to accompany T HOMAS ’ C ALCULUS E LEVENTH E DITION and T HOMAS ’ C ALCULUS E ARLY T RANSCENDENTALS E LEVENTH E DITION B ASED ON THE O RIGINAL W ORK BY George B. Thomas, Jr. Massachusetts Institute of Technology AS R EVISED BY Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis Frank R. Giordano Naval Postgraduate School O NLINE TI ® G RAPHING C ALCULATOR M ANUAL L UZ D E A LBA Drake University 1 Introduction This calculator manual accompanies the textbook Thomas’ Calculus , Eleventh Edition, by Maurice Weir, Joel Hass and Frank Giordano. The graphing calculators that are featured in this manual include the TI-83+/84+, TI-85/86, and TI-89/92+. These calculators were selected because they are most widely used in calculus and aid students in the understanding of concepts. Although the three calculators contain similar features, these vary in notation and require different keystrokes (depending on 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 calculators described here. In order to retain consistency among the parts of this manual, we have not attempted to describe all the details (including symbolic features) of this calculator. For details consult the guidebook 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 class or assigned as labs. Each part is divided into sections appearing in the same order that they appear in a traditional calculus sequence. Each section features a particular topic and provides examples showing all necessary calculator commands. As indicated above, calculators are very useful in the study of mathematics, and in particular of the calculus. However, one must always exercise caution when performing numerical calculations. Many computations done by calculators contain round-off errors, mainly due to the implementation of the algorithms used. It is always advisable to double-check answers. In Section 1.7 “Graphing with Calculators and Computers” of your textbook the authors provide a variety of examples of graphs of functions. You can graph these functions with your Texas Instruments calculator and confirm the graphs and discussions, or note any differences with your calculator. 2 PART I TI-83+/84+ 2.1 Home Screen Topics 2.1.1 Built-in Functions and Constants If you are not familiar with the basic operations of addition, subtraction, multiplication and division on the 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 used extensively in calculus. These include the following functions: x squared ( X,T, θ ,n x 2 ), square root ( 2nd [ √ ] ), the trigonometric functions sine ( SIN X,T, θ ,n ), cosine ( COS X,T, θ ,n ), tangent ( TAN X,T, θ ,n ) and their inverses arcsine ( 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 ) and logarithm to the base ten ( LOG X,T, θ ,n ), nat- ural exponential ( 2nd [e x ] X,T, θ ,n) and exponential to the base ten ( 2nd [10 x ] X,T, θ ,n ). The multi- plicative inverse or reciprocal of a number x , 1 x , is obtained by X,T, θ ,n [ x − 1 ] ENTER . The third power, cubic root and x -th root are found under the MATH menu. For example, to compute the fourth root of 21 enter the sequence 4 MATH 5 ( x √ ) 2 1 ENTER . Some powers of numbers other than 2 and 3, including negative and fractional powers , are computed using a sequence such as X,T, θ ,n ∧ ( 5 ÷ 8 ) ENTER , which is the computation of x 5 / 8 . Notice the use of the parenthesis around the entire exponent. The absolute value function | x | is listed in the MATH NUM menu. Press MATH NUM , abs( is the first item, select it and, press ENTER X,T, θ ,n ) ENTER . In each of these cases the variable x must 1 Contents 1 Introduction 1 2 PART I TI-83+/84+ 1 2.1 Home Screen Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1.1 Built-in Functions and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.1.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.3 Recalling an Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.4 Decimal to Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Mode Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3.1 The Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3.2 Solve( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4.1 Entering Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4.2 Graph Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4.3 Viewing Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4.4 Graphing a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4.5 ZOOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4.6 TRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.7 TABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4.8 Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.9 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.10 Piecewise-defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.11 Polar Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.12 Parametric Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.13 Split Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Calculus Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.2 Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.3 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5.4 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.5 DRAW menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Regression and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.1 Entering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.2 Plotting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.3 Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.4 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 PART II TI-85/86 28 3.1 Home Screen Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Built-in Functions and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.3 Recalling an Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.4 Decimal to Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 i 3.2 Mode Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1 The Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Solver( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.3 POLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4 SIMULT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 Entering Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.2 Graph Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.3 Viewing Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.4 Graphing a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.5 ZOOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.6 TRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.7 TABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.8 Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.9 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4.10 Piecewise-defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.11 Polar Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.12 Parametric Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5 Calculus Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5.2 Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5.3 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5.4 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.5 GRAPH DRAW menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5.6 GRAPH MATH menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Regression and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6.1 Entering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6.2 Plotting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6.3 Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.6.4 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.7.