Plotting to find guesses, At2 x = v0t +, $/3+$@X

Plotting to find guesses

The main reason for plotting in the Solve aplet is to help you find initial guesses and solutions for those equations that have difficult-to-find or multiple solutions.

Consider the equation of motion for an accelerating body:

at2 x = v0t + ------

where x is distance, v0 is initial velocity, t is time, and a is

acceleration. This is actually two equations, y = x and y = v0 t + (at2) / 2.

Since this equation is quadratic for t, there can be both a positive and a negative solution. However, we are concerned only with positive solutions, since only positive distance makes sense.

1.Select the Solve aplet and enter the equation.

>$3/(7@ Select Solve 67$57a

>$/3+$@X _

>$/3+$@V >;@>$/3+$@T >@>$/3+$@A >;@>2.a$/3+$@T >;@ >j@2

2.Find the solution for T (time) when X =30, V =2, and

A =4. Enter the values for X, V, and A; then highlight the independent variable, T.

>180@

30 >(17(5@

2>(17(5@ e,4>(17(5@ e,*e, to highlight T

7-8	Solve aplet

Page 90

Image 90

HP 40G manual Plotting to find guesses, At2 x = v0t +, $/3+$@X, 180@ 30 175@ 2175@ *e,4175@ *e,*e, to highlight T

