HP 48gII Instrukcja Użytkownika

hp 48gII graphing calculator user’s manual K

Page TOC-7 Curl, 13-2 Reference, 13-2 Chapter 14 – Differential Equations, 14-1 The CALC/DIFF menu, 14-1 Solution to linear and non-linear

Page 6-9 '(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

Page 6-10 In RPN mode, enter the equation between apostrophes and activate command STEQ. Thus, function STEQ can be used as a shortcut to store an

Page 6-11 Solution to simultaneous equations with MSLV Function MSLV is available in the ‚Ï menu. The help-facility entry for function MSLV is shown

Page 6-12 You may have noticed that, while producing the solution, the screen shows intermediate information on the upper left corner. Since the

Page 7-1 Chapter 7 Operations with lists Lists are a type of calculator’s object that can be useful for data processing. This chapter presents exa

Page 7-2 Addition, subtraction, multiplication, division Multiplication and division of a list by a single number is distributed across the list, fo

Page 7-3 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.

Page 7-4 ABS INVERSE (1/x) Lists of complex numbers You can create a complex number list, say, L5 = L1 ADD i⋅L2 (type the i

Page 7-5 With system flag 117 set to SOFT menus, the MTH/LIST menu shows the following functions: The operation of the MTH/LIST menu i

Page 7-6 The SEQ function The SEQ function, available through the command catalog (‚N), takes as arguments an expression in terms of an index, th

Page TOC-8 Chapter 17 – Numbers in Different Bases, 17-1 The BASE menu, 17-1 Writing non-decimal numbers, 17-1 Reference, 17-2 Warranty – W-

Page 8-1 Chapter 8 Vectors This Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as wel

Page 8-2 (‚í) or spaces (#). Notice that after pressing ` , in either mode, the calculator shows the vector elements separated by spaces. Storin

Page 8-3 The @EDIT key is used to edit the contents of a selected cell in the matrix writer. The @VEC@@ key, when selected, will produce a vector

Page 8-4 The @ROW@ key will add a row full of zeros at the location of the selected cell of the spreadsheet. The @ROW key will delete the row

Page 8-5 (3) Move the cursor up two positions by using ——. Then press @ROW. The second row will disappear. (4) Press @ROW@. A row of three

Page 8-6 Attempting to add or subtract vectors of different length produces an error message: Multiplication by a scalar, and division

Page 8-7 The MTH/VECTOR menu The MTH menu („´) contains a menu of functions that specifically to vector objects: The VECTOR menu contains the

Page 8-8 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 vector

Page 9-1 Chapter 9 Matrices and linear algebra This chapter shows examples of creating matrices and operations with matrices, including linear al

Page 9-2 Press ` once more to place the matrix on the stack. The ALG mode stack is shown next, before and after pressing , once more: If you

Page 1-1 Chapter 1 Getting started This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aim

Page 9-3 Operations with matrices Matrices, like other mathematical objects, can be added and subtracted. They can be multiplied by a scalar, or am

Page 9-4 In RPN mode, you can try a few more exercises: A ` `+ A ` `- A ` `+ A ` `- A ` `+ A ` `

Page 9-5 Vector-matrix multiplication, on the other hand, is not defined. This multiplication can be performed, however, as a special case of matr

Page 9-6 The identity matrix The identity matrix has the property that A⋅I = I⋅A = A. To verify this property we present the following examples usi

Page 9-7 Characterizing a matrix (The matrix NORM menu) The matrix NORM (NORMALIZE) menu is accessed through the keystroke sequence „´ . This menu

Page 9-8 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: mnnm

Page 9-9 .61313,,422831532321−−==−−−=bxA andxxx This system has the same number of equations as of unknowns, and will

Page 9-10 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 e

Page 10-1 Chapter 10 Graphics In this chapter we introduce some of the graphics capabilities of the calculator. We will present graphics of functio

Page 10-2 Plotting an expression of the form y = f(x) As an example, let's plot the function, )2exp(21)(2xxf −=π • First, enter the PLOT SET

