12More about Solving, !1%ª1%-22 | HP 33s Scientific instruction

Now attempt to find a negative root by entering guesses 0 and –10. Notice that the function is undefined for values of x between 0 and –0.3 since those values produce a positive denominator but a negative numerator, causing a negative square root.

Keys:Display:Description:

0 IX
10 ^	._
H	!1%ª1%-)22
X	!

X	%/
	)

Selects Equation mode; displays the left end of the equation.

No root found for f(x).

Clears error message; cancels Equation mode.

Displays the final estimate of x.

Example: A Local "Flat" Region.

Find the root of the function f(x) = x + 2 if x < –1,

f(x) = 1 for –1 ≤ x ≤ 1 (a local flat region), f(x) = –x+ 2 if x >1.

In RPN mode, enter the function as the program:

! - % º6¸@ ! . -+. º5¸@

D–12More about Solving

Image 336

HP 33s Scientific manual 12More about Solving, !1%ª1%-22, Example a Local Flat Region

Contents

Hp 33s scientific calculator Printing History Contents RPN The Automatic Memory Stack Clearing the X-Register Storing Data into Variables Fractions in Programs How You Can Use Equations Summary of Equation Operations Complex Stack Complex Operations Contents Solving an EquationFor More Information Accuracy of Integration Programming 12-4 Data Input and Output 12-4 Entering a Program 12-512-6 12-7 13-1 13-213-3 Branching GTO 13-4 Managing Calculator Memory Contents Part 3. Appendixes and ReferenceVector Operations 15-1 15-12 Resetting the Calculator Clearing Memory Round-Off Error ContentsDisabling Operations Neutral Operations About ALG Underflow How the Integral Is EvaluatedConditions That Prolong Calculation Time Page Part Basic OperationPage Adjusting Display Contrast Getting StartedImportant Preliminaries Turning the Calculator On and Off Highlights of the Keyboard and Display Shifted Keys2Getting Started Alpha Keys Cursor Keys Silver Paint Keys Backspacing and Clearing4Getting Started Keyboard-entry mode Are described in Using Menus on page 1-7. IfEquation-entry mode ¾ when number entry is complete 6Getting Started Using Menus HP 33s Menus Chapter NameProgramming Instructions Numeric Functions Other functions 8Getting StartedDescription To use a menu function Exiting Menus KeysDisplay RPN mode ALG mode To select RPN modeTo select ALG mode RPN and ALG Keys Display and Annunciators First Line Second Line HP 33s Annunciators Meaning Chapter 12Getting Started2 3 RAD or §,¨ Keying in Numbers Making Numbers NegativeExponents of Ten Keys Display Description Understanding Digit Entry Keying in Exponents of TenOther Exponent Functions Doing Arithmetic Range of Numbers and OverflowKeys Display 16Getting Started One-Number Functions Two-Number Functions, \, , Qor T Key in the number. You dont need to press ‘ Controlling the Display Format Periods and Commas in Numbers‘3 z D, , \, ,Q Number of Decimal Places Fixed-Decimal Format %Scientific Format SHOWing Full 12-Digit Precision 20Getting StartedEngineering Format ALL Format Fractions Entering FractionsKeysDisplayDescription 22Getting Started Messages Displaying FractionsDisplayDescription Calculator Memory Checking Available MemoryClearing All of Memory 24Getting Started RPN The Automatic Memory Stack What the Stack IsRPN The Automatic Memory Stack X and Y-Registers are in the Display Clearing the X-Register2RPN The Automatic Memory Stack Reviewing the Stack ¶ Roll DownRµ Roll Up Arithmetic How the Stack Does It Exchanging the X- and Y-Registers in the Stack4RPN The Automatic Memory Stack How Enter Works Lost Using a Number Twice in a Row How Clear x Works6RPN The Automatic Memory Stack Filling the stack with a constant Last X Register Correcting Mistakes with Last , \, , Qor T8RPN The Automatic Memory Stack Wrong One-Number Function Reusing Numbers with Last ›Mistake 96.704 + 10RPN The Automatic Memory Stack Chain Calculations in RPN mode Work from the Parentheses Out 12RPN The Automatic Memory Stack Solution ExercisesOrder of Calculation 16.3805x5 = 181.0000 More Exercises 14RPN The Automatic Memory Stack‘14 ‘ ⋅ 0.8 ⎟ 12.5 − 0.73 = Page Storing Data into Variables Storing Data into Variables