Contents

USER’S Guide Page Contents Parametric aplet Inference aplet Importing Sample Statistics from the Statistics aplet Variables and memory management Programming Index @, &26@, +20@, etc PrefaceManual conventions +,7@CLEAR, 6+,7@MODES, 6+,7@ACOS, etc Preface On/off, cancel operations Getting startedHome To adjust the contrast To clear the display DisplayParts of the display Keyboard Menu keysAnnunciator Description Aplet control keys Key Meaning 175@ +,7@$/3+$@ 75@ Key Description Shifted keystrokesHelpwith +,7@$/3+$@ Program commands Inactive keys Math keys To cancel a menu MenusTo search a menu Setting Options Mode settingsReset input form values Input forms Setting Options Setting a mode Aplets E-lessonsPress+20@ to return to Home Aplet Use this aplet to explore Name Quad Explorer aplet N T Trig Explorer aplet *,*A, keys Aplet views Symbolic view Plot view Numeric viewAplet library Aplets are stored in the Aplet library Views menu contains the Plot-Table view Configuration Aplet view configurationPlot Setup Numeric Setup Symbolic To change views Long results Negative numbers Mathematical calculationsEntering expressions Example Scientific notation powers Explicit and implicit multiplication@4 6+,7@EEX @13@ @@6 6+,7@EEX 23@ j@ 3 6+,7@EEX 175@ +,7@√ 85@9 Entering Calculates@ 45 @6+,7@π +,7@√ @85 @9 @ Clearing numbers Using previous results 175@ updates ANS from 50 to 75 to 50175@ @25 175@175@ Accessing the display history $/3+$@ a 175@Key Function See Setting fraction precision below for more information Using fractionsSetting Fraction mode Clearing the display history Setting fraction precision @@@2@3 Fraction calculations Converting decimals to fractionsReturn to Home and enter the calculation Evaluate the calculation Complex results To enter complex numbers Complex numbersConverting a number to a fraction + iy Catalog/Editor Contents Catalogs and editorsStoring complex numbers Statistics Pred function Differences between the HP 38G and the HP 39G/40GMemory manager Plot Goto function Trig Explorer and Quadratic Explorer apletsPage Aplets and their views About the Symbolic viewDefining an expression Symbolic view Aplet views Aplets and their views Evaluating expressions ApletsHome $7+@ Symb view keys$56@ +,7@CLEAR About the Plot view Setting up the plot Plot view setupPlot view settings 175@ or Reset plot settings Exploring the graph Plot view keys+,7@*A To jump directly to a value To turn trace on/ off Trace a graphZoom within a graph Zoom options Divides vertical scale only, using Divides horizontal scale only, usingMultiplies horizontal scale, using Multiplies vertical scale only, using Press *k, to move to the bottom of the Zoom list 018Zoom=220Un-zoom =220Zoom Out =220Y-Zoom =220X-Zoom=220X-Zoom Out =220Y-Zoom Out To box zoom To set zoom factors Other views for scaling and splitting the graph Options Menu key copies the right plot to the left Decimal scaling About the numeric viewOverlay plots Integer scaling Trigonometric scaling Setting up the table numeric view setup Numeric view settingsReset numeric settings Exploring the table of numbers NUM view menu keysZoom within a table Column, then enter the value to jump to Building your own table of numbersArrow keys to place the cursor in the independent variable 175@, the values for the dependent variables are Build Your Own menu keys Clear data Example plotting a circle $3/7@ Select+,7@√@9 Getting started with the Function aplet Function apletAbout the Function aplet Open the Function aplet @75@175@ Define the expressionsPlot the functions Plot the functions Trace the linear function Change the scale Trace a graphWith FCN functions Times Find the greater of the two roots of the quadratic function SelectRootRoot value is displayed at the bottom of the screen Choose the end value for 2Select.SlopeMove the cursor to x = 1 by pressing the *A,or @.2 To find the extremum of the quadratic Display the numeric view Set up the tableDisplay the numeric setup Display the numeric view Display a table of numeric values Match the table settings to the pixel columns in the graphExplore the table Move to X = Function aplet interactive analysis Area Extremum Isect Root Slope Access FCN variables FCN functions Function Description Symbolic view. Displays Cursor. You need to have at leastTwo selected expressions Plotting a piecewise defined function example = $3/ @ Select Page Getting started with the Parametric aplet Parametric apletAbout the Parametric aplet Open the Parametric aplet Define the expressions Set angle measure Set up the plot Plot the expression +,7@PLOT360 +,7@ Plot Overlay plot Display the numbersPlot a triangle graph over the existing circle graph 120Page Open the Polar aplet Polar apletGetting started with the polar aplet +,7@CLEAR 46+,7@π 2.a Explore the graph Display the numbers About the Sequence aplet Sequence aplet starts in the Symbolic viewSequence aplet Getting started with the Sequence aplet Define the expression Keys to assist in the entry of equations Plot the Fibonacci sequence Plot Setup, set the Seqplot option to CobwebPlot the sequence Display the table Select CobwebPage Solve aplet About the Solve aplet Getting started with the Solve aplet Open the Solve apletDefine the equation Define known