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This chapter gives you instructions for solving for unknowns or roots over a wide range of applications. Appendix D contains more detailed information about how the algorithm for SOLVE works, how to interpret results, what happens when no solution is found, and conditions that can cause incorrect results.

Solving Equations 7–11

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HP 33s Scientific manual For More Information

Contents

Hp 33s scientific calculator Printing History Contents Clearing the X-Register RPN The Automatic Memory Stack Storing Data into Variables How You Can Use Equations Summary of Equation Operations Fractions in Programs Solving an Equation Complex Stack Complex Operations ContentsFor More Information Accuracy of Integration Programming Data Input and Output 12-4 Entering a Program 12-5 12-412-6 12-7 13-2 13-113-3 Branching GTO 13-4 Part 3. Appendixes and Reference Managing Calculator Memory ContentsVector Operations 15-1 15-12 Round-Off Error Contents Resetting the Calculator Clearing MemoryDisabling Operations Neutral Operations About ALG How the Integral Is Evaluated UnderflowConditions That Prolong Calculation Time Page Basic Operation PartPage Getting Started Adjusting Display ContrastImportant Preliminaries Turning the Calculator On and Off Shifted Keys Highlights of the Keyboard and Display2Getting Started Cursor Keys Alpha Keys Backspacing and Clearing Silver Paint Keys4Getting Started Are described in Using Menus on page 1-7. If Keyboard-entry modeEquation-entry mode ¾ when number entry is complete 6Getting Started HP 33s Menus Chapter Name Using MenusProgramming Instructions Numeric Functions 8Getting Started Other functionsDescription To use a menu function KeysDisplay Exiting Menus To select RPN mode RPN mode ALG modeTo select ALG mode RPN and ALG Keys First Line Second Line Display and Annunciators 12Getting Started HP 33s Annunciators Meaning Chapter2 3 RAD or §,¨ Making Numbers Negative Keying in NumbersExponents of Ten Keys Display Description Keying in Exponents of Ten Understanding Digit EntryOther Exponent Functions Range of Numbers and Overflow Doing ArithmeticKeys Display 16Getting Started Two-Number Functions One-Number Functions, \, , Qor T Key in the number. You dont need to press ‘ Periods and Commas in Numbers Controlling the Display Format‘3 z D, , \, ,Q Fixed-Decimal Format % Number of Decimal PlacesScientific Format 20Getting Started SHOWing Full 12-Digit PrecisionEngineering Format ALL Format Entering Fractions FractionsKeysDisplayDescription 22Getting Started Displaying Fractions MessagesDisplayDescription Checking Available Memory Calculator MemoryClearing All of Memory 24Getting Started What the Stack Is RPN The Automatic Memory StackRPN The Automatic Memory Stack Clearing the X-Register X and Y-Registers are in the Display2RPN The Automatic Memory Stack ¶ Roll Down Reviewing the StackRµ Roll Up Exchanging the X- and Y-Registers in the Stack Arithmetic How the Stack Does It4RPN The Automatic Memory Stack Lost How Enter Works How Clear x Works Using a Number Twice in a Row6RPN The Automatic Memory Stack Filling the stack with a constant Last X Register , \, , Qor T Correcting Mistakes with Last8RPN The Automatic Memory Stack Wrong One-Number Function › Reusing Numbers with LastMistake 96.704 + 10RPN The Automatic Memory Stack Work from the Parentheses Out Chain Calculations in RPN mode 12RPN The Automatic Memory Stack Exercises SolutionOrder of Calculation 16.3805x5 = 181.0000 14RPN The Automatic Memory Stack More Exercises‘14 ‘ ⋅ 0.8 ⎟ 12.5 − 0.73 = Page Storing Data into Variables Storing Data into Variables 2Storing Data into Variables Storing and Recalling NumbersTo recall a copy of a number from a variable to the display Example Storing Numbers Reviewing Variables in the VAR Catalog Viewing a Variable without Recalling ItTo review the values at any or all non-zero variables Arithmetic with Stored Variables Clearing VariablesStorage Arithmetic 4Storing Data into