Storing and Recalling Numbers 2Storing Data into VariablesTo recall a copy of a number from a variable to the display Example Storing Numbers Viewing a Variable without Recalling It Reviewing Variables in the VAR CatalogTo review the values at any or all non-zero variables Clearing Variables Arithmetic with Stored VariablesStorage Arithmetic 4Storing Data into Variables Recall Arithmetic Result 3 That is, x Exchanging x with Any Variable LzE6Storing Data into Variables Variable Page Real-Number Functions Exponential and Logarithmic FunctionsReal-Number Functions Power Functions Quotient and Remainder of DivisionTo Calculate Press 2Real-Number Functions Trigonometry Entering πTo Calculate Press Result Setting the Angular Mode Trigonometric FunctionsOption Description Annunciator 4Real-Number Functions NzR Programming Note Hyperbolic Functions Percentage Functions6Real-Number Functions ‘x Q 16.12 ‘ 15.76 T . Physics Constants Const Menu Items Description Value8Real-Number Functions Conversion Functions Items Description ValueTo insert a constant Coordinate Conversions 10Real-Number FunctionsTo convert between rectangular and polar coordinates Example Polar to Rectangular Conversion Example Conversion with Vectors Time Conversions 12Real-Number FunctionsTo convert between decimal fractions and minutes-seconds Example Converting Time Formats Angle Conversions Unit ConversionsTo Convert Press Displayed Results To convert an angle between degrees and radians Probability Functions FactorialGamma Probability Keys Example Combinations of PeopleDisplay Parts of Numbers 16Real-Number Functions Names of Functions Page Fractions Entering Fractions Fractions in the Display Display Rules2Fractions Accuracy Indicators Entered Value Internal Value Displayed Fraction Changing the Fraction Display Longer Fractions4Fractions Setting the Maximum Denominator Choosing a Fraction Format Examples of Fraction Displays To Get This Fraction Format Change These FlagsFraction How 2.77 Is Displayed Format 6Fractions Rounding Fractions Fraction Number Entered and Fraction Displayed Format ¼Denominator Fixed Fractions in Equations 8Fractions Fractions in Programs Page Entering and Evaluating Equations How You Can Use EquationsEntering and Evaluating Equations Example Calculating with an Equation ZNz ZLL2Entering and Evaluating Equations Enters and leaves Equation mode Summary of Equation OperationsKey Operation Entering Equations into the Equation List Variables in Equations4Entering and Evaluating Equations To enter an equation Numbers in Equations Functions in Equations Displaying and Selecting Equations Parentheses in EquationsLR d ZLC z Editing and Clearing Equations ———To view a long equation To select an equation 8Entering and Evaluating Equations Types of Equations Evaluating Equations Type of Equation Result for ‘ 10Entering and Evaluating EquationsFx ¼ To evaluate an equation Using Enter for Evaluation Example Evaluating an Equation with Enter Using XEQ for Evaluation Responding to Equation Prompts12Entering and Evaluating Equations Example Evaluating an Equation with XEQ Syntax of Equations Operator Precedence‘5 g „ To display digits hidden by the prompt, press Order Operation Example Equations Meaning14Entering and Evaluating Equations Equation Functions Σxy Operation Equation function 16Entering and Evaluating EquationsExample Perimeter of a Trapezoid Perimeter = a + b + h HLA ’.25 zLN zLD 2 zR NqLN `qONqLN `‘ Cos π / nExample Area of a Polygon Area = Syntax Errors Verifying Equations18Entering and Evaluating Equations Example Checksum and Length of an Equation Solving Equations Solving an EquationSolving Equations Markup ⋅ Cost = Price LD dLV ZLT ›2Solving Equations Example Solving the Equation of Linear Motion Value HL P z ZL T4Solving Equations Example Solving the Ideal Gas Law Equation LV q Understanding and Controlling Solve Verifying the Result 6Solving Equations Interrupting a Solve Calculation Choosing Initial Guesses for Solve Example Using Guesses to Find a Root 8Solving Equations 40 H ZL H 10Solving Equations For More Information Page Integrating