Interpreting Results, More about Solving D-3 | HP 35s Scientific

Interpreting Results

The SOLVE operation will produce a solution under either of the following conditions:

If it finds an estimate for which f(x) equals zero. (See figure a, below.)

If it finds an estimate where f(x) is not equal to zero, but the calculated root is a 12–digit number adjacent to the place where the function's graph crosses the x–axis (see figure b, below). This occurs when the two final estimates are neighbors (that is, they differ by 1 in the 12th digit), and the function's value is positive for one estimate and negative for the other. Or they are (0, 10– 499) or (0, –10–499). In most cases, f(x) will be relatively close to zero.

To obtain additional information about the result, press see the previous estimate of the root (x), which was left in the Y–register. Press again to see the value of f(x), which was left in the Z–register. If f(x) equals zero or is relatively small, it is very likely that a solution has been found. However, if f(x) is relatively large, you must use caution in interpreting the results.

Example: An Equation With One Root.

Find the root of the equation:

–2x3+ 4x2 – 6x + 8 = 0

Enter the equation as an expression:

More about Solving D-3

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HP 35s Scientific manual Interpreting Results, More about Solving D-3

Contents

Users guide HP 35s scientific calculator Printing History Contents RPN The Automatic Memory Stack Viewing a Variable ContentsStoring Data into Variables Quotient and Remainder of Division Real-Number FunctionsTrigonometry Entering π Hyperbolic Functions Entering and Evaluating Equations Fractions Integrating Equations Solving EquationsOperations with Complex Numbers Vector Arithmetic 10-1 Statistical Operations 12-1 Base Conversions and Arithmetic and Logic 11-110-7 Creating Vectors from Variables or Registers 10-8 Simple Programming 13-1 Part 2. Programming Programming Techniques 14-1 Part 3. Appendixes and Reference Support, Batteries, and ServiceSolving and Integrating Programs 15-1 Statistics Programs 16-1 ALG Summary User Memory and the StackMore about Solving More about Integration Round-Off ErrorMessages Operation Index How the Integral Is Evaluated Contents Part Page Getting Started Adjusting Display ContrastImportant Preliminaries Turning the Calculator On and Off Shifted Keys Highlights of the Keyboard and Display2Getting Started Cursor Keys Alpha KeysFunction Key Left-shifted function Right-shifted Letter for alphabetic Backspacing and Clearing  Clear or CancelKeys for Clearing 4Getting Started Clear menu       Key Description ˆ ˆ    Using MenusHP 35s Menus Chapter Name 6Getting Started Other functions Programming Instructions or  Exiting Menus 8  RPN and ALG ModesKeys Display KeysDisplay 10Getting Started RPN mode ALG mode Undo key First Line Second Line Display and AnnunciatorsAnnunciators 12Getting Started EQN HP 35s Annunciators Meaning Chapter 14Getting Started Making Numbers Negative Keying in NumbersExponents of Ten z 16Getting Started Range of Numbers and Overflow Understanding Entry Cursor  Single Argument or Unary Operations Performing Arithmetic Calculations18Getting Started  Two Argument or Binary OperationsKey RPN,RPN Program ALG, Equation, ALG Program   Õ Key RPN, RPN Program ALG, Equation, ALG Program Controlling the Display Format 22Getting Started Periods and Commas in Numbers   8 Complex number display format  ,  , 8 24Getting Started 8   SHOWing Full 12-Digit Precision8×  Entering Fractions Fractions26Getting Started  Messages É  Checking Available Memory Calculator Memory28Getting Started Clearing All of Memory 30Getting Started What the Stack Is RPN The Automatic Memory StackRPN The Automatic Memory Stack 2RPN The Automatic Memory Stack Clearing the X-Register X and Y-Registers are in the DisplayReviewing the Stack Press  Exchanging the X- and Y-Registers in the Stack4RPN The Automatic Memory Stack Arithmetic How the Stack Does It 6RPN The Automatic Memory Stack How Enter Works Filling the stack with a constant How to Clear the Stack 8RPN The Automatic Memory Stack Last X Register Correcting Mistakes with Last  Ù Reusing Numbers with Last    Work from the Parentheses Out Chain Calculations in RPN Mode     Order of Calculation Exercises       More Exercises       18RPN