Variables Result 3 That is, x Recall Arithmetic LzE Exchanging x with Any Variable6Storing Data into Variables Variable Page Exponential and Logarithmic Functions Real-Number FunctionsReal-Number Functions Quotient and Remainder of Division Power FunctionsTo Calculate Press 2Real-Number Functions Entering π TrigonometryTo Calculate Press Result Trigonometric Functions Setting the Angular ModeOption Description Annunciator 4Real-Number Functions Programming Note NzR Percentage Functions Hyperbolic Functions6Real-Number Functions ‘x Q 16.12 ‘ 15.76 T . Const Menu Items Description Value Physics Constants8Real-Number Functions Items Description Value Conversion FunctionsTo insert a constant 10Real-Number Functions Coordinate ConversionsTo convert between rectangular and polar coordinates Example Conversion with Vectors Example Polar to Rectangular Conversion 12Real-Number Functions Time ConversionsTo convert between decimal fractions and minutes-seconds Example Converting Time Formats Unit Conversions Angle ConversionsTo Convert Press Displayed Results To convert an angle between degrees and radians Factorial Probability FunctionsGamma Probability Example Combinations of People KeysDisplay 16Real-Number Functions Parts of Numbers Names of Functions Page Entering Fractions Fractions Display Rules Fractions in the Display2Fractions Entered Value Internal Value Displayed Fraction Accuracy Indicators Longer Fractions Changing the Fraction Display4Fractions Choosing a Fraction Format Setting the Maximum Denominator To Get This Fraction Format Change These Flags Examples of Fraction DisplaysFraction How 2.77 Is Displayed Format 6Fractions Fraction Number Entered and Fraction Displayed Format ¼ Rounding FractionsDenominator Fixed 8Fractions Fractions in Equations Fractions in Programs Page How You Can Use Equations Entering and Evaluating EquationsEntering and Evaluating Equations Example Calculating with an Equation ZLL ZNz2Entering and Evaluating Equations Summary of Equation Operations Enters and leaves Equation modeKey Operation Variables in Equations Entering Equations into the Equation List4Entering and Evaluating Equations To enter an equation Functions in Equations Numbers in Equations Parentheses in Equations Displaying and Selecting EquationsLR d ZLC z ——— Editing and Clearing EquationsTo view a long equation To select an equation 8Entering and Evaluating Equations Evaluating Equations Types of Equations 10Entering and Evaluating Equations Type of Equation Result for ‘Fx ¼ To evaluate an equation Example Evaluating an Equation with Enter Using Enter for Evaluation Responding to Equation Prompts Using XEQ for Evaluation12Entering and Evaluating Equations Example Evaluating an Equation with XEQ Operator Precedence Syntax of Equations‘5 g „ To display digits hidden by the prompt, press Equations Meaning Order Operation Example14Entering and Evaluating Equations Σxy Equation Functions 16Entering and Evaluating Equations Operation Equation functionExample Perimeter of a Trapezoid Perimeter = a + b + h Cos π / n HLA ’.25 zLN zLD 2 zR NqLN `qONqLN `‘Example Area of a Polygon Area = Verifying Equations Syntax Errors18Entering and Evaluating Equations Example Checksum and Length of an Equation Solving an Equation Solving EquationsSolving Equations Markup ⋅ Cost = Price ZLT › LD dLV2Solving Equations Example Solving the Equation of Linear Motion Value ZL T HL P z4Solving Equations Example Solving the Ideal Gas Law Equation Understanding and Controlling Solve LV q 6Solving Equations Verifying the Result Choosing Initial Guesses for Solve Interrupting a Solve Calculation 8Solving Equations Example Using Guesses to Find a Root H 40 ZL H 10Solving Equations For More Information Page Integrating Equations Integrating Equations J0 x = π1 ≥0π cos x sin t dt Integrating Equations ≥ FN2Integrating Equations Value ≥ 0π f t Example Sine Integral 4Integrating Equations= ≥ t Accuracy of Integration Interpreting Accuracy Specifying Accuracy6Integrating Equations Example Specifying Accuracy Example Changing the