Equations Integrating Equations Integrating Equations ≥ FN J0 x = π1 ≥0π cos x sin t dt2Integrating Equations ≥ 0π f t Value 4Integrating Equations Example Sine Integral= ≥ t Accuracy of Integration Specifying Accuracy Interpreting Accuracy6Integrating Equations Example Specifying Accuracy Example Changing the Accuracy 8Integrating Equations Its uncertainty indicates Operations with Complex Numbers Complex Stack+ iy To enter a complex number Complex Operations 2Operations with Complex NumbersTo do an operation with one complex number Functions for One Complex Number, z To Calculate Press To do an arithmetic operation with two complex numbers ‘2 ‘ ‘4 2 5 ‘4Operations with Complex Numbers Using Complex Numbers in Polar Notation 2 3 ‘‘2 6Operations with Complex Numbers Example Vector Addition Operations with Complex Numbers Page Base Conversions and Arithmetic Base MenuMenu label Description Base Conversions and Arithmetic Arithmetic in Bases 2, 8, 10-2Base Conversions and Arithmetic Keys Display Representation of Numbers Negative Numbers˜˜ 10-4Base Conversions and Arithmetic Range of Numbers 7FFFFFFFF Windows for Long Binary Numbers 10-6Base Conversions and Arithmetic Statistical Operations Entering Statistical DataStatistical Operations Correcting Errors in Data Entry Entering One-Variable DataEntering Two-Variable Data 11-2Statistical Operations Initial x, y Corrected x, y 400‘20 Statistical Calculations MeanStatistics Menus Key Description 11-4Statistical Operations Example Weighted Mean Two Variables · Price per Part Number of Parts y Sample Standard Deviation Population Standard DeviationUº 11-6Statistical Operations Linear Regression σºKeys DisplayDescription Linear Regression Menu 11-8Statistical Operations Example Curve FittingNitrogen Applied Grain Yield Limitations on Precision of Data 70, y Summation Values and the Statistics Registers Summation Statistics11-10Statistical Operations Normalizing Close, Large Numbers Access to the Statistics Registers Statistics Registers in Calculator MemoryExample Viewing the Statistics Registers # ™ ´º¸/ ™ ´¸/ ™ ´º/ ™ ´¸/ ™ ´º/ ‡ 11-12Statistical Operations Programming Page RPN mode 5 !Nz ALG mode 5 !zN‘ Simple ProgrammingSimple Programming Example a Simple Program RPN mode ALG mode RPN modeDisplay Description 12-2Simple Programming Selecting a Mode Designing a ProgramProgram Boundaries LBL and RTN Program Labels Using RPN, ALG and Equations in Programs Data Input and OutputStrengths of RPN Operations ALG Operations Entering a Program To enter a program into memory Keys That Clear 12-6Simple ProgrammingTo enter an equation in a program line RPN mode Function Names in ProgramsExample Entering a Labeled Program HN ZLR12-8Simple Programming Example Entering a Program with an Equation Testing a Program Running a ProgramExecuting a Program XEQ NzX Example Testing a Program 12-10Simple Programming Using Input for Entering Data Entering and Displaying Data@ 12-12Simple Programming To use the Input function in a program Using View for Displaying Data To respond to a prompt Using Equations to Display Messages 12-14Simple ProgrammingExample INPUT, VIEW, and Messages in a Program NzLR LR z › LH ›p = 2π R2 + 2π RH = 2π R R + H Displaying Information without Stopping 12-16Simple Programming Error Stops Stopping or Interrupting a ProgramProgramming a Stop or Pause STOP, PSE Interrupting a Running Program Editing a Program 12-18Simple ProgrammingTo delete a program line To insert a program line Program Memory Viewing Program Memory Memory Usage Catalog of Programs MEMClearing One or More Programs 12-20Simple Programming Checksum To clear all programs from memory Selecting a Base Mode in a Program Nonprogrammable FunctionsProgramming with Base V Decimal mode set Hexadecimal mode set Polynomial Expressions and Horners MethodNumbers Entered in Program Lines Ax4 + Bx3 + Cx2 + Dx + E Keys Display ALG mode ‰X12-24Simple Programming Checksum and length E41A Page Programming Techniques Routines in ProgramsProgramming Techniques Calling Subroutines XEQ, RTN 13-2Programming Techniques Nested Subroutines Example a Nested Subroutine Branching GTO 13-4Programming TechniquesA2 + B2 Using GTO from the Keyboard Programmed GTO Instruction Conditional Instructions 13-6Programming Techniques Tests of Comparison x?y, x?0 Test Menus Program Lines Description RPN mode Flags13-8Programming Techniques !