∞ xe− xdx | HP 15c Scientific manual

250 Appendix E: A Detailed Look at f

With this number of sample points, the algorithm will calculate the same approximation for the integral of any of the functions shown. The actual integrals of the functions shown with solid lines are about the same, so the approximation will be fairly accurate if f(x) is one of these functions. However, the actual integral of the function shown with a dashed line is quite different from those of the others, so the current approximation will be rather inaccurate if f(x) is this function.

The f algorithm comes to know the general behavior of the function by sampling the function at more and more points. If a fluctuation of the function in one region is not unlike the behavior over the rest of the interval of integration, at some iteration the algorithm will likely detect the fluctuation. When this happens, the number of sample points is increased until successive iterations yield approximations that take into account the presence of the most rapid, but characteristic, fluctuations.

For example, consider the approximation of

∫0∞ xe− xdx.

Image 250

HP 15c Scientific manual ∞ xe− xdx

Contents

HP-15C Owner’s Handbook HP Part Number 00015-90001 Edition 2.4, Sep Legal Notice Introduction Contents Contents Display and Continuous Memory Program Editing Program Branching and Controls Subroutines Calculating With Complex Numbers Indirect Display ControlCalculating With Matrices Numerical Integration Appendix D a Detailed Look at Contents Appendix a Error ConditionsAppendix C Memory Allocation Appendix E a Detailed Look at f Programming Summary and Index Contents Appendix F BatteriesFunction Summary and Index Subject Index HP-15C Problem Solver Quick Look at Manual Solutions To Compute Keystrokes Display Programmed Solutions Keystrokes Display 002 003 004 005 006 007 008 009 8313 KeystrokesDisplay001-42,21,11 300.51 HP-15C a Problem Solver Part l HP-15C Fundamentals Keyboard Operation Power On and OffGetting Started Section Keying in Exponents Prefix KeysChanging Signs I O m ´ P I l F T s ? t H b Clear Keys Digit entry not terminated Display Clearing ` and −Clears only the last digit Two-Number Functions CalculationsOne-Number Functions 6532 13.0000 17 +26.0000 22.0000 5000 78.0000 Numeric Functions Number Alteration Functions One-Number Functions General Functions Time and Angle Conversions Trigonometric OperationsPressing Calculates Radians Degrees/Radians Conversions7069 40.5000 Logarithmic Functions Hyperbolic Functions Percentages Power FunctionTwo-Number Functions To Calculate Keystrokes Display Calculates 3% of $15.76 the tax Polar and Rectangular Coordinate ConversionsEnters the base number the price Polar Conversion. Pressing Keystrokes Display Automatic Memory Stack Registers Automatic Memory Stack Last X, and Data StorageAutomatic Memory Stack Stack Manipulation Always displayed Memory Stack, Last X, and Data Storage Stack Manipulation FunctionsLost Lost 22.2481 Last X Register and K287.0000 12.9000 13.9 + Calculator Functions and the Stack20.6475 Order of Entry and the v Key +15 X15 65.0000 Nested Calculations7 + 69.0000 Arithmetic Calculations With Constants 5 ‛15 Keys 1000 Keystrokes Display Growth factor000 322.5000 Storage Register OperationsStoring and Recalling Numbers 520.8750 Clearing Data Storage Registers Storage and Recall Arithmetic For storage arithmetic For recall arithmetic 24 l-0 ProblemsOverflow and Underflow 15.0000 Memory Stack, Last X, and Data Storage Probability Calculations Statistics Functions60.0000 5764 Random Number Generator270,725.0000 3422 Accumulating Statistics Registers Register Contents 20 z 61v 40 z 7.21 60 z 7.78 80 z l 20.00 40.00 60.00 80.00 Kg per hectareMetric tons per Hectare, y Σy2 Correcting Accumulated Statistics 20 w 20 z Standard Deviation Mean40.00 Standard deviation about the mean nitrogen Linear Regression31.62 Application Linear Estimation and Correlation Coefficient Statistics Functions Other Applications 70 ´j Display Control Display Continuous MemoryFixed Decimal Display 234568 Scientific Notation DisplayEngineering Notation Display 234567 Mantissa Display Round-Off ErrorSpecial Displays Annunciators 12,345.67 Error DisplayDigit Separators 12.345.6700 Continuous Memory Low-Power IndicationStatus Resetting Continuous Memory Page Part ll HP-15C Programming Creating a Program Programming BasicsMechanics Loading a Program Programming Basics ´b