Solution Manual For Thomas' Calculus, Media Upgrade, 11th Edition

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to accompanyTHOMAS’ CALCULUSELEVENTHEDITIONandTHOMAS’ CALCULUSEARLYTRANSCENDENTALSELEVENTHEDITIONBASED ON THEORIGINALWORK BYGeorge B. Thomas, Jr.Massachusetts Institute of TechnologyASREVISED BYMaurice D. WeirNaval Postgraduate SchoolJoel HassUniversity of California, DavisFrank R. GiordanoNaval Postgraduate SchoolONLINETI®GRAPHINGCALCULATORMANUALLUZDEALBADrake University

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1IntroductionThis calculator manual accompanies the textbook Thomas’Calculus, Eleventh Edition, by MauriceWeir, Joel Hass and Frank Giordano.The graphing calculators that are featured in this manualinclude the TI-83+/84+, TI-85/86, and TI-89/92+. These calculators were selected because they aremost widely used in calculus and aid students in the understanding of concepts. Although the threecalculators contain similar features, these vary in notation and require different keystrokes (dependingon the calculator), and therefore need to be explained individually.The capabilities of the TI-89/92+ make it the most powerful and complete of the three calculatorsdescribed here. In order to retain consistency among the parts of this manual, we have not attemptedto describe all the details (including symbolic features) of this calculator.For details consult theguidebook that accompanies your TI-89/92+.The manual is divided into four parts: Part I corresponds to the TI-83+/84+, Part II to the TI-85/86,Part III to the TI-89/92+, and Part IV contains sample calculator exercises that can be done in classor assigned as labs. Each part is divided into sections appearing in the same order that they appear ina traditional calculus sequence. Each section features a particular topic and provides examples showingall necessary calculator commands.As indicated above, calculators are very useful in the study of mathematics, and in particular of thecalculus. However, one must always exercise caution when performing numerical calculations. Manycomputations done by calculators contain round-off errors, mainly due to the implementation of thealgorithms used.It is always advisable to double-check answers.In Section 1.7 “Graphing withCalculators and Computers” of your textbook the authors provide a variety of examples of graphsof functions. You can graph these functions with your Texas Instruments calculator and confirm thegraphs and discussions, or note any differences with your calculator.2PART I TI-83+/84+2.1Home Screen Topics2.1.1Built-in Functions and ConstantsIf you are not familiar with the basic operations of addition, subtraction, multiplication and division onthe TI-83+/84+ calculator, we recommend that you review the guidebook that came with the calcu-lator. In addition to the basic operations, the TI-83+/84+ has several built-in functions that are usedextensively in calculus. These include the following functions:xsquared(X,T,θ,nx2),square root(2nd[√]), the trigonometric functionssine(SIN X,T,θ,n),cosine(COS X,T,θ,n),tangent(TAN X,T,θ,n)and their inversesarcsine(2nd [SIN−1] X,T,θ,n),arccosine(2nd [COS−1] X,T,θ,n),arctangent(2nd[TAN−1] X,T,θ,n),natural logarithm(LN X,T,θ,n) andlogarithm to the base ten(LOG X,T,θ,n),nat-ural exponential(2nd [ex]X,T,θ,n) andexponential to the base ten(2nd [10x] X,T,θ,n). Themulti-plicative inverseorreciprocalof a numberx,1x, is obtained byX,T,θ,n[x−1]ENTER. Thethird power,cubic root andx-th rootare found under theMATHmenu.For example, to compute the fourth rootof 21 enter the sequence4 MATH 5 (x√) 2 1 ENTER.Somepowersof numbers other than 2 and 3,includingnegative and fractional powers, are computed using a sequence such asX,T,θ,n∧( 5÷8 ) ENTER, which is the computation ofx5/8.Notice the use of the parenthesis around the entireexponent.Theabsolute value function|x|is listed in theMATH NUMmenu.PressMATH NUM,abs(isthe first item, select it and, pressENTER X,T,θ,n ) ENTER. In each of these cases the variablexmust1