variables Solve the unknown variable Plot the equation Solve for the unknown variable a@ Select Auto Solve aplet’s NUM view keys ≈20 times Use an initial guess Number format Interpreting results Message Condition Root- Finder at work $/3+$@X Plotting to find guessesAt2 x = v0t + 180@ 30 175@ 2175@ *e,4175@ *e,*e, to highlight T Plot the graph To move cursor to the intersection Initial guess Using variables in equationsHome variables 3175@*k,*k,*k Getting started with the Statistics aplet Statistics apletAbout the Statistics aplet 1400 920 1100 2265 2890 2200 To move to the next column Open the Statistics aplet Enter data175@1 175@ 175@5 175@ 175@4 175@ 1400 175@920 175@ 1100 175@2265 175@ 2890 175@2200 175@ Choose fit and data columns Explore statistics Setup plot Select a fit in the Symbolic setup viewSelect2. Linear Display the equation for the best linear fit To move to the FIT1 +2field To highlight Entering and editing statistical data$7+@ S to highlight 6175@ Statistics aplet’s NUM view keys +,7@ FXUVRU 160 175@ 165 175@ 170 175@ 175 175@ 180 175@ $56@ Sets will need to be selected again Before re-useHeight cm Frequency Angle Setting Save dataDelete data To choose the fit Defining a regression model 2VARInsert data Sort data values Select the Sort Order option. You can choose either = m lnx + b Fit models To define your own fitFit model Meaning = ax3+bx2+cx+d. Needs at least Computed statistics One-variableStatistic Definition Two-variable Columns for a linear fit only Plotting To plot statistical dataYou can plot Plot types Histogram Box and Whisker Plot Scatter Plot Correlation Relative ErrorFitting a curve to0182VAR data Coefficient Setting up the plot Plot setup view Plotting mark 2VAR Connected points 2VAR Trouble-shooting a plot Statistics aplet’s Plot view keys Calculating predicted values Find predicted valuesPage About the Inference aplet Default the input form contains example data. This exampleInference aplet Example data Inference aplet’s Symb view keys Getting started with the Inference apletOpen the Inference aplet Hypothesis Tests Confidence Intervals Define the inferential method ∝.a∝ Field name Definition Plot test results Open Statistics aplet. Note Reset current settingsImporting Sample Statistics from the Statistics aplet Open the Statistics aplet Calculate statistics Open Inference aplet @529 175@ @295 175@ @952 175@ @259 175@ @925 175@ @592 175@ Choose inference method and type Import the data2one.a E, to move to Field Display Numeric view Display Plot viewSpecify a 90% confidence interval in the C field Display the confidence interval in the Plot view Menu name Hypothesis testsOne-Sample Z-Test Inputs Two-Sample Z-Test ResultsResult Description One-Proportion Z-Test Two-Proportion Z-Test Successes in the two populations is equalH0 π1 = π2 Selected hypothesis against the null hypothesis. The null One-Sample T-TestTest 1 ∝ Hypothesis is that the sample mean has some assumed value Two-Sample T-Test Sample 1 standard deviation Sample 2 standard deviation Perform are based on the Normal Z-distribution or Student’s Confidence intervalsConfidence interval calculations that the HP 39G/40G can Confidence level Confidence interval for the difference between the means Two populations, ∝1 ∝2, when the population standardTwo-Sample Z-Interval Deviations, σ1 and σ2, are known One-Proportion Z-Interval Sample success count Sample 1 success count Two-Proportion Z-IntervalSuccesses in two populations Sample 2 success count One-Sample T-Interval Two-Sample T-Interval Page Using mathematical functions Math functionsMath menu To select a function Function categoriesCalculus Loop Stat-Two Complex Matrices Math functions by category SyntaxFunctions common to keyboard and menus Keyboard functions +,7@ATAN +,7@ASIN+,7@ACOS +,7@√ ABSvalue Valuepower+,7@ABS +,7@ n Variableexpression Calculus functionsTaylor Lower,upper,expression,variable Complex number functions ARGConj Constants Hyperbolic trigonometry EXP List functionsAlog EXPM1 Iterate Loop functionsMatrix functions Recurse Polyeval Polynomial functionsPolycoef Polyform Perm Probability functionsComb Random Utpn UtpcUtpf Utpt Real-number functions INT HMS→→HMS Mant Total MODChange RAD→DEG Truncate Statistics-TwoSign Xpon LINEAR? Symbolic functionsIsolate Quad Quote Trigonometry functions Using formal variables Symbolic calculationsSymbolic calculations in the Function aplet Home GG@$/3+$@S1 Finding derivativesTo find derivatives Evaluate the function To find derivatives in the Function aplet’s Symbolic view GG@@$/3+$@ Show the result format To find the indefinite integral using formal variables+,7@GG@0 @ $/3+$@ S1 @ 3 @ Thus, substituting X for S1, it can be seen that 10-26 Variables and memory management Introduction Storing and recalling variables Numeric PrecisionTo store a value To store the results of a calculation Calculations Vars menuTo use variables 65 @$/3+$@A 175@ Names or values of variables into programs +,7@LIST $/3+$@L3175@ 2contents,.a is copied to the command line$56@ *e,*e,*e,*A List names directly from the keyboard Category Available names To access an aplet variable Aplet variablesTo copy the value of the variable into the edit line Is displayed Memory ManagerStart the Memory Manager. a list of variable categories +,7@MEMORYPage Matrices MatricesVectors Matrix Variables + 7@CLEAR Creating and storing matricesMatrix Catalog keys +,7@*e, or To create a matrix in the matrix catalog Select the type of matrix to create Clears all elements from the matrix Working with matricesTo transmit a matrix +,7@*k, *e 728 672? 