Page 1-2 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 p

Page 10-3 VIEW, then press @AUTO to generate the V-VIEW automatically. The PLOT WINDOW screen looks as follows: • Plot the graph: @ERASE

Page 10-4 • 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 en

Page 10-5 •• The @@IG@ key simply changes the font in the table from small to big, and vice versa. Try it. •• The @OOM key, when pressed,

Page 10-6 • Press „ô, simultaneously if in RPN mode, to access to the PLOT SETUP window. • Change TYPE to Fast3D. ( @CHOOS!, find Fast3D, @@OK@

Page 10-7 • When done, press @EIT. • Press @CACL to return to the PLOT WINDOW environment. • Change the Step data to read: Step In

Page 10-8 • Press LL@)PICT to leave the EDIT environment. • Press @CACL to return to the PLOT WINDOW environment. Then, press $ , or L@@@OK@@@

Page 11-1 Chapter 11 Calculus Applications In this Chapter we discuss applications of the calculator’s functions to operations related to Calculus,

Page 11-2 command catalog (‚N~„l) or through option 2. LIMITS & SERIES… of the CALC menu (see above). Function lim is entered in ALG mode as 

Page 11-3 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 rep

Page 11-4 Please notice that functions SIGMAVX and SIGMA are designed for integrands that involve some sort of integer function like t

Page 1-3 RAD XYZ HEX R= 'X' For details on the meaning of these specifications see Chapter 2 in the calculator’s user's guide. The

Page 11-5 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, ∑∞=−⋅=0)(

Page 11-6 Function SERIES produces a Taylor polynomial using as arguments the function f(x) to be expanded, a variable name alone (for Maclau

Page 11-7 In the right-hand side figure above, we are using the line editor to see the series expansion in detail. Reference Additional de

Page 12-1 Chapter 12 Multi-variate Calculus Applications Multi-variate calculus refers to functions of two or more variables. In this Chapter we di

Page 12-2 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 volum

Page 13-1 Chapter 13 Vector Analysis Applications This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vec

Page 13-2 Alternatively, use function DERIV as follows: Divergence The divergence of a vector function, F(x,y,z) = f(x,y,z)i +g(x,y,z)j +h(x,y,z)

Page 14-1 Chapter 14 Differential Equations In this Chapter we present examples of solving ordinary differential equations (ODE) using calculator f

Page 14-2 Function LDEC The calculator provides function LDEC (Linear Differential Equation Command) to find the general solution to a linear ODE o

Page 14-3 'X' ` 'XXX' ` DE The solution is: which can be simplified to y = K1⋅e–3x + K2⋅e5x + K3⋅e2x +

Page 1-4 @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 in

Page 14-4 The variable ODETYPE You will notice in the soft-menu key labels a new variable called @ODET (ODETYPE). This variable is produced with

Page 14-5 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

Page 14-6 and you will notice that the CAS default variable X in the equation writer screen replaces the variable s in this definition. Therefore,

Page 14-7 Next, we move t o the CASDIR sub-directory under HOME to change the value of variable PERIOD, e.g. [Note: not all lines will be visible

Page 14-8 Thus, 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

Page 15-1 Chapter 15 Probability Distributions In this Chapter we provide examples of applications of the pre-defined probability distributions in t

Page 15-2 We can calculate combinations, permutations, and factorials with functions COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The ope

Page 15-3 start lists of random numbers is presented in detail in Chapter 17 of the user's guide. The MTH/PROB menu - part 2 In this section w

Page 15-4 The Student-t distribution The Student-t, or simply, the t-, distribution has one parameter ν, known as the degrees of freedom of the di

Page 16-1 Chapter 16 Statistical Applications The calculator provides the following pre-programmed statistical features accessible through the keyst

Page 1-5 the blue ALPHA key, key (7,1), can be combined with some of the other keys to activate the alternative functions shown in the keyboard.

