Integrating Equations

Integrating Equations ( FN).. 82 Accuracy of Integration... 86 Specifying Accuracy.. 86 Interpreting Accuracy.. 87
For More Information.. 89
Operations with Comb Numbers
The Complex Stack... 91 Complex Operations.. 93 Using Complex Number in Polar Notation.. 96
10. Base Conversions and Arithmetic
Arithmetic in Bases 2, 8, and 16. 102 The Representation of Numbers.. 104 Negative Numbers.. 104 Range of Numbers.. 105 Windows for Long Binary Numbers.. 106 SHOWing Partially Hidden Numbers. 106
11. Statistical Operations
Entering Statistical Data.. 111 Entering OneVariable Data. 112 Entering TwoVariable Data.. 112 Correcting Errors in Data Entry.. 113 Statistical Calculations.. 114 Mean.. 114 Sample Standard Deviation.. 116 Population Standard Deviation.. 117 Linear regression.. 117 Limitations on Precision of Data. 1110 Summation Values and the Statistics Registers. Contents
Summation Statistics... 1111 The Statistics Registers in Calculator Memory. 1112 Access to the Statistics Registers.. 1113

Part 2.

Programming

12. Simple Programming

Designing a Program... 122 Program Boundaries (LBL and RTN).. 123 Using RPN and Equations in Programs.. 124 Data Input and Output.. 124 Entering a Program.. 125 Keys That Clear.. 126 Function Names in Programs. 127 Running a Program.. 128 Executing a Program (XEQ).. 129 Testing a Program... 129 Entering and Displaying Data.. 1211 Using INPUT for Entering Data. 1211 Using VIEW for Displaying Data.. 1214 Using Equations to Display Messages.. 1214 Displaying Information without Stopping. 1217 Stopping or Interrupting a Program. 1218 Programming a Stop or Pause (STOP, PSE). 1218 Interrupting a Running Program.. 1218 Error Stops... 1218 Editing Program.. 1219
Program Memory.. 1220 Viewing Program Memory.. 1220 Memory Usage.. 1220 The Catalog of Programs (MEM).. 1221 Clearing One or More Programs.. 1222 The Checksum.. 1222 Nonprogrammable Functions.. 1223 Programming with BASE.. 1223 Selecting a Base Mode in a Program.. 1224 Numbers Entered in Program Lines. 1224 Polynomial Expressions and Horner's Method. 1225
13. Programming Techniques
Routines in Programs.. 131 Calling Subroutines (XEQ, RTN).. 132 Nested Subroutines.. 133 Branching (GTO)... 135 A Programmed GTO Instruction.. 135 Using GTO from the Keyboard.. 136 Conditional Instructions.. 137 Tests of Comparison (x?y, x?0).. 138 Flags... 139 Loops... 1316 Conditional Loops (GTO).. 1316 Loops With Counters (DSE, ISG).. 1317 Indirectly Addressing Variables and Labels.. 1320 The Variable "i"... 1320

Keying in Numbers

You can key in a number that has up to 12 digits plus a 3digit exponent up to 499. If you try to key in a number larger than this, digit entry halts and the annunciator briefly appears. If you make a mistake while keying in a number, press to backspace and delete the last digit, or press to clear the whole number.

Making Numbers Negative

The key changes the sign of a number. To key in a negative number, type the number, then press. To change the sign of a number that was entered previously, just press. (If the number has an exponent, affects only the mantissa the nonexponent part of the number.)

Exponent of Ten

Exponents in the Display Numbers with exponents of ten (such as 4.are displayed with an preceding the exponent (such as ). A number whose magnitude is too large or too small for the display format will automatically be displayed in exponential form. For example, in FIX 4 format for four decimal places, observe the effect of the following keystrokes:

.000062

Shows number being entered. Rounds number to fit the display format.

.000042

Automatically uses scientific
notation because otherwise no
significant digits would appear.
Keying in Exponents of Ten Use (exponent) to key in numbers multiplied by powers of ten. For example, take Planck's constant, 6.6262 1034: 1. Key in the mantissa (the nonexponent part) of the number. If the mantissa is negative, press after keying in its digits.