Accuracy Its uncertainty indicates 8Integrating Equations Complex Stack Operations with Complex Numbers+ iy To enter a complex number 2Operations with Complex Numbers Complex OperationsTo do an operation with one complex number To do an arithmetic operation with two complex numbers Functions for One Complex Number, z To Calculate Press 2 5 ‘ ‘2 ‘ ‘4 4Operations with Complex Numbers 2 3 ‘ Using Complex Numbers in Polar Notation‘2 Example Vector Addition 6Operations with Complex Numbers Operations with Complex Numbers Page Base Menu Base Conversions and ArithmeticMenu label Description Base Conversions and Arithmetic 10-2Base Conversions and Arithmetic Arithmetic in Bases 2, 8, Keys Display Negative Numbers Representation of Numbers˜˜ 10-4Base Conversions and Arithmetic 7FFFFFFFF Range of Numbers 10-6Base Conversions and Arithmetic Windows for Long Binary Numbers Entering Statistical Data Statistical OperationsStatistical Operations Entering One-Variable Data Correcting Errors in Data EntryEntering Two-Variable Data 11-2Statistical Operations 400 Initial x, y Corrected x, y‘20 Mean Statistical CalculationsStatistics Menus Key Description 11-4Statistical Operations · Example Weighted Mean Two VariablesPrice per Part Number of Parts y Population Standard Deviation Sample Standard DeviationUº 11-6Statistical Operations σº Linear RegressionKeys DisplayDescription Linear Regression Menu Example Curve Fitting 11-8Statistical OperationsNitrogen Applied Grain Yield 70, y Limitations on Precision of Data Summation Statistics Summation Values and the Statistics Registers11-10Statistical Operations Normalizing Close, Large Numbers Statistics Registers in Calculator Memory Access to the Statistics RegistersExample Viewing the Statistics Registers # ™ ´º¸/ ™ ´¸/ ™ ´º/ ™ ´¸/ ™ ´º/ ‡ 11-12Statistical Operations Programming Page Simple Programming RPN mode 5 !Nz ALG mode 5 !zN‘Simple Programming Example a Simple Program RPN mode RPN mode ALG modeDisplay Description 12-2Simple Programming Designing a Program Selecting a ModeProgram Boundaries LBL and RTN Program Labels Data Input and Output Using RPN, ALG and Equations in ProgramsStrengths of RPN Operations ALG Operations To enter a program into memory Entering a Program 12-6Simple Programming Keys That ClearTo enter an equation in a program line Function Names in Programs RPN modeExample Entering a Labeled Program ZLR HN 12-8Simple Programming Example Entering a Program with an Equation Running a Program Testing a ProgramExecuting a Program XEQ NzX 12-10Simple Programming Example Testing a Program Entering and Displaying Data Using Input for Entering Data@ To use the Input function in a program 12-12Simple Programming To respond to a prompt Using View for Displaying Data 12-14Simple Programming Using Equations to Display MessagesExample INPUT, VIEW, and Messages in a Program LR z › LH NzLR ›p = 2π R2 + 2π RH = 2π R R + H 12-16Simple Programming Displaying Information without Stopping Stopping or Interrupting a Program Error StopsProgramming a Stop or Pause STOP, PSE Interrupting a Running Program 12-18Simple Programming Editing a ProgramTo delete a program line To insert a program line Viewing Program Memory Program Memory Catalog of Programs MEM Memory UsageClearing One or More Programs 12-20Simple Programming To clear all programs from memory Checksum Nonprogrammable Functions Selecting a Base Mode in a ProgramProgramming with Base V Polynomial Expressions and Horners Method Decimal mode set Hexadecimal mode setNumbers Entered in Program Lines Ax4 + Bx3 + Cx2 + Dx + E ‰X Keys Display ALG mode12-24Simple Programming Checksum and length E41A Page Routines in Programs Programming TechniquesProgramming Techniques 13-2Programming Techniques Calling Subroutines XEQ, RTN Example a Nested Subroutine Nested Subroutines 13-4Programming Techniques Branching GTOA2 + B2 Programmed GTO Instruction Using GTO from the Keyboard 13-6Programming Techniques Conditional Instructions Test Menus Tests of Comparison x?y, x?0 Flags Program Lines Description RPN mode13-8Programming Techniques !