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Contents1Introduction12PART I TI-83+/84+12.1Home Screen Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.1.1Built-in Functions and Constants. . . . . . . . . . . . . . . . . . . . . . . . . .12.1.2Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22.1.3Recalling an Entry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22.1.4Decimal to Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22.2Mode Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32.3Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32.3.1TheSolver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32.3.2Solve(. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.4Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.4.1Entering Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.4.2Graph Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52.4.3Viewing Window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.4.4Graphing a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.4.5ZOOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.4.6TRACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.4.7TABLE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.4.8Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.4.9Composition of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.4.10Piecewise-defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.4.11Polar Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122.4.12Parametric Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132.4.13Split Screen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.5Calculus Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.5.1Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.5.2Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162.5.3Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182.5.4Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202.5.5DRAWmenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212.6Regression and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232.6.1Entering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232.6.2Plotting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242.6.3Regressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242.6.4Residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252.7Sequences and Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .262.7.1Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .262.7.2Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263PART II TI-85/86283.1Home Screen Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283.1.1Built-in Functions and Constants. . . . . . . . . . . . . . . . . . . . . . . . . .283.1.2Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283.1.3Recalling an Entry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283.1.4Decimal to Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29i

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3.2Mode Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293.3Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293.3.1TheSolver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303.3.2Solver(. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313.3.3POLY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313.3.4SIMULT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313.4Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323.4.1Entering Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323.4.2Graph Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333.4.3Viewing Window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333.4.4Graphing a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .343.4.5ZOOM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .343.4.6TRACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .343.4.7TABLE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353.4.8Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363.4.9Composition of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .383.4.10Piecewise-defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .393.4.11Polar Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .393.4.12Parametric Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .413.5Calculus Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423.5.1Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423.5.2Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .433.5.3Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463.5.4Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483.5.5GRAPH DRAWmenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .493.5.6GRAPH MATHmenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .513.6Regression and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .523.6.1Entering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .523.6.2Plotting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .533.6.3Regressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543.6.4Residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .563.7Sequences and Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573.7.1Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573.7.2Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .574PART III TI-89/92+584.1Home Screen Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .584.1.1Built-in Functions and Constants. . . . . . . . . . . . . . . . . . . . . . . . . .584.1.2Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .584.1.3Recalling an Entry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594.1.4Decimal to Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594.2Mode Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594.3Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604.3.1solve(. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604.4Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604.4.1Entering Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614.4.2Graph Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614.4.3Viewing Window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61ii

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4.4.4Graphing a Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .624.4.5Zoom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .624.4.6Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .624.4.7Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .634.4.8Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .634.4.9Composition of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .664.4.10Piecewise-defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674.4.11Polar Graphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .684.4.12Parametric Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .694.4.13Split Screen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .704.5Calculus Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .714.5.1Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .714.5.2Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .724.5.3Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .754.5.4Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .764.5.5GRAPH Drawmenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .784.5.6GRAPH Mathmenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .794.6Regression and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .814.6.1Entering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .814.6.2Plotting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .824.6.3Regressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .834.6.4Residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .844.7Sequences and Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .854.7.1Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .854.7.2Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .855PART IVSample Calculator Labs875.1Calculator Lab 1 – Preliminaries, Trigonometry. . . . . . . . . . . . . . . . . . . . . .875.2Calculator Lab 2 – A Study of1fversusf−1. . . . . . . . . . . . . . . . . . . . . . . .905.3Calculator Lab 3 – Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . .915.4Calculator Lab 4 – Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .935.5Calculator Lab 5 – Applications of the Derivative. . . . . . . . . . . . . . . . . . . . .945.6Calculator Lab 6 – L’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . .965.7Calculator Lab 7 – Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .975.8Calculator Lab 8 – Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1005.9Calculator Lab 9 – Sequences, Series, and Taylor Polynomials. . . . . . . . . . . . . .102iii