6.6262 2. Press _

. Notice that the cursor moves behind the : _
3. Key in the exponent. (The largest possible exponent is 499.) If the exponent is negative, press after you key in the E or after you key in the value of the exponent: 34 _ 34. The
For a power of ten without a multiplier, such as 1034, just press calculator displays. Other Exponent Functions
To calculate an exponent of ten (the base 10 antilogarithm), use. To calculate the result of any number raised to a power (exponentiation), use (see chapter 4).
Understanding Digit Entry
As you key in a number, the cursor (_) appears in the display. The cursor shows you where the next digit will go; it therefore indicates that the number is not complete.
Digit entry not terminated: the number is not complete.
If you execute a function to calculate a result, the cursor disappears because the number is complete digit entry has been terminated.

Displays the current value of D.
Clears the VIEW display; displays X-register again. Suppose the variables D, E, and F contain the values 2, 3, and 4 from the last example. Divide 3 by D, multiply it by E, and add F to the result.

3 E F D

Calculates 3 D. 3 D E. 3DE+F
Exchanging x with Any Variable
The key allows yon to exchange the contents of (the Displayed X register with 1 contents of any variable. Executing this function does not effect the Y, Z, or Tregisters

Example:

A A A _

Display x.

Stores 12 in variable A. Exchange contents of the Xregister and variable A. Exchange contents of the Xregister and variable A.

t z y 12

The Variable "i"
There is a 27th variables that you can access directlythe variable i. The key is labeled "i", and it means i whenever the A.Z annunciator is on. Although it stores numbers as other variables do, i is special in that it can be used to refer to other variables, including the statistics registers, using the (i) function. This is a programming technique called indirect addressing that is covered under "Indirectly Addressing variables and labels" in chapter 13.
This chapter covers most of the calculator's functions that perform computations on real numbers, including some numeric functions used in programs (such as ABS, the absolutevalue function): Exponential and logarithmic functions. Power functions. ( Hyperbolic functions. Percentage functions. Conversion functions for coordinates, angles, and units. Probability functions. Parts of numbers (numberaltering functions). Arithmetic functions and calculations were covered in chapters 1 and 2. Advanced numeric operations (rootfinding, integrating, complex numbers, base conversions, and statistics) are described in later chapters. All the numeric functions are on keys except for the probability and partsofnumbers functions. The probability functions ( (press [PROB]). Thepartsof numbers functions( [PARS]). , , , , and, , and ) are in the PROB menu ) are in PARTS menu (press and ) Trigonometric functions.
Exponential and Logarithmic Functions

Names of Function

You might have noticed that the name of a function appears in the display when you press and hold the key to execute it. (The name remains displayed for as long as you hold the key down.) For instance, while pressing , the display shows. "SQRT" is the name of the function as it will appear in program lines (and usually in equations also).
"Fractions" in chapter 1 introduces the basics about entering, displaying, and calculating with fractions: To enter a fraction, press twiceafter the integer part, and between the numerator and denominator. To enter 2 3/8, press 8. To 5/8, press enter or 5 8. To turn Fractiondisplay mode on and off, press. When you turn off Fractiondisplay mode, the display goes back to the previous display format. (FIX, SCI, ENG, and ALL also turn off Fractiondisplay mode.) Functions work the same with fractions as with decimal numbersexcept for RND, which is discussed later in this chapter. This chapter gives more information about using and displaying fractions.
You can type almost any number as a fraction on the keyboard including an improper fraction (where the numerator is larger than the denominator). However, the calculator displays if you disregard these two restrictions. The integer and numerator must not contain more than 12 digits total. The denominator must not contain more than 4 digits.
Turns on Fractiondisplay mode. Enters 1.5; shown as a fraction. Enters 1 3/4. Displays x as a decimal number. Displays x as a fraction.
If you didn't get the same results as the example, you may have accidentally changed how fractions are displayed. (See "Changing the Fraction Display" later in this chapter.) The next topic includes more examples of valid and invalid input fractions. You can type fractions only if the number base is 10 the normal number base. See chapter 10 for information about changing the number base.