a Running a Program Intermediate Program Stops002 003 004 005 006 007 008 How to Enter Data 300.51 300.51 ´A Program Memory Radius, r Height, h Base Area Volume Surface Area Totals 007-44,40 002004 005 010 Or G a Program Instructions Further InformationInstruction Coding Memory Configuration Keycode 25 second row, fifth key Initial Memory Configuration 60 ´ m% 19 ´ m% Program Boundaries´ m % 19.0000 Abbreviated Key Sequences Unexpected Program Stops´bA ´b3 End of memory ¤ @ y ∕ User ModePolynomial Expressions and Horners Method LOG % 002 003 004 005 006 007 008 009 0000 Nonprogrammable Functions001-42,21,12 12,691.0000 Problems Program Editing Moving to a Line in Program Memory Deleting Program Lines ExamplesInserting Program Lines Or use Â Single-Step Operations Release Line PositionÂhold Result Insertions and Deletions Initializing Calculator Status + i n InterestPV 1 + i n 100 270 ´bA D ´4 O0 2* O1 2÷ * ´ ´ l0 l1 ´r * n Program Branching Controls Branching Conditional Tests Test Flags  n will clear flag number n Example Branching and Looping 014 010-45,20013-43,30 016-44,40 Example Flags Formula is 005-43, 4 002-43004-42,21,15 006-42,21 48.0000 Go to250.0000 10,698.3049 Looping Conditional Branching System Flags Flags 8 Program Branching and Controls Subroutine Execution SubroutinesGo To Subroutine and Return ´b.1 Subroutine Limits 003- O0 000 001- ´b9002- R 004 ´ b.4 ´b.5 Subroutine Return Nested Subroutines V and % Keys Index Register Loop Control106 Index Register Storage and Recall Indirect Program Control With the Index RegisterProgram Loop Control Index Register and Loop Control Exchanging the X-Register Index Register ArithmeticIndirect Branching With Indirect Display Format Control With Indirect Flag Control WithLoop Control With Counters I and e Nnnnn x x x y y 5 0 0 Start count at zero Count by twos Count up to Storing and Recalling Keystrokes Display Examples Register OperationsIterations 12.3456 Exchanging the X-Register Example Loop Control with eStorage Register Arithmetic 012-42, 5 Loop control number in R2−− 011- 42 013- 22 15 O Example Display Format Control64.8420 0000 50.0000 Index Register Contents Indirect Display Control Index Register and Loop Control 118 Part lll HP-15C Advanced Functions Creating the Complex Stack Complex Stack and Complex ModeCalculating With Complex Numbers 120 Complex Numbers and the Stack Deactivating Complex ModeEntering Complex Numbers ´ % hold 8.0000 release Z 8 Y 7 X Keys Stack Lift in Complex Mode Manipulating the Real and Imaginary Stacks Clearing a Complex Number Or other operation Continue with any operation − 4 v Continue with any operation ´ %hold release Entering Complex Numbers with −. The clearing functions −0000 17.0000 144.0000 Entering a Real Number Followed by another number Entering a Pure Imaginary Number ´ Continue with any operation ´ O Operations With Complex NumbersStoring and Recalling Complex Numbers L 2 ´ ¤x N o ∕ @ a + * ÷ y 0428 20007000 0491 5708 Polar and Rectangular Coordinate ConversionsComplex Results from Real Numbers ´ % hold Release1.5708 Cos θ + i sin θ = re iθ Polar + ib = ∠ θ 2981 8452+ 3.1434 2361 352.0000872.0000 4721 For Further Information Calculating With Matrices 138 Keystrokes Display Deactivates Complex Mode = A-1B 11.2887 Matrix DimensionsRunning 2496 Dimensioning a Matrix Number Rows Columns ´mA Displaying Matrix DimensionsChanging Matrix Dimensions Keystrokes l B Display Storing and Recalling Matrix Elements Storing and Recalling All Elements in Order ⎡ a Checking and Changing Matrix Elements Individually Keystrokes Display Storing a Number in All Elements of a Matrix Matrix OperationsMatrix Descriptors Result Matrix Copying a Matrix One-Matrix Operations Calculating with Matrices Scalar Operations LB b Elements of Result Matrix LA a Keystrokes Display Subtracts 1 from the elements Arithmetic OperationsLB b 2 LA a 2 Matrix Multiplication = AT B Keystrokes Display l a a Solving the Equation AX = B 86 OA 24 OA2400 8600 274 OB 233 OB 331 OB 120.32 OB 112.96 OB 151.36 OB ´Á Calculating the Residual Week Cabbage kg 186 141 215 Broccoli kg 116 Using Matrices in LU Form Calculations With Complex Matrices Storing the Elements of a Complex Matrix Then Z can be represented in the calculator by Pressing Transforms Into = ⎢ LA a Complex Transformations Between ZP and Z Inverting a Complex Matrix Multiplying Complex Matrices ´ a Keystrokes lA lB Display Displays descriptor of matrix a ´U