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have a numerical value assigned to it before the operation can be executed and you must pressENTERto compute the value of an expression or execute a command. Two mathematical constants that areused in calculus frequently are the numberse, andπ. These are also built into the TI-83+/84+. Thesequence2nd [e] ENTERgivese, and2nd [π] ENTERgivesπ.In calculus, you will be using menus such asGRAPH, CALCandMATH. You must scroll down to an itemin the menu and pressENTERto select it.The sequence2nd [Quit]will clear the menu and returnyou to the Home Screen.2.1.2ExpressionsAfter you input a mathematical expression directly into the TI-83+/84+, pressENTERto evaluate it.When entering an expression, use the arrow keys to move the cursor within the expression, then use thedelete (DEL) and insert (2nd [INS]) keys to edit the expression as needed. The calculator automaticallysaves the answer in the system variableAns. The TI-83+/84+ also allows you to save a value into anamed variable. For example, if you want to compute√2 and save it under the nameR(you can onlyuse one letter for each variable name) execute the sequence2nd [√] 2 ) sto ALPHA [R] ENTER.(Figure 1).Figure 1: Storing a value to a variable2.1.3Recalling an EntryTo retrieve your most recent entry and edit it, press the2nd [ENTRY]keys, position the cursor where youdesire and then edit the expression, pressENTERto execute the command. This feature is particularlyuseful if you are evaluating similar expressions repeatedly. The key sequence2nd [ANS]will retrievethe value of the variableAns, that is, the most recently computed value. You can useAnsas input ina new expression.2.1.4Decimal to FractionThe TI-83+/84+ has a very useful feature, in theMATHmenu, that allows you (in many cases) toconvert your most recently computed value into fractions. Press theMATH 1keys to selectFrac, thenpressENTER. See Figures 2 and 3.Figure 2: TheMATHmenu2

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Figure 3: Converting to fractions2.2Mode SettingsThe Mode Settings control how a calculator interprets and displays numbers and graphs of functions.You may use a variety of these settings in your study of calculus.To access the mode settings on the TI-83+/84+ press theMODEkey. The mode screen is displayed inFigure 4; the current settings are highlighted. To change the settings use the ‘up’ or ‘down’ keys toscroll up or down, then use the ‘left’ or ‘right’ arrow keys to select a setting, then pressENTER. For moredetail on Mode Settings refer to the guidebook that came with your TI-83+/84+ calculator. Specificsettings may be required for certain calculus topics. Your instructor may request that you change modesettings as needed. For now, make sure that your calculator has the same settings as shown in Figure 4.Figure 4: Mode Settings on the TI-83+/84+2.3Solving EquationsWhen you solve an equation you find values for the variables in the equation that make the equationtrue. When you solve an equation given as a function,y=f(x), you find values forxandywhich makey=f(x) true; geometrically, this is equivalent to finding points (x, y) on the graph of the functionf. When you solve the equation,f(x) = 0, you find thezeros of the functionf; geometrically, this isequivalent to finding the points of intersection of the graph of the function with thex-axis.2.3.1TheSolverTheSolveris a feature that allows you to solve an equation,eqn, for any variable. You can accessthe solver from theMATHmenu. Press theMATHkey, then use the up or down arrow keys to scroll toSolver. PressENTERto access the equation editor. (See Figures 5 and 6.) If the equation editor doesnot appear, scroll up using the up arrow key. After you input an equation, which is always assumed to3

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equal zero, pressENTERto activate theSolveras shown in Figure 7.Figure 5: Selecting theSolverFigure 6: The Equation EditorFigure 7: TheSolverThe solver displays all the variables in the equation. You can edit these values by scrolling to the valueof the variable and entering a new value. You also need to provide a guess for the variable for whichyou are solving.Also, make sure you edit thebound ={lower, upper}values.This is not alwaysnecessary but may help you find a solution more quickly, since the TI-83+/84+ searches for a solutionin the interval [lower, upper].To solve, position the cursor at the variable for which you wish to solve, then press theALPHA [SOLVE]keys. The solution is displayed in Figure 8. The solution window contains the solution as well as thevalueleft-rt, which is the difference between the left and right sides of the equation.Figure 8: The equation solvedFor a more detailed discussion of theSolverwith the TI-83+/84+, consult the guidebook that camewith your calculator.4