Fractions in the Display

In Fractiondisplay mode, numbers are evaluated internally as decimal numbers, then they're displayed using the most precise fractions allowed. In addition, accuracy annunciators show the direction of any inaccuracy of the fraction compared to its 12digit decimal value. (Most statistics registers are exceptions they're always shown as decimal numbers.)

Display Rules

The fraction you see may differ from the one you enter. In its default condition, the calculator displays a fractional number according to the following rules. (To change the rules, see "Changing the Fraction Display" later in this chapter.) The number has an integer part and, if necessary, a proper fraction (the numerator is less than the denominator).
The denominator is no greater than 4095. The fraction is reduced as far as possible.

Examples:

These are examples of entered values and the resulting displays. For comparison, the internal 12digit values are also shown. The and annunciators in the last column are explained below.

14 A A.

Calculates e14. Shows all decimal digits. Stores value in A. Views A. Clears x.
Changing the Fraction Display
In its default condition, the calculator displays a fractional number according to certain rules. (See "Display Rules" earlier in this chapter.) However, you can change the rules according to how you want fractions displayed: You can set the maximum denominator that's used. You can select one of three fraction formats. The next few topics show how to change the fraction display.
Setting the Maximum Denominator
For any fraction, the denominator is selected based on a value stored in the calculator. If you think of fractions as a b/c, then /c corresponds to the value that controls the denominator. The /c value defines only the maximum denominator used in Fractiondisplay mode the specific denominator that's used is determined by the fraction format (discussed in the next topic). To set the /c value, press n , where n is the maximum denominator you want. n can't exceed 4095. This also turns on Fraction display mode. To recall the /c value to the Xregister, press 1 To restore the default value or 4095, press 0. (You also restore
the default if you use 4095 or greater.) This also turns on Fractiondisplay mode. The /c function uses the absolute value of the integer part of the number in the Xregister. It doesn't change the value in the LAST X register.

Choosing Fraction Format

The calculator has three fraction formats. Regardless of the format, the displayed fractions are always the closest fractions within the rules for that format. Most precise fractions. Fractions have any denominator up to the /c value, and they're reduced as much as possible. For example, if you're studying math concepts with fractions, you might want any denominator to be possible (/c value is 4095). This is the default fraction format. Factors of denominator. Fractions have only denominators that are factors of the /c value, and they're reduced as much as possible. For example, if you're calculating stock prices, you might want to see and (/c value is 8). Or if the /c value is 12, possible denominators are 2, 3, 4, 6, and 12. Fixed denominator. Fractions always use the /c value as the denominatorthey're not reduced. For example, if you're working with time measurements, you might want to see (/c value is 60). To select a fraction format, you must change the states of two flags. Each flag can be "set" or "clear," and in one case the state of flag 9 doesn't matter.
To Get This Fraction Format:

Change These Flags: 8 9

Clear Set

Clears the statistics registers. Stores the first data pair (1,2). Stores the second data pair (3,4). Displays VAR catalog and views

xy register.

Views y2 register. Views x2 register. Views y register. Views x register. Views n register. 2.0000 Leaves VAR, catalog.
The Statistics Registers in Calculator Memory
The memory space (48 bytes) for the statistics registers is automatically allocated (if it doesn't already exist) when you press or. The registers are deleted and the memory deallocated when you execute { }.
1112 Statistical Operations
If not enough calculator memory is available to hold the statistics registers when you first press (or ), the calculator displays. You will rived to clear variables, equations, or programs (or a combination) to make room for the statistics registers before you can enter statistical data. Refer to "Managing Calculator Memory" in appendix B.
Access to the Statistics Registers
The statistics register assignments in the HP 32SII are shown in the following table.
Statistics Registers Register

n x y x2 y2 xy

Number
Number of accumulated data pairs. Sum of accumulated xvalues. Sum of accumulated yvalues. Sum of squares of accumulated xvalues. Sum of squares of accumulated yvalues. Sum of products of accumulated xand yvalues.
You can load a statistics register with a summation by storing the numb r (28 through 33) of the register you want in i (number and then storing the summation (value. Similarly, you can press to view a register valuethe display is labeled with the register name. The SUMS menu contains functions for recalling the register values. See "Indirectly Addressing Variables and Labels" in chapter 13 for more information.
Statistical Operations 1113