- % ! ! ! º¸@ ! Flag Fraction-Control Flags Status ClearSet 13-10Programming Techniques Using Flags Flags Menu Menu Key DescriptionAnnunciators for Set Flags Example Using Flags 13-12Programming Techniques Program Lines Description RPN mode $ @ $ $ @ $ 13-14Programming Techniques Example Controlling the Fraction Display Program Lines Description Loops 13-16Programming Techniques Conditional Loops GTO ! ! . ! º6¸@ ! #$ ! Loops with Counters DSE, ISG 13-18Programming TechniquesLoop-Control Number $ $ ! $$ Indirectly Addressing Variables and Labels Variable13-20Programming Techniques Indirect Address If i contains Then i will address Program Control with 13-22Programming TechniquesExample Choosing Subroutines With ! L If i holds Then XEQi calls Program lines- L & %1L2 Example Loop Control With Equations with 13-24Programming Techniques L 1L2 Page Solving and Integrating Programs Solving a ProgramSolving and Integrating Programs To solve a programmed function Example Program Using ALG 14-2Solving and Integrating ProgramsV= N x R x T Pressure ⋅ volume. Pressure ⋅ volume Example Program Using Equation LP z LV ’ LN z LR z LT ‘14-4Solving and Integrating Programs @ @ ‘10 !@ # / Using Solve in a Program 14-6Solving and Integrating Programs/ label #variable Example Solve in a Program Program Lines In RPN mode Integrating a ProgramTo integrate a programmed function 14-8Solving and Integrating Programs To write a program for ≥ FN Using Integration in a Program Sin/ label 14-10Solving and Integrating Programs ≥ G variableExample ≥ FN in a Program Restrictions on Solving and Integrating Page Mathematics Programs Vector OperationsMathematics Programs = R cosP = arctan 15-2Mathematics Programs = XU + YV + ZWV1 = X i + Y j + Z k V2=U i + V j + W k X2 + Y2 + Z2 and P Program Listing Program Lines Description ALG mode 15-4Mathematics Programs Goes to the vector-addition routine 15-6Mathematics Programs Flags Used RemarksProgram Instructions 15-8Mathematics Programs Variables Used XR 15-10Mathematics Programs @ 07 g !@ 125 g @ Solutions of Simultaneous Equations 15-12Mathematics ProgramsAX + DY + GZ = J BX + EY + HZ = K CX + FY + IZ = L % 15-14Mathematics Programs Mathematics Programs 15-16Mathematics Programs Flags Used 15-18Mathematics Programs Scratch @ Polynomial Root Finder 15-20Mathematics Programs@ @ @ Where J = a3/2 = y0 /2 X2 + J + Lx + K + M =X2 + J Lx + K M = ⋅ the sign of JK a1/2 15-22Mathematics Programs Polynomial 15-24Mathematics Programs ! Checksum and length C7A6 B0 =a04a0 a32 a12B2= -a2 B1 = a3a1 4a0 15-26Mathematics Programs JK. a1 + L + M 15-28Mathematics Programs Terms and Coefficients Order 15-30Mathematics Programs %/ Coordinate Transformations 15-32Mathematics Programs Old coordinate System 15-34Mathematics Programs Checksum and length 8C82 15-36Mathematics Programs Remark Ÿ @ 15-38Mathematics Programs Statistics Programs Curve FittingStatistics Programs = B + MIn 16-2Statistics Programs Description 16-4Statistics Programs #$ #$ 16-6Statistics Programs 16-8Statistics Programs XS%@ 40.5 g &@ value 104.5 g %@ 38.6 g &@ %@ 37.9 g &@ 16-10Statistics Programs // / %@ Logarithmic Exponential Power Normal and Inverse-Normal Distributions 16-12Statistics Programs Function 16-14Statistics Programs ª º ª -+. H % ! Variable used temporarily to pass the value S ⋅ 2 π to Inverse programXS@ 16-16Statistics Programs Grouped Standard Deviation 2f − 16-18Statistics Programs Updates ƒxi 2fi in register Increments or decrements N 16-20Statistics Programs Group /%@ @ XU/ %@ 13 g @ / 16-22Statistics Programs %@ %@ %@ Miscellaneous Programs and Equations Time Value of MoneyMiscellaneous Programs and Equations …LN ` QLI ›LF z›LI 17-2Miscellaneous Programs and Equations Solve instructions Keys Display Description RPN mode šas17-4Miscellaneous Programs and Equations Retains 5750 in B calculates