SD card
Accessing an object on the SD card is similar to when an object is located
in ports 0, 1, or 2. However, port 3 will not appear in the menu when you
are using the LIB function (‚á). The SD files can only be managed
using the Filer, or File Manager („¡). When starting the Filer, the
Tree view will show:
Long names of files on an SD card are supported, but are displayed in 8.3
format in the Filer (that is, their names are truncated to 8 characters and a
three character extension is added as a suffix). The type of each object
will be displayed, unless it is a PC object or an object of unknown type. (In
these cases, its type is listed as String.)
In addition to using File Manager operations, you can use STO and RCL to
store objects on, and recall objects from, the SD card.
Storing objects on the SD card
To store an object, use function STO as follows:
• In algebraic mode:
Enter object, press K, type the name of the stored object using port 3
(e.g., :3:VAR1), press `.
• In RPN mode:
Enter object, type the name of the stored object using port 3 (e.g.,
:3:VAR1), press K.
Page 18-2
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1:31 PM
Note that if the name of the object you intend to store on an SD card is
longer than eight characters, it will appear in 8.3 DOS format in port 3 in
the Filer once it is stored on the card.
Recalling an object from the SD card
To recall an object from the SD card onto the screen, use function RCL, as
follows:
• In algebraic mode:
Press „©, type the name of the stored object using port 3 (e.g.,
:3:VAR1), press `.
• In RPN mode:
Type the name of the stored object using port 3 (e.g., :3:VAR1),
press „©.
With the RCL command, it is possible to recall variables by specifying a
path in the command, e.g., in RPN mode: :3: {path}`RCL. The
path, like in a DOS drive, is a series of directory names that together
specify the position of the variable within a directory tree. However, some
variables stored within a backup object cannot be recalled by specifying a
path. In this case, the full backup object (e.g., a directory) will have to be
recalled, and the individual variables then accessed on the screen.
Note that in the case of objects with long files names, you can specify the
full name of the object, or its truncated 8.3 name, when issuing an RCL
command.
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.
Note that in the case of objects with long files names, you can specify the
full name of the object, or its truncated 8.3 name, when issuing a PURGE
command.
Page 18-3
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1:31 PM
Purging all objects on the SD card (by
reformatting)
You can purge all objects from the SD card by reformatting it. When an SD
card is inserted, @FORMA appears an additional menu item in File Manager.
Selecting this option reformats the entire card, a process which also deletes
every object on the card.
Specifying a directory on an SD card
You can store, recall, evaluate and purge objects that are in directories on
an SD card. Note that to work with an object at the root level of an SD
card, the ³ key is used. But when working with an object in a
subdirectory, the name containing the directory path must be enclosed
using the …Õ keys.
For example, suppose you want to store an object called PROG1 into a
directory called PROGS on an SD card. With this object still on the first
level of the stack, press:
!ê3™…Õ~~progs…/
prog1`K
This will store the object previously on the stack onto the SD card into the
directory named PROGS into an object named PROG1.
NOTE: If PROGS does not exist, the directory will be automatically
created.
You can specify any number of nested subdirectories. For example, to refer
to an object in a third-level subdirectory, your syntax would be:
:3:”DIR1/DIR2/DIR3/NAME”
Note that pressing ~…/ produces the forward slash character.
Page 18-4
SG49A.book
Page 1
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1:31 PM
Chapter 19
Equation Library
The Equation Library is a collection of equations and commands that
enable you to solve simple science and engineering problems. The library
consists of more than 300 equations grouped into 15 technical subjects
containing more than 100 problem titles. Each problem title contains one
or more equations that help you solve that type of problem.
NOTE: the examples in this chapter assume that the operating mode is
RPN and that flag –117 is set. (Flag –117 should be set whenever you
use the numeric solver to solve equations in the equations library.)
Example: Examine the equation set for Projectile Motion.
Step 1:
Fix the display to 2 decimal places and then open the Equation
Library application. (If #SI# and #UNIT# aren’t flagged with small
squares, press each of the corresponding menu keys once.)
H˜~f™2`
G—`#EQLIB #EQNLI
Step 2:
Select the Motion subject area and open its catalog.
~m˜`
Step 3:
Select Projectile Motion and look at the diagram that
describes the problem.