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2.3.2Solve(This feature is only available from theCATALOG. Press the yellow2ndkey, then[CATALOG]; scroll downuntil you findsolve(, then pressENTER. The format for this command issolve(expression,variable,guess,lower,upper).expressionis assumed to be equal to zero. All variables, except the one for which you wish to solve,should have values assigned to them.guessis an initial guess for the value ofvariable, andlowerandupperare bounds for the solution sought. Once you enter all the necessary information pressENTERtocompute the solution.Your TI-83+/84+ contains other features with which you can find the solution to an equation. Theseare described in Section 2.4 of this manual.2.4FunctionsCalculus is an area of mathematics in which you can study functions of one or more real variables in avariety of ways. The topics below will help you to enter functions into your calculator and to analyzetheir values and graphs. First, make sure that your calculator is set to Function Mode, that isFuncshould be highlighted in the mode settings screen. (See Section 2.2 of this manual.)2.4.1Entering FunctionsThe TI-83+/84+ allows you to store ten functions in its memory. To store a function press theY=keyto access theY= Editor. Figure 9 shows how to enter the functionsy1= 5x−2 andy2=x2−y1(x) =x2−5x+ 2.Figure 9: TheY= EditorYou can use the arrow keys to scroll up or down to select a function or to scroll left and right if youare editing a function.Use theCLEARkey to erase an entire line.In function mode, theX,T,θ,nkey producesX, which is used as the independent variable; the sequenceVARS Y-VARS Function Y1copiesY1onto the screen. When you enter the first character of the function the ‘=’ sign is highlightedindicating that the function is selected and that its graph will be shown in the graph window. If youwish to deselect the function, position the cursor over the ‘=’ sign and pressENTER. One nice thingabout the TI-83+/84+ is that you can use numbers, variables, matrices, lists, and other functions todefine new functions, these features can be particularly useful when studying calculus.2.4.2Graph StyleFunctions can be graphed in different styles. Two such styles and the necessary keystrokes to displaythem are described in this section. For additional information see the guidebook that came with yourcalculator.The standard style for drawing graphs is calledLine. This is the default style setting. With this setting5

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the calculator plots certain points of the graph and then joins them with tiny line segments, creatinga continuous-looking graph.InDotstyle, the calculator simply plots certain points on the graph ofthe function. In this setting, points are not joined together by line segments. To change the style ofthe graph you must be in theY= Editor. Use the arrow keys to place the cursor in the extreme leftposition. Press theENTERkey to change the style. A diagonal segment with three dots is the Dot style.Move the cursor away from the style marker and the new style will be selected. Figure 10 shows thefunctiony1= 5x−2 entered inDotmode. See Example 3 in Section 1.7 of your text.Figure 10: TheDotstyle selected2.4.3Viewing WindowThe viewing window of your calculator only represents a portion of the Cartesian plane. The standardviewing window is within the bounds−10≤x≤10, and−10≤y≤10. In many cases you will needto draw graphs of functions that are outside this range, but this is not a problem if you are using aTI-83+/84+, since you can set the viewing window as needed.Press theWINDOWkey to access theviewing window feature (Figure 11).Figure 11:WINDOWThe values ofXmin,Xmax,Ymin, andYmaxdetermine the portion of the Cartesian plane that will beshown. You must enter values that satisfyXmin<Xmax, andYmin<Ymax. The numbersXsclandYscldetermine the distance between tickmarks. Setting these numbers equal to ten will result in a tickmarkat every ten units; setting these numbers equal to zero will result in no tickmarks. The numberXressets pixel resolution, for our purposes we wantXres=1. See Examples 1, 2 and 4 in Section 1.7 of yourtext.2.4.4Graphing a FunctionPress theGRAPHkey to display the graphs of the functions that you have selected.Your calculatorallows you to analyze graphs in a variety of ways. The remainder of the section contains descriptions ofseveral of the features connected to functions and their graphs. See Section 2.5 for topics that requireknowledge of calculus.2.4.5ZOOMTheZOOMkey allows you to change the viewing window in ten specific ways. See Examples 4 and 5 inSection 1.7 of your text. Select the first item by highlightingZbox. After the graph is drawn use the6