Page 16-2 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 w

Page 16-3 Obtaining frequency distributions The application 2. Frequencies.. in the STAT menu can be used to obtain frequency distributions for a

Page 16-4 This information indicates that our data ranges from -9 to 9. To produce a frequency distribution we will use the interval (-8,8) dividi

Page 16-5 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.

Page 16-6 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 d

Page 16-7 • Press @@@OK@@@ to obtain the following results: Confidence intervals The application 6. Conf Interval can be accessed by using ‚Ù

Page 16-8 5. T-INT: 1 µ.: Single sample confidence interval for the population mean, µ, for small samples with unknown population variance. 6. T-IN

Page 16-9 The graph shows the standard normal distribution pdf (probability density function), the location of the critical points ±zα/2, the mean

Page 16-10 1. Z-Test: 1 µ.: Single sample hypothesis testing for the population mean, µ, with known population variance, or for large samples with

Page 16-11 Select µ ≠ 150. Then, press @@@OK@@@. The result is: Then, we reject H0: µ = 150, against H1: µ ≠ 150. The test z value is z0 = 5

Page 1-6 ~„p ALPHA-Left-Shift function, to enter the lower-case letter p ~…p ALPHA-Right-Shift function, to enter the symbol π Of the six functi

Page 17-1 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

Page 17-2 OCT(al), or BIN(ary) in the BASE menu. For example, if @HE! is selected, binary integers will be a hexadecimal numbers, e.g., #53, #A5B

Page W-1 Warranty hp 48gII Graphing Calculator; Warranty period: 12 months 1. HP warrants to you, the end-user customer, that HP hardware, accesso

Page W-2 7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE REMEDIES. EXCEPT AS INDICATED A

Page W-3 Sweden +46-851992065 Switzerland +41-1-4395358 (German) +41-22-8278780 (French) +39-0422-303069 (Italian) Turkey +420-5-41422523 UK

Page W-4 Regulatory information This section contains information that shows how the hp 48gII graphing calculator complies with regulations in certa

Page 1-7 Press the !!@@OK#@ F soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. Opera

Page 1-8 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 `a

Page 1-9 different levels are referred to as the stack levels, i.e., stack level 1, stack level 2, etc. Basically, what RPN means is that, instea

Page 1-10 5.232333153e+⋅−⋅ 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

Page 1-11 more about reals, see Chapter 2 in this Guide. To illustrate this and other number formats try the following exercises: • Standard form

Page 1-12 Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Notice how the number is rounded

Page 1-13 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,

Page 1-14 • Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with

Page 1-15 • Grades: There are 400 grades (400 g) in a complete circumference. The angle measure affects the trig functions like SIN, COS, TAN and

Page 1-16 Selecting CAS settings CAS stands for Computer Algebraic System. This is the mathematical core of the calculator where the symbolic mat

Page 1-17 options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Appro

Page 1-18 The calculator display can be customized to your preference by selecting different display modes. To see the optional display settings us

Preface You have in your hands a compact symbolic and numerical computer that will facilitate calculation and mathematical analysis of problems in

Page 1-19 (D) to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0:system 8 is selected. This is the defa

Page 1-20 Selecting properties of the Stack First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES in

Page 1-21 Selecting properties of the equation writer (EQW) First, press the H button to activate the CALCULATOR MODES input form. Within the CAL

Page 1-22

Page 2-1 Chapter 2 Introducing the calculator In this chapter we present a number of basic operations of the calculator including the use of the Equ

Page 2-2 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 i

Page 2-3 To evaluate the expression we can use the EVAL function, as follows: µ„î If the CAS is set to Exact, you will be asked to approve chan

Page 2-4 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 st

Page 2-5 Entering this expression when the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise. For additional in

Page 2-6 The cursor is shown as a left-facing key. The cursor indicates the current edition location. For example, for the cursor in the locatio

Page TOC-1 Table of Contents Chapter 1 – Getting Started, 1-1 Basic Operations, 1-1 Batteries, 1-1 Turning the calculator on and off, 1-2

Page 2-7 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 ent

Page 2-8 Creating algebraic expressions An algebraic expression is very similar to an arithmetic expression, except that English and Greek letters