The Automatic Memory Stack Keys DisplayDescription AStoring Data into Variables 2Storing Data into Variables Storing and Recalling NumbersPress letter-key  Press letter-key  G  A  Viewing a Variable Using the MEM CatalogVAR catalog 4Storing Data into Variables D E C Storage Arithmetic Arithmetic with Stored Variables6Storing Data into Variables Storage arithmetic uses , , , or  Recall Arithmetic  D Exchanging x with Any VariableD E  Variables I and JA 10Storing Data into Variables Real-Number Functions Exponential and Logarithmic Functions Quotient and Remainder of Division Power Functions    Entering π Trigonometry Trigonometric Functions Setting the Angular ModeOption Description Annunciator 4Real-Number Functions   Percentage Functions Hyperbolic Functions     Const Menu Items Description Value Physics Constants8Real-Number Functions Items Description Value Rectangular/Polar Conversions Conversion Functions10Real-Number Functions 30o 6 812Real-Number Functions Angle Conversions Time Conversions Unit Conversions Factorial Probability FunctionsGamma Probability  16Real-Number Functions Integer part Parts of NumbersFractional part Absolute value PressDisplay To calculate18Real-Number Functions Entering Fractions Fractions  Display Rules Fractions in the Display2Fractions Entered Value Internal Value Displayed Fraction Accuracy Indicators Changing the Fraction Display Setting the Maximum Denominator4Fractions  6Fractions Choosing a Fraction Format To Get This Fraction Format Change These Flags   j Examples of Fraction Displays Rounding Fractions Fractions in Equations 10Fractions Fractions in Programs How You Can Use Equations Entering and Evaluating EquationsEntering and Evaluating Equations = .25 π d2 l 2Entering and Evaluating Equations    D Value   Key Operation Summary of Equation Operations Variables in Equations Entering Equations into the Equation List4Entering and Evaluating Equations Functions in Equations Numbers in Equations Parentheses in Equations Displaying and Selecting Equations   Ö  8Entering and Evaluating Equations Editing and Clearing Equations Types of Equations 10Entering and Evaluating Equations Evaluating Equations Type of Equation Result for  Result for  Using Enter for Evaluation       Using XEQ for Evaluation12Entering and Evaluating Equations  Responding to Equation Prompts Operator Precedence Syntax of Equations14Entering and Evaluating Equations Equations Meaning Order Operation Example 16Entering and Evaluating Equations Equation FunctionsΣxy Perimeter = a + b + h sin1θ + sin1φ RPN Operation Equation function Area = 18Entering and Evaluating Equations Verifying Equations Syntax Errors Õ 20Entering and Evaluating Equations Solving an Equation Solving EquationsSolving Equations Markup ⋅ Cost = Price = v0 t + 1/2 g t 2Solving Equations  G    P      P    Solving built-in Equation 6Solving Equations Verifying the Result Understanding and Controlling Solve Choosing Initial Guesses for Solve Interrupting a Solve Calculation8Solving Equations Solving Equations = 40 H ⋅ 20 H ⋅ 4 ⋅ H 10Solving EquationsType in the equation Equation H   4H  value 12Solving Equations For More Information Integrating Equations Integrating Equations 2Integrating Equations Integrating Equations FN   J0 x = π 0 cosx sin t dt = t 4Integrating Equations X  SinxX Specifying Accuracy Accuracy of IntegrationInterpreting Accuracy 6Integrating Equations 8        8Integrating Equations 8 Form  Operations with Complex Numbers Complex Operations Complex Stack2Operations with Complex Numbers Functions for One Complex Number, z To Calculate Press 8  6 6 6 8  6Display Description 4Operations with Complex Numbers Using Complex Numbers in Polar Notation 6Operations with Complex Numbers ?  ?  ?  