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arrow keys to move the cursor to a position that you want to become one corner of the viewing window.PressENTER, and move the cursor to the opposite corner of the window. PressENTER, and the graphwill be redrawn within the boundaries of the window you selected. TheZoom InandZoom Outfeaturesallow you to look at a graph closer or further away, respectively. To select one of these items, highlightZoom InorZoom Outand pressENTER. A cursor will appear in the graph, which will determine thecenter of the new viewing window. Move the cursor to the desired center and pressENTER. The graphwill be redrawn. The viewing windowXmin=-10, Xmax=10, Xscl=1, Ymin=-10, Ymax=10, Yscl=1is the default set at the factory. You can restore this window by selectingZStandard. A square viewingarea is sometimes necessary,ZSquaresets the dimensions of the viewing window so that a circle willlook like a circle, not like an ellipse. When plotting statistical data pointsZoomStatsets the viewingwindow so all data points are visible in the window.ZoomFitresizes the window, changing only theYvalues in such a way that the graph is displayed within the prespecified values ofX. The other items intheZoommenu are discussed in the guidebook that came with your calculator.2.4.6TRACETheTRACEkey allows you to move the cursor along the graph of a function as the calculator displaysthe values of the coordinates of the points on the graph. Press theTRACEkey, and you will see yourgraph displayed and the trace cursor will appear on the graph. Use the left and right arrow keys tomove the cursor along the graph. You can also move the cursor to a specific point by entering thex-value of the point and pressing theENTERkey. If the values ofxandyare within the viewing window,the cursor will immediately move to the point on the graph that has the givenx-coordinate and thecalculator will display both coordinates.Figure 12 shows the cursor on the graph of the functiony2=x2−y1(x) =x2−5x+ 2 and the coordinates of the point where the cursor is positioned. Use theup and down arrows to move from function to function.Figure 12:TRACE2.4.7TABLEIf you have entered a function intoY1(or any other dependent variable), the table feature will allowyou to compute values for this function for many values of the independent variable. First, press2nd[TBLSET]to set the starting value ofX,TblStart=-1, and the increment ofX, ∆Tbl=.5.Set bothIndpntandDependtoAUTO, pressENTERto save the values (Figure 13). Press2nd [TABLE]to viewa table in which the values forY1are computed automatically.Figure 14 displays a table of valuesfor the functiony1= 5x−2. You can scroll through the table of values using the up and down arrowkeys.When setting the options for the table, you can also setIndpnttoASKandDependtoAUTO.PressENTERto save these options, then press2nd [TABLE]. Enter a value forX, pressENTERand thecorresponding value forY1will be computed. For more information on tables, see the guidebook that7

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came with your calculator.Figure 13:TBLSETFigure 14:TABLE2.4.8Finding Zeros of FunctionsThis section contains methods for finding zeros of functions, that is, points where the graph of thefunction crosses thex-axis.Your calculator has built-in algorithms, that make use of graphs andtables, for finding zeros of functions.The values obtained with these methods may be very roughapproximations, depending on your calculator. (See Section 2.3 for other methods of finding zeros offunctions.)Trace.Enter and graph the functiony1=x3+ 2.55x2−2.655x−5.13 in the viewing windowXmin=-3,Xmax=3, Xscl=1, Ymin=-4, Ymax=2, Yscl=1. Press theTRACEkey and use the arrow keys to movethe cursor to the point where the graph meets thex-axis.Once you establish anx-value that givesyou ay-value close to zero, you can experiment with the graph and zoom in to reach otherx-valuesthat may give ay-value closer to zero (Figure 15). In many cases, you may not be able to arrive at anx-value that lies exactly on thex-axis.Figure 15: Finding zeros of a function withTRACETable.Enter the functiony1=x3+ 2.55x2−2.655x−5.13 and construct a table of values for thefunction (see Section 2.4.7). You might want to take a peek at the graph to see if there is a zero between1 and 2. If this is the case, it’s a good idea to setTblStart=1and ∆Tbl=0.1, and bothIndpntandDependtoAUTO. Scroll through the values in the table to find values of the dependent variable closeto zero. Once you establish anx-value that gives you ay-value close to zero, you can experiment with8

