HP 50g graphing calculator

users manual

Edition 1 HP part number F2229AA-90001
FrontPageQS49_E.backup.fm Page 2
Friday, February 24, 2006

4:54 PM

Notice
REGISTER YOUR PRODUCT AT: www.register.hp.com THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE PROVIDED AS IS AND ARE SUBJECT TO CHANGE WITHOUT NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS FOR A PARTICULAR PURPOSE. HEWLETT-PACKARD CO. SHALL NOT BE LIABLE FOR ANY ERRORS OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES IN CONNECTION WITH THE FURNISHING, PERFORMANCE, OR USE OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN. Copyright 2003, 2006 Hewlett-Packard Development Company, L.P. Reproduction, adaptation, or translation of this manual is prohibited without prior written permission of Hewlett-Packard Company, except as allowed under the copyright laws.
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Printing History

Edition 1 April 2006

SG49A.book

Page 1
Friday, September 16, 2005

1:31 PM

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 users 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.

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 calculators display, 1-3 Menus, 1-3 The TOOL menu, 1-3 Setting time and date, 1-4 Introducing the calculators 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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@@RCL@ C ReCaLl the contents of a variable @@STO@ 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 calculators users guide to learn how to set time and date.
Introducing the calculators keyboard
The figure on the next page shows a diagram of the calculators keyboard with the numbering of its rows and columns. Each key has three, four, or five functions. The main key function correspond to the most prominent label in the key. Also, the 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 N ~p ~p Main function, to activate the SYMBolic menu Left-shift function, to activate the MTH (Math) menu Right-shift function, to activate the CATalog function ALPHA function, to enter the upper-case letter P ALPHA-Left-Shift function, to enter the lower-case letter p

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ALPHA-Right-Shift function, to enter the symbol
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.

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

Calculator objects

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

1.0 7.5 5.0 3.0 2.1.0 +

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

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

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 () can be used to group operations, as well as to enclose arguments of functions. In ALG mode: 5+3.2/72.2` In RPN mode, you do not need the parenthesis, calculation is done directly on the stack: 5`3.2+7`2.2-/ In RPN mode, typing the expression between single quotes will allow you to enter the expression like in algebraic mode: 5+3.2/ 7-2.2` For both, ALG and RPN modes, using the Equation Writer: O5+3.2/7-2.2 The expression can be evaluated within the Equation writer, by using @EVAL@ or, @EVAL@ The absolute value function, ABS, is available through. Example in ALG mode: \2.32` Example in RPN mode: 2.32\ The square function, SQ, is available through. Example in ALG mode: \2.3` Example in RPN mode: 2.3\ 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., 327` 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 (function LOG) while its inverse function (ALOG, or antilogarithm) is calculated by using. In ALG mode, the function is entered before the argument: 2.45` \2.3` In RPN mode, the argument is entered before the function 2.45 2.3\

Using powers of 10 in entering data
Powers of ten, i.e., numbers of the form -4.-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 (function LN) while the exponential function (EXP) is calculated by using. In ALG mode, the function is entered before the argument: 2.45` \2.3` In RPN mode, the argument is entered before the function

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2.45` 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 (), arccosine (), and arctangent (). The answer from these functions will be given in the selected angular measure (DEG, RAD, GRD). Some examples are shown next: In ALG mode: 0.25` 0.85` 1.35` In RPN mode: 0.25 0.85 1.35 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 @@OK@@, 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: 4@@OK@@ 5@@OK@@ 2.5` In the RPN mode, the keystrokes to perform this calculation are the following: 2.5`4@@OK@@ 5@@OK@@ 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 )@@HYP@ , to produce:

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Finally, in order to select, for example, the hyperbolic tangent (tanh) function, simply press @@TANH@. 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: @@HYP@ @@TANH@ 2.5` In RPN mode, the same value is calculated using: 2.5`)@@HYP@ @@TANH@ As an exercise of applications of hyperbolic functions, verify the following values: SINH (2.5) = 6.05020. COSH (2.5) = 6.13228. TANH(2.5) = 0.98661. EXPM(2.0) = 6.38905. ASINH(2.0) = 1.4436 ACOSH (2.0) = 1.3169 ATANH(0.2) = 0.2027 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 (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 to get the following menus. Press L to move to the next menu page.