Page 2-9 Also, you can always copy special characters by using the CHARS menu (…±) if you don’t want to memorize the keystroke combination that prod

Page 2-10 Variables Variables are similar to files on a computer hard drive. One variable can store one object (numerical values, algebraic express

Page 2-11 To unlock the upper-case locked keyboard, press ~ Try the following exercises: ³~~math1~` ³~~m„a„t„h~` ³~~m„~at„h~` The calculator disp

Page 2-12 Press ` to create the variable. The variable is now shown in the soft menu key labels: The following are the keystrokes required to

Page 2-13 • RPN mode (Use I\@@OK@@ to change to RPN mode). Use the following keystrokes to store the value of –0.25 into variable α: 0.25\` ~‚a`.

Page 2-14 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 a

Page 2-15 Using the right-shift key ‚ followed by soft menu key labels This approach for viewing the contents of a variable works the same in both

Page 2-16 Deleting variables The simplest way of deleting variables is by using function PURGE. This function can be accessed directly by using the

Page TOC-2 Creating algebraic expressions, 2-4 Using the Equation Writer (EQW) to create expressions, 2-5 Creating arithmetic expressions, 2-5

Page 2-17 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 s

Page 2-18 @@OK@@ ˜˜˜˜ Show the MEMORY menu list and select DIRECTORY @@OK@@ —— Show the DIRECTORY menu list and select ORDER @@OK@@

Page 2-19 Press the @CHECK! soft menu key to set flag 117 to soft MENU. The screen will reflect that change: Press twice to return to normal ca

Page 2-20 To activate the ORDER command we press the C(@ORDER) soft menu key. References For additional information on entering and manipulating e

Page 3-1 Chapter 3 Calculations with real numbers This chapter demonstrates the use of the calculator for operations and functions related to real n

Page 3-2 6.3` 8.5 - 4.2` 2.5 * 2.3` 4.5 / Alternatively, in RPN mode, you can separate the operands with a space (#) before pressing the operato

Page 3-3 „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

Page 3-4 enter the function XROOT followed by the arguments (y,x), separated by commas, e.g., ‚»3‚í 27` In RPN mode, enter the argument y, first,

Page 3-5 2.45` ‚¹ 2.3\` „¸ • Three trigonometric functions are readily available in the keyboard: sine (S), cosine (T), and tangent (U). Argumen

Page 3-6 Real number functions in the MTH menu The MTH („´) menu include a number of mathematical functions mostly applicable to real numbers. Wit

Page TOC-3 Operations with units, 3-12 Unit conversions, 3-14 Physical constants in the calculator, 3-14 Defining and using functions, 3-16

Page 3-7 For example, in ALG mode, the keystroke sequence to calculate, say, tanh(2.5), is the following: „´4 @@OK@@ 5 @@OK@@ 2.5` In t

Page 3-8 Finally, in order to select, for example, the hyperbolic tangent (tanh) function, simply press @@TAH@. Note: To see additional option

Page 3-9 Option 1. Tools.. contains functions used to operate on units (discussed later). Options 3. Length.. through 17.Viscosity.. con

Page 3-10 Pressing on the appropriate soft menu key will open the sub-menu of units for that particular selection. For example, for the @)

Page 3-11 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 forc

Page 3-12 ____________________________________________________ Prefix Name x Prefix Name x ____________________________________________________

Page 3-13 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: R

Page 3-14 These operations produce the following output: Unit conversions The UNITS menu contains a TOOLS sub-menu, which provides the following

Page 3-15 The soft menu keys corresponding to this CONSTANTS LIBRARY screen include the following functions: SI when selected, constants values ar

Page 3-16 To copy the value of Vm to the stack, select the variable name, and press !STK, then, press @UIT@. For the calculator set to the ALG,

Page TOC-4 The PROPFRAC function, 5-11 The PARTFRAC function, 5-11 The FCOEF function, 5-11 The FROOTS function, 5-12 Step-by-step ope

Page 3-17 and get the result you want without having to type the expression in the right-hand side for each separate value. In the following exampl

Page 3-18 between quotes that contain that local variable, and show the evaluated expression. To activate the function in ALG mode, type the name

Page 4-1 Chapter 4 Calculations with complex numbers This chapter shows examples of calculations and application of functions to complex numbers. D

Page 4-2 Entering complex numbers Complex numbers in the calculator can be entered in either of the two Cartesian representations, namely, x+iy, or

Page 4-3 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.