- % ! ! ! º¸@ ! Fraction-Control Flags Status Clear FlagSet 13-10Programming Techniques Flags Menu Menu Key Description Using FlagsAnnunciators for Set Flags 13-12Programming Techniques Example Using Flags Program Lines Description RPN mode$ @ $ $ @ $ Example Controlling the Fraction Display 13-14Programming Techniques Program Lines Description 13-16Programming Techniques Loops ! ! Conditional Loops GTO . ! º6¸@ ! #$ ! 13-18Programming Techniques Loops with Counters DSE, ISGLoop-Control Number $ ! $ $ $ Variable Indirectly Addressing Variables and Labels13-20Programming Techniques If i contains Then i will address Indirect Address 13-22Programming Techniques Program Control withExample Choosing Subroutines With ! L Program lines If i holds Then XEQi calls- L & %1L2 Example Loop Control With 13-24Programming Techniques Equations with L 1L2 Page Solving a Program Solving and Integrating ProgramsSolving and Integrating Programs To solve a programmed function 14-2Solving and Integrating Programs Example Program Using ALGV= N x R x T Pressure ⋅ volume. Pressure ⋅ volume LP z LV ’ LN z LR z LT ‘ Example Program Using Equation14-4Solving and Integrating Programs @ @ ‘10 !@ # / 14-6Solving and Integrating Programs Using Solve in a Program/ label #variable Example Solve in a Program Integrating a Program Program Lines In RPN modeTo integrate a programmed function To write a program for ≥ FN 14-8Solving and Integrating Programs Sin Using Integration in a Program/ label ≥ G variable 14-10Solving and Integrating ProgramsExample ≥ FN in a Program Restrictions on Solving and Integrating Page Vector Operations Mathematics ProgramsMathematics Programs = R cosP = arctan = XU + YV + ZW 15-2Mathematics ProgramsV1 = X i + Y j + Z k V2=U i + V j + W k Program Listing X2 + Y2 + Z2 and P 15-4Mathematics Programs Program Lines Description ALG mode Goes to the vector-addition routine 15-6Mathematics Programs Remarks Flags UsedProgram Instructions Variables Used 15-8Mathematics Programs XR @ 15-10Mathematics Programs 07 g !@ 125 g @ 15-12Mathematics Programs Solutions of Simultaneous EquationsAX + DY + GZ = J BX + EY + HZ = K CX + FY + IZ = L % 15-14Mathematics Programs Mathematics Programs 15-16Mathematics Programs Flags Used Scratch 15-18Mathematics Programs @ 15-20Mathematics Programs Polynomial Root Finder@ @ @ X2 + J + Lx + K + M = Where J = a3/2 = y0 /2X2 + J Lx + K M = ⋅ the sign of JK a1/2 15-22Mathematics Programs Polynomial ! 15-24Mathematics Programs B0 =a04a0 a32 a12 Checksum and length C7A6B2= -a2 B1 = a3a1 4a0 JK. a1 15-26Mathematics Programs + L + M 15-28Mathematics Programs Order Terms and Coefficients 15-30Mathematics Programs %/ 15-32Mathematics Programs Coordinate Transformations Old coordinate System 15-34Mathematics Programs Checksum and length 8C82 Remark 15-36Mathematics Programs Ÿ @ 15-38Mathematics Programs Curve Fitting Statistics ProgramsStatistics Programs 16-2Statistics Programs = B + MIn Description 16-4Statistics Programs #$ #$ 16-6Statistics Programs 16-8Statistics Programs %@ 37.9 g &@ XS%@ 40.5 g &@ value 104.5 g %@ 38.6 g &@ / 16-10Statistics Programs/ / %@ Normal and Inverse-Normal Distributions Logarithmic Exponential Power 16-12Statistics Programs Function ª º ª -+. H % ! 16-14Statistics Programs Inverse program Variable used temporarily to pass the value S ⋅ 2 π toXS@ 16-16Statistics Programs 2f − Grouped Standard Deviation 16-18Statistics Programs Updates ƒxi 2fi in register Increments or decrements N 16-20Statistics Programs / Group%@ @ XU/ %@ 13 g @ / %@ 16-22Statistics Programs%@ %@ Time Value of Money Miscellaneous Programs and EquationsMiscellaneous Programs and Equations QLI ›LF z …LN `›LI 17-2Miscellaneous Programs and Equations Solve instructions šas Keys Display Description RPN mode17-4Miscellaneous Programs and Equations Retains 5750 in B calculates monthly interest rate 17-6Miscellaneous Programs and Equations Prime Number Generator LBL Y #$ 17-8Miscellaneous Programs and Equations No test is made to ensure that the input is greater than 