175@ 2175@*e 175@ 4175@ Matrix arithmeticTo multiply and divide by a scalar 175@ 8 175@ $/3+$@M1@ $/3+$@M2 175@ $/3+$@M1 j@$/3+$@M2 175@ 175@7 175@ Solving systems of linear equations+,7@MATRIX1 175@ +,7@MATRIX Matrix functions and commands About functions Argument conventions Idenmat EigenvalEigenvv Inverse Rref RankRownorm Schur TRN ExamplesTrace Identity Matrix Transposing a Matrix Reduced-Row Echelon Form Create a list in the List Catalog ListsCreating lists Open the List catalog List catalog keys Create a list List edit keys To display one element To edit a list Displaying and editing listsTo display a list @ /@ 16a*e To insert an element in a list To store one element To delete a list Deleting listsTransmitting lists To delete all lists List functions $7+@L *A,Select $/3+$@L 5175@ $7+@L *A$/3+$@L5175@ $/3+$@A @ Calculates the number of elements in a list Calculates the product of all elements in listPOSlist, element Sorts elements in ascending order Finding statistical values for list elements 89@65@ 0@89$/3+$@ L1 175@ 180@67$76a Go to the Numeric view To display calculated statisticsPage +,7@letter Aplet note viewTo write a note in Note view Your work is automatically saved. Press any view key +,7@%.63$= +,7@ Cmds Aplet sketch view Sketch keysTo draw a line To draw a box Where you want any corner of the box to beYou can adjust the size of the box by moving the cursor Draw keys Keys Press 2.again to affix the labelHighlight2. the variable name you want to use and press +,7@NOTEPAD To import a graphics variableTo create a note Name of the note is ‘MYNOTE’ Transmits the selected note to Opens the selected note forBegins a new note, and asks for Another HP 39G/40G or PC 14-8 Contents of a Program Structured Programming Programming Program catalog Open Program catalog Program catalog keys +,7@*k, or *e Creating and editing programs Create a new program A,*e Enter commands Edit a program+,7@CMDS To enter functions more to come Editing keys Using programs Debug a program Stop a programRun a program From HOME, type RUN programname. or Program Working with programsCopy a Transmit a program Delete a program Delete all Delete the contents of a programAbout customizing an aplet Programs Aplet naming convention Customizing an aplet example$3/7@Select Configuring the Setviews menu option programs Setviews ’’’’’’’’18 Programming 15-13 Check Programming commandsAplet commands Select Command syntax Syntax for the command is as follows Auto-run programs IF...THEN...END Branch commandsUncheck If test-clause ELSE... END CASE...END Iferr Then Iferr trap-clause Then error-clause ENDIF... then END Drawing commands Graphic commands Grobor →GROBGrobnot Grobxor Loop commands Matrix commands Scale RandmatRedim Scaleadd Print commands Prompt commands Disp linenumbertextitem DispDisptime Editmat matrixname Input nametitle,labelhelpdefault GetkeyInput Wait MsgboxPrompt Wait seconds Stat-One and Stat-Two commands Stat-One commandsStat-Two commands Area Storing and retrieving variables in programsPlot-view variables Axes FastRes CoordExtremum Grid InvCross HwidthIndep Isect Root Nmin / NmaxRecenter S1mark-S5mark StatPlot SimultSlope Umin/Umax Tstep Tmin / TmaxTracing Xcross Xmin / Xmax XtickYtick Ymin / Ymax Yzoom Symbolic-view variablesXzoom Angle U1...U9, U0 X1, Y1...X9,Y9 X0,Y0R1...R9, R0 E1...E9, E0 Numeric-view variables C1...C9, C0Digits NumFont FormatNumCol NumIndep NumStep NumRowNumStart NumType PageNum Extending aplets Creating new aplets based on existing aplets Aplet Keys $/3+$@ H 175@ $1@ $/3+$@ θ @$/3+$@ C @ 175@ $3/ @Select 180@ 35 175@ Solve for the missing Resetting an apletDownloading e-lessons from the web Length of the ladder is approximately 8.72 metres Sending and receiving aplets To transmit an aplet Sorting items in the aplet library menu list To sort the aplet listTo delete an aplet Reference information Regulatory informationCanada LED safety Warranty Reference information Resetting the HP 39G/40G To reset using the keyboard To erase all memory and reset defaults If the calculator does not turn on21@ key Glossary Cross matrix1,matrix2 Operating temperature 0 to 45C 32 to 113F Operating detailsBatteries Menu maps of the Vars menu Home variables areCategory Available name Modes Function aplet variablesUser-named Function aplet variables are Parametric aplet variables Plot-FCNParametric aplet variables are Polar aplet variables Polar aplet variables are Sequence aplet variables Sequence aplet variables are Solve aplet variables Solve aplet variables are Statistics aplet variables Statistics aplet variables areStat-One Math functions are Menu maps of the Math menuMath functions Calculus Polynom = = Program constants Program commands Category Command Selected status messages Message Meaning+,7@ Memory Function value, root, extremum Or intersection is not visibleCurrent screen Index Index Index Index Index Index Index Index Index Index Index Index Index