Part 2

Statistics Programs
File name 32sii-Manual-E-0424Page: 14/162 Printed Date : 2003/4/24 Size : 17 x 25.2 cm.7

Simple Programming

Part 1 of this manual introduced you to functions and operations that you can use manually, that is, by pressing a key for each individual operation. And you saw how you can use equations to repeat calculations without doing all of the keystrokes each time. In part 2, you'll learn how you can use programs for repetitive calculations calculations that may involve more input or output control or more intricate logic. A program lets you repeat operations and calculations in the precise manner you want. In this chapter you will learn how to program a series of operations. In the next chapter, "Programming Techniques," you will learn about subroutines and conditional instructions.
Example: A Simple Program. To find the area of a circle with a radius of 5, you would use the formula A = r2 and press 5 to get the result for this circle, 78.5398. But what if you wanted to find the area of many different circles? Rather than repeat the given keystrokes each time (varying only the "5" for the different radii), you can put the repeatable keystrokes into a program:

Numbers Entered in Program Lines
Before starting program entry, set the base mode. The current setting for the base mode determines the base of the numbers that are entered into program lines. The display of these numbers changes when you change the base mode. Program line numbers always appear in base 10. An annunciator tells you which base is the current setting. Compare the program lines below in the left and right columns. All nondecimal numbers are right justified in the calculator's display. Notice how the number 13 appears as "D" in Hexadecimal mode.

Decimal mode set:

: : PRGM

Hexadecimal mode set:

PRGM PRGM

HEX HEX

1224 Simple Programming
Polynomial Expressions and Horner's Method
Some expressions, such as polynomials, use the same variable several times for their solution. For example, the expression Ax4 + Bx3 + Cx2 + Dx + E uses the variable x four different times. A program to calculate such an expression using RPN operations could repeatedly recall a stored copy of x from a variable. A shorter RPN programming method, however, would be to use a stack which has been filled with the constant (see "Filling the Stack with a Constant" in chapter 2). Rorer's Method is a useful means of rearranging polynomial expressions to cut calculation steps and calculation time. It is especially expedient with SOLVE and FN, two relatively complex operations that use subroutines. This method involves rewriting a polynomial expression in a nested fashion to eliminate exponents greater than 1: Ax4 + 13x3 + Cx2+D x + E (Ax3 + Bx2 + Cx + D ) x + E ((Ax2 + Bx + C ) x + D )x + E (((Ax + B )x + C ) x + D )x + E
Example: Write a program using RPN operations for 5x4 + 2x3 as (((5x + 2)x)x)x, then evaluate it for x = 7.
P X Fills the stack with x.

Simple Programming 1225

5 5x. 2 5x + 2. (5x + 2)x. (5x + 2)x2. (5x + 2)x3. { } Displays label P, which takes 19.5 bytes. Checksum and length. Cancels program entry. Now evaluate this polynomial x = 7.

value Result.

Prompts for x.
A more general form of this program for any equation (((Ax + B) + C) + D) + E would be:

1524 Mathematics Programs
b2= a2. Stores b2. a3. a3 a1.
4a0. b1 = a3a1 4a0. Stores b1. To enter lines D21 and D22 Press 4 3.
Creates 7.004 as a pointer to the cubic coefficients. Solves for real root and puts a0 and a1 for secondorder polynomial on stack. Discards polynomial function value. Solves for remaining roots of cubic and stores roots. Gets real root of cubic. Stores real root. Complex roots? Calculate four roots of remaining fourthorder polynomial. If not complex roots, determine largest real root (y0)
Stores largest real root of cubic. Checksum and length: C333 060.0 Starts fourthorder solution routine. J = a3/2
Mathematics Programs 1525