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other values ofTblStartand ∆Tblto see if you can achieve ay-value of closer to zero (Figure 16). Inmany cases, you may not be able to arrive at anx-value that yields ay-value of exactly zero.Figure 16: Finding zeros of a function usingTABLEZero.Enter the functiony1=x3+ 2.55x2−2.655x−5.13 and graph it using the viewing windowXmin=-3, Xmax=3, Xscl=1, Ymin=-8, Ymax=4, Yscl=2. Press2nd [CALC]to access theCALCULATEmenu, selectzeroby pressing2. Use the arrow keys to move the cursor to select the left bound, theright bound, and a guess, as prompted by the calculator. PressENTERto save each of your selections.The cursor will move to the zero of the function, and the calculator will display the values ofxandyat that point (Figures 17–19).Figure 17: Left boundFigure 18: Right boundFigure 19: The zeroIntersection.Suppose you want to solvee3x−5x−7 = 0 forx. This problem is equivalent to findingthex-value of the point where the graphs ofy1=e3xandy2= 5x+ 7 meet. Enter both functions intomemory and graph them. Use the viewing windowXmin= -5, Xmax=5, Xscl=1, Ymin=-3, Ymax=15,Yscl=1. Press2nd [CALC]to access the[CALCULATE]menu, selectintersectby pressing5. Use thearrow keys to move the cursor to select the first curve, the second curve, and a guess, as prompted by9

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the calculator. PressENTERto save each one. The cursor will move to the point of intersection of thecurves, and the calculator will display the values ofxandyat that point (Figures 20–22).Figure 20: First curveFigure 21: Second curveFigure 22: The intersection2.4.9Composition of FunctionsFunctions defined in the TI-83+/84+ can be combined to form new functions, one such combinationis the composition of two functions. Enter the functionsy1= 1−xandy2=exinto your calculator.Both functions have domain equal to the set of real numbers, therefore the compositionsy1(y2(x)),andy2(y1(x)) can both be formed without restrictions.EnterY3=Y1(Y2(x)) as shown in Figure 23.This is the functiony3= 1−ex; its graph is shown in Figure 24, using the viewing windowXmin= -5,Xmax=5, Xscl=1, Ymin=-5, Ymax=5, Yscl=1.EnterY4=Y2(Y1(x)), this is the functiony4=e1−x;its graph is shown in Figure 25. (Recall that the symbolY1(Y2, respectively) is obtained by means ofthe keystroke sequenceVARS Y-VARS Function Y1(VARS Y-VARS Function Y2, respectively).Figure 23: The functiony3=y1(y2(x)) entered10

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Figure 24: The graph ofy3= 1−exFigure 25: The graph ofy4=e1−x2.4.10Piecewise-defined functionsIn many applications, functions cannot be given by one unique formula.Instead, functions relatedto applications are given in parts.Such functions are calledpiecewise-defined functions.The TI-83+/84+ allows you to enter and graph piecewise-defined functions.Consider the functionf(x) ={ex+ 1−2≤x≤0x2−2x+ 20< x≤32.In order to avoid any vertical lines, you must first change the GraphStyle toDot, (see Section 2.4.2) then enter the function as shown in Figure 26. The symbols ‘≤’, ‘≥’and ‘>’ are found in the2nd [TEST]menu. Set the viewing window toXmin=-2.5, Xmax=2, Xscl=1,Ymin=-1, Ymax=4, Yscl=1, and pressGRAPH. The graph of the piece-wise defined function is shownin Figure 27. Notice that the graph is limited to the interval [−2,32].Figure 26: Entering a piecewise-defined functionFigure 27: The graph of a piecewise-defined function11

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