6 ? ? Complex Numbers in Equations Complex Number in Program Program lines ALG mode          Vector operations Vector ArithmeticVector Arithmetic    Õ 3   3  Absolute value of the vector    Dot product3 3 10-4Vector Arithmetic  Õ Angle between vectors  3   3   Vectors in Equations10-6Vector Arithmetic Program lines Description Vectors in Programs Then hC Õ hA Creating Vectors from Variables or Registers10-8Vector Arithmetic Base Menu Base Conversions and Arithmetic and LogicMenu label Description Base Conversions and Arithmetic and Logic    11-2Base Conversions and Arithmetic and Logic Õ  Logic Menu Arithmetic in Bases 2, 8,11-4Base Conversions and Arithmetic and Logic          Negative Numbers Representation of Numbers   Range of Numbers Windows for Long Binary Numbers Using base in program and equations    Entering Statistical Data Statistical OperationsStatistical Operations Entering One-Variable Data Correcting Errors in Data EntryEntering Two-Variable Data 12-2Statistical Operations Initial x, y Corrected x, y   Mean Statistical CalculationsStatistics Menus Menu Key Description   Sample Standard Deviation        Linear Regression Population Standard Deviation Linear Regression Menu  ˆ Õ ˆ12-8Statistical Operations         Limitations on Precision of Data12-10Statistical Operations Summation Statistics Summation Values and the Statistics Registers 12-12Statistical Operations Access to the Statistics Registers Statistical Operations 12-14Statistical Operations Programming Page Simple Programming Simple Programming 13-2Simple Programming Area = πx2 KeysDisplay In ALG mode Selecting a ModeDesigning a Program  X  Program Boundaries LBL and RTN Using RPN, ALG and Equations in Programs13-4Simple Programming  letter-key Strengths of RPN Operations Data Input and Output 13-6Simple Programming Entering a Program Clear functions and backspace key A Function Names in Programs13-8Simple Programming E RPN modeR  R  Executing a Program XEQ Running a Program 13-10Simple Programming Keys Display Description RPN mode Testing a Program 13-12Simple Programming Entering and Displaying Data RPN mode ALG mode Using Input for Entering Data 13-14Simple Programming Using View for Displaying Data   ÇR ÇH Using Equations to Display Messages13-16Simple Programming  R H V  R R H  Keys RPN mode V OL  A RE A    V  S       Displaying Information without Stopping13-18Simple Programming C  value Stopping or Interrupting a Program Error StopsProgramming a Stop or Pause STOP, PSE Interrupting a Running Program 13-20Simple Programming Editing a Program Viewing Program Memory Program Memory Catalog of Programs MEM Memory Usage13-22Simple Programming Checksum Clearing One or More Programs Programming with Base Nonprogrammable Functions   Decimal mode set Selecting a Base Mode in a ProgramNumbers Entered in Program Lines 13-26Simple Programming Polynomial Expressions and Horners MethodAx4 + Bx3 + Cx2 + Dx + E A    A X     13-28Simple Programming Routines in Programs Programming TechniquesCalling Subroutines XEQ, RTN Programming Techniques 14-2Programming Techniques Nested Subroutines A2 + b2 + c2 + d2 14-4Programming Techniques Branching GTOA2 + B2 Programmed GTO Instruction Using GTO from the Keyboard 14-6Programming Techniques Conditional Instructions Test Menus Tests of Comparison x?y, x?0 14-8Programming Techniques Program Lines Description RPN mode Meanings of Flags Flags Clear Flag Fraction-Control Flags14-10Programming Techniques Programming Techniques 14-12Programming Techniques Flags Menu … takes the natural log of the X-input 14-14Programming Techniques  S  value   value Number F      Loops14-16Programming Techniques Conditional Loops GTO 14-18Programming Techniques Loops with Counters DSE, ISG Programming Techniques Variables I and J Indirectly Addressing Variables and Labels14-20Programming Techniques Indirect Address, I and J 14-22Programming Techniques If I/J contains Then I/J will address Equations with I/J Program Control with I/JUnnamed indirect variables 14-24Programming Techniques Solving a Program Solving and Integrating ProgramsSolving and Integrating Programs V= N x R x T 15-2Solving and Integrating Programs Program Lines Description ALG mode Keys Display Description ALG modeG P  value   value   H  P   L  15-6Solving and Integrating Programs Using Solve in a Program label  variable Integrating a Program Program Lines In RPN mode 15-8Solving and Integrating Programs Sin  S   X 15-10Solving and Integrating Programs Using Integration in a Program label   variable Restrictions on Solving and Integrating 15-12Solving and Integrating Programs Curve Fitting Statistics ProgramsStatistics Programs 16-2Statistics Programs = B + MIn Program Listing 16-4Statistics Programs    16-6Statistics Programs Determines if D001 or B001 should be run 16-8Statistics Programs   S          Normal and Inverse-Normal Distributions Logarithmic Exponential Power  16-12Statistics Programs Qx = 0.5 − σ 12π xx e−x −x ⎟σ 2 ⎟2dx This routine initializes the normal distribution program 16-14Statistics Programs Program Lines Program Lines 16-16Statistics Programs S To the inverse Program D  16-18Statistics Programs Grouped Standard Deviation∑ x f   16-20Statistics Programs Statistics register Index variable used to indirectly address the correctRegister Summation Σ f Summation Σ x i f    U     16-24Statistics Programs Time Value of Money Miscellaneous Programs and EquationsMiscellaneous Programs and Equations TVM equation is 4 PI NÕ I     KeysDisplay In RPN mode17-4Miscellaneous Programs and Equations P  value     I 17-6Miscellaneous Programs and Equations   Prime Number Generator 17-8Miscellaneous Programs and Equations Tests to see whether all possible factors have been tried 17-10Miscellaneous Programs and Equations   Cross Product in Vectors 17-12Miscellaneous Programs and Equations z 17-14Miscellaneous Programs and Equations Appendixes and Reference Page Support, Batteries, and Service A-1 Support, Batteries, and ServiceCalculator Support Answers to Common Questions Environmental Limits 2Support, Batteries, and Service Changing the Batteries Support, Batteries, and Service A-3 4Support, Batteries, and Service Testing Calculator Operation Self-Test Support, Batteries, and Service A-5 6Support, Batteries, and Service Support, Batteries, and Service A-7 Warranty Customer Support 8Support, Batteries, and Service Emea Support, Batteries, and Service A-9 10Support, Batteries, and Service Invent Support, Batteries, and Service A-11 12Support, Batteries, and Service Yyyy+ Support, Batteries, and Service A-13 14Support, Batteries, and Service Managing Calculator Memory User Memory and the StackUser Memory and the Stack B-1 2User Memory and the Stack Resetting the Calculator User Memory and the Stack B-3 Clearing Memory Status of Stack Lift DefaultCategory 4User Memory and the Stack Neutral Operations Disabling OperationsUser Memory and the Stack B-5 ˆ ˆ Status of the Last X Register6User Memory and the Stack Accessing Stack Register Contents 8User Memory and the Stack About ALG ALG Summary, ALG Summary C-1 Simple Arithmetic Doing Two argument Arithmetic in ALG Percentage Calculations Power Functions Permutations and Combinations Parentheses CalculationsQuotient and Remainder Of Division xÕ Õ 4   Hyperbolic functions Trigonometric Functions ALG Summary C-7 Reviewing the Stack Operations with Complex Numbers Integrating an Equation8ALG Summary ALG Summary C-9 6 4 66 8Ë 6Õ4 6 Entering Statistical Two-Variable Data Z6 z Σ12ALG Summary ALG Summary C-13   ˆ Õ ˆ 14ALG Summary 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 Results X  X X  X 4More about Solving X XX   XÕ  6More about Solving X4 X  X Õ  More about Solving D-7  X   When Solve Cannot Find a Root8More about Solving    XMore about Solving D-9  X   X  10More about Solving More about Solving D-11 X4 X ÕÕ 12More about Solving X    X   Round-Off ErrorMore about Solving D-13 14More about Solving How the Integral Is Evaluated More about IntegrationMore about Integration 2More about Integration Conditions That Could Cause Incorrect Results ∞ xe−xdx 4More about Integration   More about Integration 6More about Integration   Conditions That Prolong Calculation Time 8More about Integration More about Integration 10More about Integration Messages F-1 Messages Or program. This might take a while 2Messages Messages F-3    Solve include EQN and PGM modecannot find 4Messages Messages F-5 FN each uses a level, they can also generate this error 6Messages Name Keys and Description Operation IndexOperation Index G-1 Ø ×2Operation Index Operation Index G-3 4Operation Index =  Denominator Operation Index G-5 6Operation Index Natural exponential Operation Index G-7label  Fractional part  8Operation Index Operation Index G-9 Input variable variable  Integer part ISG variable variable 10Operation Index LBL label label Natural logarithm RCL variable Operation Index G-11 Pause Degrees to radians. Returns 2π/360 12Operation Index Operation Index G-13 Hyperbolic Tangent 14Operation IndexView variable variable XEQ label label ≠ ÕÕÕ≤ Operation Index G-15 =y? 16Operation Index14-7 Menu ≠0? 11-4 Operation Index G-17 Power 18Operation Index Index-1 Index Index-2 EQN List TOP 6-7, F-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Angle between two vectors Index-11 Index-12