K = y0/2

Creates 109 as a lower bound for M2 K K2. M2 = K2 a0. If M2 < 10 9, use 0 for M2. M = K 2 a0 Stores M. J. JK. a1. a1/2 JK a1/2. Use 1 if JK a1/2 = 0 Stores 1 or JK a1/2. Calculates sign of C. J. J2 J2 - a2. J2 - a2 +y0. C = J 2 a2 + y 0. Stores C with proper sign. J. J + L. K. K + M. Calculate and display two roots of the fourthorder
1526 Mathematics Programs
polynomial. J. J L. K. K M. Checksum and length: 9133 061.5
Starts routine to calculate and display two roots. Uses quadratic routine to calculate two roots. Checksum and length: 0019 003.0 Starts routine to display two real roots or two roots. Gets the first real root. Stores the first real root. Displays real root or real part of complex root. Gets the second real root or imaginary part of complex root. Were there any complex roots? Displays complex roots if any. Stores second real root. Displays second real root. Returns to calling routine. Checksum and length: BE87 015.0 Starts routine to display complex roots. Stores the imaginary part of the first complex root. Displays the imaginary part of the first complex root. Displays the real part of the second complex root. Gets the imaginary part of the complex roots. Generates the imaginary part of the second complex root. Stores the imaginary part of the second complex root. Displays the imaginary part of the second complex root. Checksum and length: OEE4 012.0
Mathematics Programs 1527
Flags Used: Flag 0 is used to remember if the root is real or complex (that is, to remember the sign of d). If d is negative, then flag 0 is set. Flag 0 is tested later in the program to assure that both the real and imaginary parts are displayed if necessary. Memory Required: 382.0 bytes: 268.5 for programs, 33.5 for SOLVE, 80 for variables. Remarks: The program accommodates polynomials of order 2, 3, 4, and 5. It does not check if the order you enter is valid. The program requires that the constant term a0 is nonzero for these polynomials. (If a0 is 0, then 0 is a real root. Reduce the polynomial by one order by factoring out x.) The order and the coefficients are not preserved by the program. Because of roundoff error in numerical computations, the program may produce values that are not true roots of the polynomial. The only way to confirm the roots is to evaluate the polynomial manually to see if it is zero at the roots. For a third or higherorder polynomial, if SOLVE cannot find a real root, the error is displayed. You can save time and memory by omitting routines you don't need. If you're not solving fifthorder polynomials, you can omit routine E. If you're not solving fourth or fifthorder polynomials, yoga can omit routines D, E, and F. If you're not solving third, fourth, or fifthorder polynomials, you can omit routines C, D, E, and F.

y for the exponential

Calculates y = BeMX. Returns to the calling routine. Checksum and length: AA19 009.0
This subroutine calculates x for the exponential model. Restores index value to its original value.
Calculates x = (ln (Y B)) M. Returns to the calling routine. Checksum and length: 7D3B 010.5 This subroutine calculates

y for the power model.

Calculates Y= B(XM). Returns to the calling routine. Checksum and length: 30CD 009.0 This subroutine calculates x for the power model. Restores index value to its original value.
Calculates x = (Y/B) 1/M Returns to the calling routine. Checksums and length: 7139 012.0
Flags Used: Flag 0 is set if a natural log is required of the X input. Flag 1 is set if a natural log is required of the Y input.
Memory Required: 270 bytes: 174 for program, 96 for data (statistic. registers 48). Program instructions: 1. Key in the program routines; press when done. and select the type of curve you wish to fit by pressing: 2. Press S for a straight line; L for a logarithmic curvy.; E for an exponential curve; or P for a power curve. 3. Key in xvalue and press 4. Key in yvalue and press. 5. Repeat steps 3 and 4 for each data pair. If you discover that you have made an error after you have pressed in step 3 (with the value prompt still visible), press again (displaying the value prompt) and press U to undo (remove) the last data pair. If you discover that you made an error after step 4, press U. In either case continue at step 3. 6. After all data are keyed in, press R to see the correlation coefficient, R. 7. Press to see the regression coefficient B. 8. Press to see the regression coefficient M. 9. Press to see the value prompt for the x , y estimation routine. 10. ff you wish to estimate y based on x, key in x at the value prompt, then press to see y ( ). 11. If you wish to estimate x based on y, press until you see the value prompt, key in y, then press to see x ( ). 12. For more estimations, go to step 10 or 11. 13. For a new case, go to step 2.