˜˜#PIC#
Page 19-1
SG49A.book
Page 2
Step 4:
Friday, September 16, 2005
1:31 PM
View the five equations in the Projectile Motion set. All five are
used interchangeably in order to solve for missing variables (see
the next example).
#EQN# #NXEQ# #NXEQ# #NXEQ# #NXEQ#
Step 5:
Examine the variables used by the equation set.
#VARS#
and —as ˜ needed
Now use this equation set to answer the questions in the following
example.
Example: You estimate that on average professional goalkeepers can
punt a soccer ball a distance (R) of 65 meters downfield at an
elevation angle (0) of 50 degrees. At what velocity (v0) do they
kick it? How high is the ball halfway through its flight? How far
could they drop kick the ball if they used the same kicking
velocity, but changed the elevation angle to 30 degrees?
(Ignore the effects of drag on the ball.)
Step 1:
Start solving the problem.
#SOLV#
Step 2:
Enter the known values and press the soft menu key
corresponding to the variable. (You can assume that x0 and y0
are zero.) Notice that the menu labels turn black as you store
values. (You will need to press L to see the variables that are
initially shown.)
Page 19-2
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1:31 PM
0 *!!!!!!X0!!!!!+ 0 *!!!!!!Y0!!!!!+ 50 *!!!!!!Ô0!!!!!+
L65*!!!!!!R!!!!!+
Step 3:
Solve for the velocity, v0. (You solve for a variable by pressing
! and then the variable’s menu key.)
!*!!!!!!V0!!!!!+
Step 4:
Recall the range, R, divide by 2 to get the halfway distance,
and enter that as the x-coordinate. Notice that pressing the
right-shifted version of a variable’s menu key causes the
calculator to recall its value to the stack. (The small square next
to the R on the menu label indicates that it was used in the
previous calculation.)
@ ##R#2/LL*!!!!!!X!!!!!+
Step 5:
Solve for the height, y. Notice that the calculator finds values for
other variables as needed (shown by the small squares) in
order to solve for the specified variable.
! *!!!!!!Y!!!!!+
Step 6:
Enter the new value for the elevation angle (30 degrees), store
the previously computed initial velocity (v0) and then solve for R.
30 ##¢0#™L *!!!!!!V0!!!!!+
! *!!!!!!!!R!!!!!!!+
Page 19-3
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Page 4
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1:31 PM
Reference
For additional details on the Equation Library, see Chapter 27 in the
calculator’s user’s guide.
Page 19-4
WarrantyQS49_E.fm Page 1 Friday, February 24, 2006
8:25 PM
Limited Warranty
HP 50g graphing calculator; Warranty period: 12 months
1. 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.
2. 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.
3. 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.
4. 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.
7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS
WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE
Page W-1
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Page 2
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1:31 PM
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.
Page W-2
SG49A.book
Page 3
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1:31 PM
Service
Europe
Country :
Telephone numbers
Austria
+43-1-3602771203
Belgium
Denmark
Eastern Europe countries
Finland
France
Germany
Greece
Holland
Italy
Norway
Portugal
Spain
Sweden
+32-2-7126219
+45-8-2332844
+420-5-41422523
+358-9-640009
+33-1-49939006
+49-69-95307103
+420-5-41422523
+31-2-06545301
+39-02-75419782
+47-63849309
+351-229570200
+34-915-642095
+46-851992065
+41-1-4395358 (German)
+41-22-8278780 (French)
+39-02-75419782 (Italian)
+420-5-41422523
+44-207-4580161
+420-5-41422523
+27-11-2376200
+32-2-7126219
Switzerland
Turkey
UK
Czech Republic
South Africa
Luxembourg
Other European countries +420-5-41422523
Asia Pacific
Page W-3
Country :
Telephone numbers
Australia
+61-3-9841-5211
Singapore
+61-3-9841-5211
SG49A.book
Page 4
Friday, September 16, 2005
L.America
1:31 PM
Country :
Telephone numbers
Argentina
Brazil
0-810-555-5520
Sao Paulo 3747-7799;
ROTC 0-800-157751
Mx City 5258-9922;
ROTC 01-800-472-6684
0800-4746-8368
800-360999
9-800-114726
0-800-10111
1-800-711-2884
Mexico
Venezuela
Chile
Columbia
Peru
Central America &
Caribbean
Guatemala
Puerto Rico
Costa Rica
N.America
1-800-999-5105
1-877-232-0589
0-800-011-0524
Country :
Telephone numbers
U.S.