monthly interest rate Prime Number Generator 17-6Miscellaneous Programs and Equations 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 Service Support, Batteries, and Service A-1Calculator Support Answers to Common Questions 2Support, Batteries, and Service Environmental LimitsChanging the Batteries Support, Batteries, and Service A-3 To install batteries Testing Calculator Operation 4Support, Batteries, and ServiceDo not mutilate, puncture, or dispose of batteries Fire. The batteries can burst or explode Support, Batteries, and Service A-5 Self-Test™Ÿ˜—š Warranty 6Support, Batteries, and Service Service Support, Batteries, and Service A-7Europe Country Telephone numbers 8Support, Batteries, and Service Asia Pacific Country Telephone numbersAmerica Country Telephone numbers Support, Batteries, and Service A-9 Regulatory InformationCanada 10Support, Batteries, and Service Japan User Memory and the Stack Managing Calculator MemoryUser Memory and the Stack Resetting the Calculator 2User Memory and the Stack Default Clearing MemoryCategory Status of Stack Lift Disabling OperationsNeutral Operations 4User Memory and the Stack Vlabel nnnn EQN FDISP Status of the Last X Register 6User Memory and the Stack€ θ ,r ALG Summary About ALGALG Summary C-1 Power Functions Doing Two-number Arithmetic in ALGSimple Arithmetic 2ALG Summary Percentage Calculations ‘27 Q…ALG Summary C-3 Permutations and Combinations Quotient and Remainder Of Division16.12 T 4ALG Summary Keys Display Description ALG Summary C-5 Reviewing the Stack 6ALG Summary Coordinate Conversions ALG Summary C-7 Integrating an Equation Q3 z8ALG Summary Operations with Complex Numbers ALG Summary C-9To view the result of complex operations Complex Operations 2 ›3 ` ` Q2 › 10ALG Summary ºy4 ÃË2 Ë5 ¹cº  ºy3 ÃË ALG Summary C-11 Initial x, y Corrected x, y 12ALG Summary 20 ALG Summary C-13Page More about Solving How Solve Finds a RootMore about Solving D-1 2More about Solving Function Whose Roots Can Be Found Interpreting Results More about Solving D-3Example An Equation With One Root 4More about Solving Example An Equation with Two Roots More about Solving D-5 / / -%. LX `’1.56More about Solving Example Discontinuous Function − 1= More about Solving D-7 When Solve Cannot Find a Root 8More about Solving More about Solving D-9 Example a Relative Minimum 10More about Solving Example An Asymptote ⎟ x + 0.3 − 0.5 = #LX q LX ›3 ``…More about Solving D-11 Example Find the root of the equation 12More about Solving !1%ª1%-22Example a Local Flat Region Round-Off Error More about Solving D-13 Underflow 14More about Solving More about Integration How the Integral Is EvaluatedMore about Integration Conditions That Could Cause Incorrect Results 2More about Integration ≥0∞ xe −x dx LX z LX ` ‡ 4More about Integration More about Integration 6More about Integration Conditions That Prolong Calculation Time 8More about Integration . More about Integration Page Messages Messages F-1 2Messages Or program. This might take a while Messages F-3 # variable 4Messages Self-Test Messages Operation Index Name Keys and DescriptionOperation Index G-1 ˜or — jh 2Operation Index Operation Index G-3 Polar to rectangular coordinates P 4Operation Index M Operation Index G-5 6Operation Index GComplex Change sign›Complex …Complex Subtraction Operation Index G-7 8Operation Index RHyperbolicVRadians to degrees DSE variable Mvariable Operation Index G-9 Natural exponentialWlabel ?Fractional part 10Operation Index GTO label Vlabellabel Nnnn ”I” Operation Index G-11Input variable ‰variable Integer part ISG variable Lvariable 12Operation Index LBL label •labelNatural logarithm Common logarithm Operation Index G-13 Permutations of n itemsFPause WDegrees to radians 14Operation Index Operation Index G-15 16Operation Index Operation Index G-17 18Operation Index Operation Index G-19 ≠0? 13-7If x=0, skips the next program Line ≤0? 13-7 11-10 Given an x-value Power Returns y raised to the xth power 20Operation Index Index Index-1 Index-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Index-11 Index-12 RPN Index-13 Index-14 View Index-15