Statistics Programs 1613

Returns to the calling routine. Checksum and length: F79E 032.0 This subroutine calculates the integrand for the normal 2 function e (( X M)S) 2
Returns to the calling routine. Checksum and length: 3DC2 015.0
Flags Used: None. Memory Required: 155.5 bytes: 107.5 for program, 48 for variables. Remarks: The accuracy of this program is dependent on the display setting. For inputs in the rare between 3 standard deviations a display of four or more significant figures is adequate for most application. At full precision, the input limit becomes 5 standard deviations. Computation time is significantly less with a lower number of displayed digits. In routine N, the constant 0.5 may be replaced by 2 and 6.5 byte at the expense of clarity. This will save

1614 Statistics Programs

Yom do riot need to key in the inverse routine (in routines I and T) if you are not interested in the inverse capability. Program Instructions: 1. Key in the program routines; press when done. 2. Press S. (If the 3. After the prompt for M, key in the population mean and press mean is zero, just press.) 4. After the prompt for S, key in the population standard deviation and press. (If the standard deviation is 1, just press ) 5. To calculate X given Q(X), skip to step 9 of these instructions. 6. To calculate Q(X) given X, D. 7. After the prompt, key in the value of X and press. The result, Q(X), is displayed. 8. To calculate Q(X) for a new X with the same mean and standard deviation, press and go to step 7. 9. To calculate X given Q(X), press I. 10. After the prompt, key in the value of Q(X) and press. The result, X, is displayed. 11. To calculate X for a new Q(X) with the same mean and standard deviation, press and go to step 10. Variables Used: D M Q S T X Dummy variable of integration. Population mean, default value zero. Probability corresponding to the uppertail area. Population standard deviation, default value of 1. Variable used temporarily to pass the value S 2 to the inverse program. Input value that defines the left side of the uppertail area.

Statistics Programs 1615

Example 1: Your good friend informs you that your blind date has "3" intelligence. You interpret this to mean that this person is more intelligent than the local population except for people more than three standard deviations above the mean. Suppose that you intuit that the local population contains 10,000 possible blind dates. How many people fall into the "3" band? Since this problem is stated in terms of standard deviations, use the default value of zero for M and 1 for S.
Starts the initialization routine. Accepts the default value of zero for M. Accepts the default value of 1 for S.

Answers to Common Questions
Q: How can I determine if the calculator is operating properly? A: Refer to page A5, which describes the diagnostic selftest. Q. My numbers contain commas instead of periods as decimal points. How do I restore the periods? A: Use the { } function (page 114).
Q: How do l change the number of decimal places in the display? A: Use the menu (page 115).
Q; How do 1 clear all or portions of memory? A: displays the CLEAR menu, which allows you to clear all variables, all programs (in program entry only), the statistics registers, or all of user memory (not during program entry). Q: What does an "E" in a number (for example, ) mean?
A: Exponent of ten; that is, 2.51 1013. Q: The calculator has displayed the message do?. What should I
A: You must clear a portion of memory before proceeding. (See appendix B.) Q: Why does calculating the sine (or tangent) of radians display a very small number instead of 0? A: cannot be represented exactly with the 12digit precision of the calculator. Q: Why do I get incorrect answers when I use the trigonometric functions? A: You must make sure the calculator is using the correct angular mode ( { }, { }, or { } ). Q. What does the symbol in the display mean? A: This is an annuncidor, and it indicates something about the status of the calculator. See "Annunciators" in chapter 1. Q: Numbers show up as fractions. How do I get decimal numbers? A: Press.

Environmental Limits

To maintain product reliability, observe the following temperature and humidity limits: Operating temperature: 0 to 45 C (32 to 113 F). Storage temperature: 20 to 65 C (4 to 149 F). Operating and storage humidity: 90% relative humidity at 40 C (104 F) maximum.