17-10Miscellaneous Programs and Equations Appendixes and Reference Page Support, Batteries, and Service A-1 Support, Batteries ServiceCalculator Support Answers to Common Questions Environmental Limits 2Support, Batteries, and ServiceChanging the Batteries To install batteries Support, Batteries, and Service A-3 4Support, Batteries, and Service Testing Calculator OperationDo not mutilate, puncture, or dispose of batteries Fire. The batteries can burst or explode Self-Test Support, Batteries, and Service A-5™Ÿ˜—š 6Support, Batteries, and Service Warranty Support, Batteries, and Service A-7 ServiceEurope Country Telephone numbers Asia Pacific Country Telephone numbers 8Support, Batteries, and ServiceAmerica Country Telephone numbers Regulatory Information Support, Batteries, and Service A-9Canada Japan 10Support, Batteries, and Service Managing Calculator Memory User Memory and the StackUser Memory and the Stack 2User Memory and the Stack Resetting the Calculator Clearing Memory DefaultCategory Disabling Operations Status of Stack LiftNeutral Operations 4User Memory and the Stack EQN FDISP Vlabel nnnn 6User Memory and the Stack Status of the Last X Register€ θ ,r About ALG ALG SummaryALG Summary C-1 Doing Two-number Arithmetic in ALG Power FunctionsSimple Arithmetic 2ALG Summary ‘27 Q… Percentage CalculationsALG Summary C-3 Quotient and Remainder Of Division Permutations and Combinations16.12 T 4ALG Summary ALG Summary C-5 Keys Display Description 6ALG Summary Reviewing the Stack ALG Summary C-7 Coordinate Conversions Q3 z Integrating an Equation8ALG Summary ALG Summary C-9 Operations with Complex NumbersTo view the result of complex operations Complex Operations ` Q2 › 2 ›3 `10ALG Summary Ë5 ¹cº  ºy3 ÃË  ºy4 ÃË2ALG Summary C-11 12ALG Summary Initial x, y Corrected x, y ALG Summary C-13 20 Page How Solve Finds a Root More about SolvingMore about Solving D-1 Function Whose Roots Can Be Found 2More about Solving More about Solving D-3 Interpreting ResultsExample An Equation With One Root Example An Equation with Two Roots 4More about Solving / More about Solving D-5/ -%. `’1.5 LX 6More about Solving Example Discontinuous Function More about Solving D-7 − 1= 8More about Solving When Solve Cannot Find a Root Example a Relative Minimum More about Solving D-9 Example An Asymptote 10More about Solving #LX q LX ›3 ``… ⎟ x + 0.3 − 0.5 =More about Solving D-11 Example Find the root of the equation !1%ª1%-22 12More about SolvingExample a Local Flat Region More about Solving D-13 Round-Off Error 14More about Solving Underflow How the Integral Is Evaluated More about IntegrationMore about Integration 2More about Integration Conditions That Could Cause Incorrect Results ≥0∞ xe −x dx LX ` LX z‡ 4More about Integration More about Integration 6More about Integration Conditions That Prolong Calculation Time . 8More about Integration More about Integration Page Messages F-1 Messages Or program. This might take a while 2Messages # variable Messages F-3 Self-Test Messages 4Messages Name Keys and Description Operation IndexOperation Index G-1 j ˜or —h 2Operation Index Polar to rectangular coordinates Operation Index G-3 4Operation Index P Operation Index G-5 M GComplex Change sign 6Operation Index›Complex …Complex Subtraction Operation Index G-7 RHyperbolic 8Operation IndexVRadians to degrees DSE variable Mvariable Natural exponential Operation Index G-9Wlabel ?Fractional part GTO label Vlabel 10Operation Indexlabel Nnnn Operation Index G-11 ”I”Input variable ‰variable Integer part ISG variable Lvariable LBL label •label 12Operation IndexNatural logarithm Common logarithm Permutations of n items Operation Index G-13FPause WDegrees to radians 14Operation Index Operation Index G-15 16Operation Index Operation Index G-17 18Operation Index ≠0? 13-7 Operation Index G-19If x=0, skips the next program Line ≤0? 13-7 11-10 Given an x-value 20Operation Index Power Returns y raised to the xth power Index-1 Index Index-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Index-11 RPN Index-12 Index-13 View Index-14 Index-15