HP 49g+ Graphing Calculator Manual do Utilizador

hp 49g+ graphing calculator user’s manual H Edition 2 HP part number F2228-90001

Page TOC-7 Chapter 14 – Differential Equations, 14-1 The CALC/DIFF menu, 14-1 Solution to linear and non-linear equations, 14-1 Function LDEC

Page 6-10 Then, enter the SOLVE environment and select Solve equation…, by using: ‚Ï@@OK@@. The corresponding screen will be shown as: The eq

Page 6-11 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 c

Page 6-12 by MSLV is numerical, the information in the upper left corner shows the results of the iterative process used to obtain a solution. 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 8-1 Chapter 8 Vectors This Chapter provides examples of entering and operating with vectors, both mathematical vectors of many elements, as wel

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 Chapter 18 – U

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 vecto

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 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 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 9-4 In RPN mode, try the following eight examples: A22 ` B22`+ A22 ` B22`- A23 ` B23`+ A23 ` B23`- A32 ` B32`+ A32 ` B32`- A33 `

Page 9-5 Matrix multiplication Matrix multiplication is defined by Cm×n = Am×p⋅Bp×n. Notice that matrix multiplication is only possible if the numbe

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: mnn

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

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

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 pl

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 ens

Page 10-5 •• The @ZOOM key, when pressed, produces a menu with the options: In, Out, Decimal, Integer, and Trig. Try the following exercises:

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

Page 10-7 • When done, press @EXIT. • Press @CANCL to return to the PLOT WINDOW environment. • Change the Step data to read: Step I

Page 10-8 • Press LL@)PICT to leave the EDIT environment. • Press @CANCL 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 where the limit is to be calculated. Function lim is available through the command catalog (‚N~„l) or through option 2. LIMITS & SER

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 repres

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

Page 11-5 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 a

Page 1-3 For details on the meaning of these specifications see Chapter 2 in the calculator’s user’s guide. The second line shows the characters

Page 11-6 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 returne

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 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

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 The solution is: which is equivalent to y = K1⋅e–3x + K2⋅e5x + K3⋅e2x + (450⋅x2+330⋅x+241)/13500. Function DESOLVE The calculator

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

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 1-4 and Chapter 2 and Appendix L in the user’s guide for more information on editing) @VIEW B VIEW the contents of a variable @@ RCL @

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 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 chan

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) ≈

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 The MTH/PROB menu - part 2 In this section we discuss four continuous probability distributions that are commonly used for problems relat

Page 15-4 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. The

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

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 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 16-4 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 4. Z-INT: p1− p2.: Confidence interval for the difference of two proportions, p1-p2, for large samples with unknown population variances.

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 =

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 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 @HEX

Page 1-6 ~p ALPHA function, to enter the upper-case letter P ~„p ALPHA-Left-Shift function, to enter the lower-case letter p ~…p ALPHA-Right-Shi

Page 18-1 Chapter 18 Using SD cards The calculator provides a memory card port where you can insert an SD flash card for backing up calculator obje

Page 18-2 Enter object, type the name of the stored object using port 3 (e.g., :3:VAR1), press K. Recalling an object from the SD card To recall an

Page W-1 Limited Warranty hp 49g+ graphing calculator; Warranty period: 12 months 1. HP warrants to you, the end-user customer, that HP hardware, a

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 +39-02-75419782 (Italian) Turkey +420-5-41422523 UK +44-207-4580161 Czech Republic +420-5-41422523 South Africa +27-11-2376200

Page W-4 Regulatory information This section contains information that shows how the hp 49g+ graphing calculator complies with regulations in certai

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

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 `ag

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 W

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 (12 significant digits).”To learn more about reals, see Chapter 2 in this guide. To illustrate this and other number formats try the fol

Page 1-12 Press the !!@@OK#@ soft menu key to complete the selection: Press the !!@@OK#@ soft menu key return to the calculator d

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 CA

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 intege

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 cha

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 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 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 2-8 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~~math` ³~~m„a„t„h` ³~~m„~at„h` The calculator display

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 H\@@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

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 A

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 2-17 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 co

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

Page 2-18 this exercise, we use the ORDER command to reorder variables in a directory, we use, in ALG mode: „°˜ Show PROG menu list and select M

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

Page 2-20 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(@O

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 i

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 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

Page TOC-3 Unit conversions, 3-14 Physical constants in the calculator, 3-14 Defining and using functions, 3-16 Reference, 3-18 Chapter 4 –

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

Page 3-9 Option 1. Tools.. contains functions used to operate on units (discussed later). Options 2. Length.. through 17.Viscosity.. co

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 !²STK, then, press @QUIT@. For the calculator set to the ALG,

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 TOC-4 The PARTFRAC function, 5-11 The FCOEF function, 5-11 The FROOTS function, 5-12 Step-by-step operations with polynomials and

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 Examples of applications of these functions are shown next in RECT coordinates. Recall that, for ALG mode, the function must precede the a

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). If the calculator is in APPROX mode, the result will be Y = 5.*(X-5.)-3. Referenc

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

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

Page TOC-5 Chapter 8 – Vectors, 8-1 Entering vectors, 8-1 Typing vectors in the stack, 8-1 Storing vectors into variables in the stack

Page 5-3 In ALG mode, the following keystrokes will show a number of operations with the algebraics contained in variables @@A1@@ and @@A2@@ (pres

Page 5-4 Functions in the ALG menu The ALG (Algebraic) menu is available by using the keystroke sequence ‚× (associated with the 4 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 Most polynomial examples above were written using variable X. This is because a variable called VX exists in the calculat

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 5-12 The FROOTS function The function FROOTS, in the ARITHMETIC/POLYNOMIAL menu, obtains the roots and poles of a fraction. As an example, ap

Page TOC-6 Solution with the inverse matrix, 9-10 Solution by “division” of matrices, 9-10 References, 9-10 Chapter 10 – Graphics, 10-1

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 (Use Complex mode in the CAS): The screen shot shown above d

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 ZERO

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 Press ` to return to stack. The stack will show the following results in ALG mode (the same result would be shown in RPN mode):

Page 6-8 Generating an algebraic expression for the polynomial You can use the calculator to generate an algebraic expression for a polynomial give

Page 6-9 Financial calculations The calculations in item 5. Solve finance.. in the Numerical Solver (NUM.SLV) are used for calculations of time valu