Page 4-4 (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)

Page 4-5 CONJ(z): Produces the complex conjugate of z Examples of applications of these functions are shown next. Recall that, for ALG mode, the fu

Page 4-6 Functions applied to complex numbers Many of the keyboard-based functions and MTH menu functions defined in Chapter 3 for real numbers (e.g

Page 4-7 Function DROITE is found in the command catalog (‚N). Reference Additional information on complex number operations is presented in Cha

Page 5-1 Chapter 5 Algebraic and arithmetic operations An algebraic object, or simply, algebraic, is any number, variable name or algebraic expressi

Page TOC-5 The SEQ function, 7-6 The MAP function, 7-6 Reference, 7-6 Chapter 8 – Vectors, 8-1 Entering vectors, 8-1 Typing vectors in

Page 5-2 After building the object, press to show it in the stack (ALG and RPN modes shown below): Simple operations with algebraic objects

Page 5-3 In ALG mode, the following keystrokes will show a number of operations with the algebraics contained in variables @@A@@ and @@A@@ (pres

Page 5-4 Functions in the ALG menu The ALG (Algebraic) menu is available by using the keystroke sequence ‚× (associated with the ‚ key). With sys

Page 5-5 Copy the examples provided onto your stack by pressing @ECHO!. For example, for the EXPAND entry shown above, press the @ECHO! soft menu k

Page 5-6 Operations with transcendental functions The calculator offers a number of functions that can be used to replace expressions containing log

Page 5-7 These functions allow to simplify expressions by replacing some category of trigonometric functions for another one. For example, the func

Page 5-8 FACTORS: SIMP2: The functions associated with the ARITHMETIC submenus: INTEGER, POLYNOMIAL, MODULO, and PERMUTAT

Page 5-9 The variable VX A variable called VX exists in the calculator’s {HOME CASDIR} directory that takes, by default, the value of ‘X’. This i

Page 5-10 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 fu

Page 5-11 The PROPFRAC function The function PROPFRAC converts a rational fraction into a “proper” fraction, i.e., an integer part added to a fracti

Page TOC-6 Function TRACE, 9-7 Solution of linear systems, 9-7 Using the numerical solver for linear systems, 9-8 Solution with the inve

Page 5-12 The FROOTS function The function FROOTS obtains the roots and poles of a fraction. As an example, applying function FROOTS to the result

Page 5-13 Reference Additional information, definitions, and examples of algebraic and arithmetic operations are presen

Page 6-1 Chapter 6 Solution to equations Associated with the 7 key there are two menus of equation-solving functions, the Symbolic SOLVer („Î), and

Page 6-2 Using the RPN mode, the solution is accomplished by entering the equation in the stack, followed by the variable, before entering function

Page 6-3 The following examples show the use of function SOLVE in ALG and RPN modes. [Note: not all lines will be visible when done with the exerc

Page 6-4 Function SOLVEVX The function SOLVEVX solves an equation for the default CAS variable contained in the reserved variable name VX.

Page 6-5 To use function ZEROS in RPN mode, enter first the polynomial expression, then the variable to solve for, and then function ZER

Page 6-6 Following, we present applications of items 3. Solve poly.., 5. Solve finance, and 1. Solve equation.., in that order. Appendix 1-A, in

Page 6-7 ‚í 1\‚í1@@OK@@ @SOLVE@ Solve equation The screen will show the solution as follows: Press ` to return to stack. The stack will

Page 6-8 Press ˜ to trigger the line editor to see all the coefficients. Generating an algebraic expression for the polynomial You can use the