1800-HP INVENT
Canada
(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.
Page W-4
SG49A.book
Page 5
Friday, September 16, 2005
1:31 PM
Regulatory information
Federal Communications Commission Notice
This equipment has been tested and found to comply 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. This equipment generates, uses,
and can radiate radio frequency energy and, if not installed and used in
accordance with the instructions, may cause harmful interference to radio
communications. However, there is no guarantee that interference will not
occur in a particular installation. If this equipment does cause harmful
interference to radio or television reception, which can be determined by
turning the equipment off and on, the user is encouraged to try to correct
the interference by one or more of the following measures:
• Reorient or relocate the receiving antenna.
• Increase the separation between the equipment and the receiver.
• Connect the equipment into an outlet on a circuit different from that to
which the receiver is connected.
• Consult the dealer or an experienced radio or television technician for
help.
Modifications
The FCC requires the user to be notified that any changes or modifications
made to this device that are not expressly approved by Hewlett-Packard
Company may void the user’s authority to operate the equipment.
Cables
Connections to this device must be made with shielded cables with metallic
RFI/EMI connector hoods to maintain compliance with FCC rules and
regulations.
Declaration of Conformity for Products Marked with FCC
Logo, United States Only
This device complies with Part 15 of the FCC Rules. Operation is subject to
the following two conditions: (1) this device may not cause harmful
interference, and (2) this device must accept any interference received,
including interference that may cause undesired operation.
For questions regarding your product, contact:
Hewlett-Packard Company
P. O. Box 692000, Mail Stop 530113
Houston, Texas 77269-2000
Page W-5
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Page 6
Friday, September 16, 2005
1:31 PM
Or, call
1-800-474-6836
For questions regarding this FCC declaration, contact:
Hewlett-Packard Company
P. O. Box 692000, Mail Stop 510101
Houston, Texas 77269-2000
Or, call
1-281-514-3333
To identify this product, refer to the part, series, or model number found
on the product.
Canadian Notice
This Class B digital apparatus meets all requirements of the
Canadian Interference-Causing Equipment Regulations.
Avis Canadien
Cet appareil numérique de la classe B respecte toutes les
exigences du Règlement sur le matériel brouilleur du Canada.
European Union Regulatory Notice
This product complies with the following EU Directives:
• Low Voltage Directive 73/23/EEC
• EMC Directive 89/336/EEC
Compliance with these directives implies conformity to applicable
harmonized European standards (European Norms) which are listed on the
EU Declaration of Conformity issued by Hewlett-Packard for this product or
product family.
This compliance is indicated by the following conformity marking placed
on the product:
This compliance is indicated by the following conformity marking placed
on the product:
xxxx*
This marking is valid for non-Telecom prodcts
and EU harmonized Telecom products (e.g.
Bluetooth).
This marking is valid for EU non-harmonized Telecom products.
*Notified body number (used only if applicable - refer to the
product label)
Page W-6
SG49A.book
Page 7
Friday, September 16, 2005
1:31 PM
Japanese Notice
こ の装置は、 情報処理装置等電波障害自主規制協議会 (VCCI) の基準に基づ く ク ラ ス B
情報技術装置です。 こ の装置は、 家庭環境で使用する こ と を目的 と し ていますが、 こ の装
置がラ ジオやテ レ ビ ジ ョ ン受信機に近接 し て使用 さ れる と 、 受信障害を引き起 こ す こ と が
あ り ます。
取扱説明書に従 っ て正 し い取 り 扱い を し て く だ さ い。
Korean Notice
Disposal of Waste Equipment by Users in
Private Household in the European Union
This symbol on the product or on its packaging indicates that
this product must not be disposed of with your other
household waste. Instead, it is your responsibility to dispose
of your waste equipment by handing it over to a designated
collection point for the recycling of waste electrical and
electronic equipment. The separate collection and recycling of your waste
equipment at the time of disposal will help to conserve natural resources
and ensure that it is recycled in a manner that protects human health and
the environment. For more information about where you can drop off your
waste equipment for recycling, please contact your local city office, your
household waste disposal service or the shop where you purchased the
product.
Page W-7
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