which shows as 65_(myd). To convert to units of the SI system, use function UBASE (find it using the command catalog, N):
NOTE: Recall that the ANS(1) variable is available through the keystroke combination (associated with the ` key). To calculate a division, say, 3250 mi / 50 h, enter it as (3250_mi)/(50_h) ` which transformed to SI units, with function UBASE, produces:

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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 @@@m@@@ `1.5 @@yd@@ `* 3250 @@mi@@ `50 @@@h@@@ `/ 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) UVAL(x) UFACT(x,y) UNIT(x,y) convert unit object x to SI units extract the value from unit object x factors a unit y from unit object x combines value of x with units of y
Examples of function CONVERT are shown below. Examples of the other UNIT/TOOLS functions are available in Chapter 3 of the calculators users guide. For example, to convert 33 watts to btus use either of the following entries: CONVERT(33_W,1_hp) ` CONVERT(33_W,11_hp) `

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Physical constants in the calculator
The calculators physical constants are contained in a constants library activated with the command CONLIB. To launch this command you could simply type it in the stack: ~~conlib`, or, you can select the command CONLIB from the command catalog, as follows: First, launch the catalog by using: N~c. Next, use the up and down arrow keys to select CONLIB. Finally, press @@OK@@. 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 ENGL UNIT VALUE STK QUIT when selected, constants values are shown in SI units (*) when selected, constants values are shown in English units (*) when selected, constants are shown with units attached (*) when selected, constants are shown without units copies value (with or without units) to the stack 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 @STK, then, press @QUIT@. 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: 2*` 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`*

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Defining and using functions
Users can define their own functions by using the DEFINE command available thought the keystroke sequence (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: ~h~x ~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 (@@@H@@). To see the contents of this variable press @@@H@@. 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 calculators users guide). The program shown above is

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Ch03_RealNumbersQS.fm Page 16 Friday, February 24, 2006
relatively simple and consists of two parts, contained between the program containers Input: Process: x x 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., @@@H@@@ 2`. Some examples are shown below:

The PEVAL function

The function PEVAL (Polynomial EVALuation) can be used to evaluate a polynomial p(x) = anxn+an-1x n-1+ + a2x2+a1x+ 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 calculators users 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 ` (or, simply , 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. ((1i*3)/2) 1. 3 1. 2 1.].

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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(x/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: ~x-S *~x/30 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: @@OK@@. 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 @SOLVE@. 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\ @@@OK@@ @SOLVE@. The solution is now X: -3.045.
Solution to simultaneous equations with MSLV
Function MSLV is available in the 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 ():

lim f ( x)

The keystrokes that generate this particular example are:

~!s`!2/S~!s`6!$OK$ $OK$

Additional definitions and applications of calculus operations are presented in Chapter 13 in the calculators users 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
for example, will be entered as x(f(x,y)) ` in ALG mode in the screen.

( f ( x, y )) , x

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
( x , y )dA = a f ( x ) ( x , y )dydx = c r ( y ) ( x , y )dydx

b g( x )

s( y )
Calculating a double integral in the calculator is straightforward. A double integral can be built in the Equation Writer (see example in Chapter 2 in the users guide), as shown below. This double integral is calculated directly in the Equation Writer by selecting the entire expression and using function @EVAL. The result is 3/2.
For additional details of multi-variate calculus operations and their applications see Chapter 14 in the calculators users guide.

Page 12-2

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

Page 13-1

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 [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])
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

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):

Page 14-4

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

Fourier series

A complex Fourier series is defined by the following expression

f (t ) =

2int ), T

f (t ) exp(

2 i n 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 ().