lC LC lC lC lC lC lC lC ´U Solving the Complex Equation AX = B ZZ −1 AX = B 200.0000 170.0000 0437 03721311 1543 Calculating with Matrices Using Matrix Descriptors in the Index Register Using a Matrix Element With Register OperationsMiscellaneous Operations Involving Matrices Stack Operation for Matrix Calculations Conditional Tests on Matrix Descriptors Calculating with Matrices Using Matrix Operations in a Program Keystrokes Results Summary of Matrix Functions´m a Calculates residual in result matrix For Further Information Finding the Roots An Equation Using180 Finding the Roots of an Equation Clear program memory 002 003 ´b0001-42,21 005 006 007 Finding the Roots of an Equation Desired root 000 001-42,21,11 Keystrokes ¥´ bA 003 004 Into X-register 5000 1 e tBrings another t-value 200 t When No Root Is Found 000 001-42,21 002 003 004 005 Error Choosing Initial Estimates Label 008 009 003 004 005 007X + 8 6 x + 8 Finding the Roots of an Equation Using in a Program Restriction on the Use Memory Requirements Numerical Integration Using f194 002 003 004 4040 1416 7652 Begin subroutine with a label $ ÷ 38254401 6054 Accuracy of f ´ i ´ f 8826 7091 Using f in a Program 382 Memory Requirements Appendix a Error ConditionsError 0 Improper Mathematics Operation 205 Error 1 Improper Matrix Operation Error 2 Improper Statistics Operation Error 5 Subroutine Level Too Deep Error 3 Improper Register Number or Matrix ElementError 4 Improper Line Number or Label Call Error 6 Improper Flag Number Pr Error Power Error Stack Lift Stack Lift Last X RegisterDigit Entry Termination Appendix B Disabling Operations Enabling Operations Appendix B Stack Lift and the Last X Register Keys Stack Stack Enabled. disabled 53.1301 No stack LiftNeutral Operations Nnn Clear u ¥ Last X Register \ k + H ∆ \ h ÷ À P* q r c ‘ / N z ∕ P\ o j Appendix C Memory AllocationMemory Space Registers Appendix C Memory Allocation Memory Status W Memory ReallocationM % Function ´m% 1.0000 Whold 1 64 Restrictions on Reallocation19 ´ m Program Memory Automatic Program Memory Reallocation If executed Memory Requirements for the Advanced FunctionsTwo-Byte Program Instructions Together Appendix C Memory Allocation Appendix D Detailed Look atHow Works 220 Appendix D a Detailed Look at Accuracy of the Root X4 = 000 1718 006 007 008 009 010-43,30 011 012-43,30 013 ´ v B Interpreting Results0681 − 45 For 0 x 3x 45x 2 + Test for x rangeBranch for x ≥ End subroutine 1358 000.0000Initial estimates Possible root Appendix D a Detailed Look at 007 008 009 010 ´ b.0 001-42,21,.0 002 003 004 005Bring x-value into X-register 013 014 015 016 017 018 10 v ´ ‛ 20 Error 0000 1250 5626 Finding Several Roots Fx = xx a3 = 002 003 004 005 006 007 6667 Stores root for deflation Same initial estimatesSecond root Deflated function value Deflation for third root Limiting the Estimation Time Counting Iterations For Advanced InformationSpecifying a Tolerance Appendix E Detailed Look at fHow f Works 240 Accuracy, Uncertainty, and Calculation Time X = π1 0π cos4θ − x sinθ dθ 0000 1416 ´ i ´ f Keystrokes ´ i Display Keystrokes Display Return approximation to´ Clear u Hold ´ f ´ Clear u hold 7858 7807 Uncertainty and the Display Format Functions values for example Δx = 0.5×10−n ×10m = aδx dxb = ab 0.5×10−n + m x dx Conditions That Could Cause Incorrect Results ∞ xe− xdx 001-42,21 002- 1 003 004 005 Appendix E a Detailed Look at f Appendix E a Detailed Look at f Conditions That Prolong Calculation Time Approximation to integral Keys lower limit intoKeys upper limit into Uncertainty Appendix E a Detailed Look at f Obtaining the Current Approximation to an Integral For Advanced Information Batteries Low-Power IndicationInstalling New Batteries Batteries Appendix F Batteries Verifying Proper Operation Self-Tests 2.C 3.H Conversions Function Summary and IndexComplex Functions Digit Entry Logarithmic Exponential Functions Display ControlIndex Register Control Mantissa. Pressing Matrix Functions Mathematics146 To ZP page164 Number AlterationTo XT Stack Manipulation PercentageProbability Clear u Statistics Storage Trigonometry Programming Summary and Index 269 Programming Summary and Index Subject Index 271 Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Subject Index Federal Communications Commission Notice Product Regulatory Environment InformationModifications Canadian Notice Avis Canadien European Union Regulatory Notice Body number is inserted between CE