Changing the Batteries

Replace the batteries as soon as possible when the low battery annunciator ( ) appears. If the battery annunciator is on, and the display dims, you may lose data. If data is lost, the message is displayed. Once you've removed the batteries, replace them within 2 minutes to avoid losing stored information. (Have the new batteries readily at hand before you open the battery compartment.) Use any brand of fresh I.E.C LR44 (or manufacturer's equivalent) buttoncell batteries. Equivalent 1.5volt, buttoncell batteries you might find from various manufacturers are LR44, A76, V13GA, KA76, 357, SP357, V357, and SR44W. 1. Have three fresh buttoncell batteries at hand. Avoid touching the battery terminals handle batteries only by their edges. 2. Make sure the calculator is OFF. Do not press ON ( ) again until the entire batterychanging procedure is completed. If the calculator is ON when the batteries are removed, the contents of Continuous Memory will be erased. 3. Remove the batterycompartment door by pressing down and outward on it until the door slides off (left illustration).

A-3 picture

4. Turn the calculator over and shake the batteries out. Warning Do not mutilate, puncture, or dispose of batteries in fire. The batteries can burst or explode, releasing hazardous chemicals.

Approximation of the integral.
The answer returned by the calculator is clearly incorrect, since the actual integral of f(x) = xex from zero to is exactly 1. But the problem is not that was represented by 10499, since the actual integral of this function from zero to 10499 is very close to 1. The reasons or the incorrect answer becomes apparent from the graph of f(x) over the interval of integration.
The graph is a spike very close to the origin. Because no sample point happened to discover the spike, the algorithm assumed that f(x) was identically equal to zero throughout the interval of integration. Even if you increased the number of sample points by calculating the integral in SCI 11 or ALL format, none of the additional sample points would discover the spike when this particular function is integrated over this particular interval. (For better approaches to problems such as this, see the next topic, "Conditions That Prolong Calculation Time.") Fortunately, functions exhibiting such aberrations (a fluctuation that is uncharacteristic of the behavior of the function elsewhere) are unusual enough that you are unlikely to have to integrate one unknowingly. A function that could lead to incorrect results can be identified in simple terms by how rapidly it and its loworder derivatives vary across the interval of integration. Basically, the more rapid the variation in the function or its derivatives, and the lower the order of such rapidly varying derivatives, the less quickly will the calculation finish, and the less reliable will be the resulting approximation. Note that the rapidity of variation in the function (or its loworder derivatives) must be determined with respect to the width of the interval of integration. With a given number of sample points, a function f(x) that has three
fluctuations can be better characterized by its samples when these variations are spread out over most of the interval of integration than if they are confined to only a small fraction of the interval. (These two situations are shown in the following two illustrations.) Considering the variations or fluctuation as a type of oscillation in the function, the criterion of interest is the ratio of the period of the oscillations to the width of the interval of integration: the larger this ratio, the more quickly the calculation will finish, and the more reliable will be, the resulting approximation.
Calculated integral of this function will be accurate.
Calculated integral of this function may be accurate.
In many cases you will be familiar enough with the function you want to integrate that you will know whether the function has any quick wiggles relative to the interval of integration. If you're not familiar with the function,
and you suspect that it may cause problems, you can quickly plot a few points by evaluating the function using the equation or program you wrote for that purpose. If, for any reason, after obtaining an approximation to an integral, you suspect its validity, there's a simple procedure to verify it: subdivide the interval of integration into two or more adjacent subintervals, integrate the function over each subinterval, then add the resulting approximations. This causes the function to be sampled at a brand new set of sample points, thereby more likely revealing any previously hidden spikes. If the initial approximation was valid, it will equal the sum of the approximations over the subintervals.

no stack effect, 12-15 stopping programs, 12-14 volume conversions, 4-12
warranty, A-6 weight conversions, 4-12 weighted means, 11-4 windows (binary numbers), 10-7
evaluating equations, 6-12, 6-14 running programs 12-10, 12-22 X-register affected by prompts, 6-16 arithmetic with variables, 3-5 clearing, 1-4, 2-2, 2-7 clearing in programs, 12-7 displayed, 2-2 during programs pause, 12-19 exchanging with variables, 3-8 exchanging with Y, 2-4 not clearing, 2-5 part of stack, 2-1 testing, 13-7 unaffected by VIEW, 12-15 X ROOT arguments, 6-18

Index15

Batteries are delivered with this product, when empty do not throw them away but correct as small chemical waste.
Bij dit produkt zijn batterijen. Wanneer deze leeg zijn, moet u ze niet weggooien maar inleveren aIs KCA.
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