HP 50g

Graphing calculator

A new more powerful HP 50g HP Graphing Calculators feature easy-to-use, powerful tools that students and professionals can rely on and trust for years of performance, including: Powerful built-in graphing functions, constants and applications Built-in lessons and step-by-step problem solving The choice of efficient RPN, Textbook or Algebraic data entry User-friendly Computer Algebra System (CAS) Expansive memory with SD card* slot Flexible connectivity and communication options Large high-contrast display with adjustable font type and size Award-winning HP quality and support 24/7, plus on-line tutorials and emulators Built for performance New display upgradethe new HP 50g display features a 30% increase in usable space over the HP 49g+ New powerful SD card slot allows you to format your card right in the calculator and expand memory Massive 2.5 MB total memory512 KB RAM plus 2 MB flash ROM for performing future upgrades** Isolate and evaluate sub-expressions using the intelligent editorplus cut, paste and copy objects New larger equation library and 2300+ built-in functionsideal for both professionals and students
The new HP 50g Graphing Calculator provides the best in power, flexibility and connectivity for math, science and engineering professionals and college students. It now features a more capable SD card slot, 2.5 MB total memory**, new RS232 and USB connectivity, and a built-in intelligent editor that gives you more capability than ever before.

Graphing Calculator

Operating features Entry Logic: Algebraic/RPN/textbook Menus, prompts Alpha messages Soft-keys Numerical precision: intermediate and internal: 15 digits, exponent: -49999 to +49999 Displayed precision: 12 digits, exponent: -499 to +499, infinite for integers (limited by memory) Additional applications available on Internet Configuration Flash ROM for future electronic software upgrades RAM 512 KB + 2 MB flash ROM Redefinable menu keys Redefinable keyboard (user mode) Peripherals Supports libraries/RPL programs Vector operations Rectangular/polar Matrix operations: Includes symbolic matrices Matrix editor Graphing features 2-D function, polar, parametric plot 3-D, differential equation, bar plot Histogram, scatter plot Find: intersect, extremum, slope, area Zoom, trace, co-ordinates, shade Math features +, , x, , , 1/x, +/, In, ex, xy yx, LOG, 10x, x2, %, , n! Fractions Degrees, radians or grads mode Trigonometric functions/inverses Hyperbolics/inverses HP Solve application (root finder) Numeric Iintegration Symbolic integration Numeric differentiation Symbolic differentiation Complex number functions Polynomial root finder, Taylor series Absolute value, round Integer & fractional part of a number Modulo function, floor, ceiling CAS system Scientific features Decimal hr/hr min sec conversions Polar/rectangular conversions Angle conversions Base conversions and arithmetic Unit conversions Bit, boolean, graphics Display and printer graphics Built-in equation library Statistical features x, x2, y, y2, xy Sample standard deviation, mean Population standard deviation Linear regression Combinations, permutations Weighed mean Edit, save, name, list Curve fit (LIN, LOG, EXP, POW) Plot statistical data Hypothesis tests Confidence intervals Programming features Root finder: HP Solve Number of steps/regs or bytes: 1.2 MB Number of programs/formulae: ++ Levels of subroutines: ++ Branching Flags: 256 Alpha prompts in programs Input forms, alpha string manipulation Indirect addressing Index looping Alpha listing HP Equation library, over 300 physics equations and graphics Electronic specifications CPU: 75 Mhz ARM9 Display size: 131 x 80 pixels Display type: Pixel Memory RAM: 512 KB, 2 MB Flash ROM** IR port: IrDA (limited to 10 cm distance) Serial port: USB, RS232 for connections to equipment such as PCs, datalogers, surveying equipment, etc. Expansion port: SD card slot* Power supply: AAA x 4 + CR2032 Material specifications Enclosure/key top material: Plastic/plastic Size (L x W x D): 7.24 x 3.43 x.93 in (18.4 x 8.7 x 2.35 cm) Weight: Approximately 220 g Contents: Calculator, batteries, User Manual, CD (connectivity software and Advanced User Guide), USB cable and premium pouch

SD card slot at bottom

Infrared communications
* Secure Digital (SD) card sold separately ** 377KB RAM and 780KB Flash ROM available to the user. A PC with an Internet connection is required for downloads. USB cable is included. HP makes no representation that a future upgrade will be available. 2006 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice. The only warranties for HP products and services are set forth in the express warranty statements accompanying such products and services. Nothing herein should be construed as constituting an additional warranty. HP shall not be liable for technical or editorial errors or omissions contained herein.

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