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.7.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 PART III TI-89/92+ 58 4.1 Home Screen Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.1 Built-in Functions and Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.2 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.3 Recalling an Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1.4 Decimal to Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Mode Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.1 solve( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.1 Entering Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.2 Graph Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.3 Viewing Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ii 4.4.4 Graphing a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.5 Zoom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.6 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.7 Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.8 Finding Zeros of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.9 Composition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4.10 Piecewise-defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.11 Polar Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.12 Parametric Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.13 Split Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.5 Calculus Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5.2 Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.5.3 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5.4 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5.5 GRAPH Draw menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.6 GRAPH Math menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.6 Regression and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.6.1 Entering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.6.2 Plotting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.6.3 Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6.4 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.7 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.7.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.7.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5 PART IV Sample Calculator Labs 87 5.1 Calculator Lab 1 – Preliminaries, Trigonometry . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Calculator Lab 2 – A Study of 1 f versus f − 1 . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 Calculator Lab 3 – Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4 Calculator Lab 4 – Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Calculator Lab 5 – Applications of the Derivative . . . . . . . . . . . . . . . . . . . . . 94 5.6 Calculator Lab 6 – L’Hˆ opital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.7 Calculator Lab 7 – Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.8 Calculator Lab 8 – Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.9 Calculator Lab 9 – Sequences, Series, and Taylor Polynomials . . . . . . . . . . . . . . 102 iii have a numerical value assigned to it before the operation can be executed and you must press ENTER to compute the value of an expression or execute a command. Two mathematical constants that are used in calculus frequently are the numbers e , and π . These are also built into the TI-83+/84+. The sequence 2nd [e] ENTER gives e , and 2nd [ π ] ENTER gives π . In calculus, you will be using menus such as GRAPH, CALC and MATH . You must scroll down to an item in the menu and press ENTER to select it. The sequence 2nd [Quit] will clear the menu and return you to the Home Screen. 2.1.2 Expressions After you input a mathematical expression directly into the TI-83+/84+, press ENTER to evaluate it. When entering an expression, use the arrow keys to move the cursor within the expression, then use the delete ( DEL ) and insert ( 2nd [INS] ) keys to edit the expression as needed. The calculator automatically saves the answer in the system variable Ans . The TI-83+/84+ also allows you to save a value into a named variable. For example, if you want to compute √ 2 and save it under the name R (you can only use one letter for each variable name) execute the sequence 2nd [ √ ] 2 ) sto ALPHA [R] ENTER . (Figure 1). Figure 1: Storing a value to a variable 2.1.3 Recalling an Entry To retrieve your most recent entry and edit it, press the 2nd [ENTRY] keys, position the cursor where you desire and then edit the expression, press ENTER to execute the command. This feature is particularly useful if you are evaluating similar expressions repeatedly. The key sequence 2nd [ANS] will retrieve the value of the variable Ans , that is, the most recently computed value. You can use Ans as input in a new expression. 2.1.4 Decimal to Fraction The TI-83+/84+ has a very useful feature, in the MATH menu, that allows you (in many cases) to convert your most recently computed value into fractions. Press the MATH 1 keys to select Frac , then press ENTER . See Figures 2 and 3. Figure 2: The MATH menu 2 Figure 3: Converting to fractions 2.2 Mode Settings The 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 the MODE key. The mode screen is displayed in Figure 4; the current settings are highlighted. To change the settings use the ‘up’ or ‘down’ keys to scroll up or down, then use the ‘left’ or ‘right’ arrow keys to select a setting, then press ENTER . For more detail on Mode Settings refer to the guidebook that came with your TI-83+/84+ calculator. Specific settings may be required for certain calculus topics. Your instructor may request that you change mode settings 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+ 2.3 Solving Equations When you solve an equation you find values for the variables in the equation that make the equation true. When you solve an equation given as a function, y = f ( x ), you find values for x and y which make y = f ( x ) true; geometrically, this is equivalent to finding points ( x, y ) on the graph of the function f . When you solve the equation, f ( x ) = 0, you find the zeros of the function f ; geometrically, this is equivalent to finding the points of intersection of the graph of the function with the x -axis. 2.3.1 The Solver The Solver is a feature that allows you to solve an equation, eqn , for any variable. You can access the solver from the MATH menu. Press the MATH key, then use the up or down arrow keys to scroll to Solver . Press ENTER to access the equation editor. (See Figures 5 and 6.) If the equation editor does not appear, scroll up using the up arrow key. After you input an equation, which is always assumed to 3 equal zero, press ENTER to activate the Solver as shown in Figure 7. Figure 5: Selecting the Solver Figure 6: The Equation Editor Figure 7: The Solver The solver displays all the variables in the equation. You can edit these values by scrolling to the value of the variable and entering a new value. You also need to provide a guess for the variable for which you are solving. Also, make sure you edit the bound = { lower, upper } values. This is not always necessary but may help you find a solution more quickly, since the TI-83+/84+ searches for a solution in the interval [lower, upper]. To solve, position the cursor at the variable for which you wish to solve, then press the ALPHA [SOLVE] keys. The solution is displayed in Figure 8. The solution window contains the solution as well as the value left-rt , which is the difference between the left and right sides of the equation. Figure 8: The equation solved For a more detailed discussion of the Solver with the TI-83+/84+, consult the guidebook that came with your calculator. 4 2.3.2 Solve( This feature is only available from the CATALOG . Press the yellow 2nd key, then [CATALOG] ; scroll down until you find solve( , then press ENTER . The format for this command is solve(expression,variable,guess,lower,upper) . expression is assumed to be equal to zero. All variables, except the one for which you wish to solve, should have values assigned to them. guess is an initial guess for the value of variable , and lower and upper are bounds for the solution sought. Once you enter all the necessary information press ENTER to compute the solution. Your TI-83+/84+ contains other features with which you can find the solution to an equation. These are described in Section 2.4 of this manual. 2.4 Functions Calculus is an area of mathematics in which you can study functions of one or more real variables in a variety of ways. The topics below will help you to enter functions into your calculator and to analyze their values and graphs. First, make sure that your calculator is set to Function Mode, that is Func should be highlighted in the mode settings screen. (See Section 2.2 of this manual.) 2.4.1 Entering Functions The TI-83+/84+ allows you to store ten functions in its memory. To store a function press the Y= key to access the Y= Editor . Figure 9 shows how to enter the functions y 1 = 5 x − 2 and y 2 = x 2 − y 1 ( x ) = x 2 − 5 x + 2. Figure 9: The Y= Editor You can use the arrow keys to scroll up or down to select a function or to scroll left and right if you are editing a function. Use the CLEAR key to erase an entire line. In function mode, the X,T, θ ,n key produces X , which is used as the independent variable; the sequence VARS Y-VARS Function Y1 copies Y1 onto the screen. When you enter the first character of the function the ‘=’ sign is highlighted indicating that the function is selected and that its graph will be shown in the graph window. If you wish to deselect the function, position the cursor over the ‘=’ sign and press ENTER . One nice thing about the TI-83+/84+ is that you can use numbers, variables, matrices, lists, and other functions to define new functions, these features can be particularly useful when studying calculus. 2.4.2 Graph Style Functions can be graphed in different styles. Two such styles and the necessary keystrokes to display them are described in this section. For additional information see the guidebook that came with your calculator. The standard style for drawing graphs is called Line . This is the default style setting. With this setting 5 the calculator plots certain points of the graph and then joins them with tiny line segments, creating a continuous-looking graph. In Dot style, the calculator simply plots certain points on the graph of the function. In this setting, points are not joined together by line segments. To change the style of the graph you must be in the Y= Editor . Use the arrow keys to place the cursor in the extreme left position. Press the ENTER key 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 the function y 1 = 5 x − 2 entered in Dot mode. See Example 3 in Section 1.7 of your text. Figure 10: The Dot style selected 2.4.3 Viewing Window The viewing window of your calculator only represents a portion of the Cartesian plane. The standard viewing window is within the bounds − 10 ≤ x ≤ 10, and − 10 ≤ y ≤ 10. In many cases you will need to draw graphs of functions that are outside this range, but this is not a problem if you are using a TI-83+/84+, since you can set the viewing window as needed. Press the WINDOW key to access the viewing window feature (Figure 11). Figure 11: WINDOW The values of Xmin , Xmax , Ymin , and Ymax determine the portion of the Cartesian plane that will be shown. You must enter values that satisfy Xmin<Xmax , and Ymin<Ymax . The numbers Xscl and Yscl determine the distance between tickmarks. Setting these numbers equal to ten will result in a tickmark at every ten units; setting these numbers equal to zero will result in no tickmarks. The number Xres sets pixel resolution, for our purposes we want Xres=1 . See Examples 1, 2 and 4 in Section 1.7 of your text. 2.4.4 Graphing a Function Press the GRAPH key to display the graphs of the functions that you have selected. Your calculator allows you to analyze graphs in a variety of ways. The remainder of the section contains descriptions of several of the features connected to functions and their graphs. See Section 2.5 for topics that require knowledge of calculus. 2.4.5 ZOOM The ZOOM key allows you to change the viewing window in ten specific ways. See Examples 4 and 5 in Section 1.7 of your text. Select the first item by highlighting Zbox . After the graph is drawn use the 6 arrow keys to move the cursor to a position that you want to become one corner of the viewing window. Press ENTER , and move the cursor to the opposite corner of the window. Press ENTER , and the graph will be redrawn within the boundaries of the window you selected. The Zoom In and Zoom Out features allow you to look at a graph closer or further away, respectively. To select one of these items, highlight Zoom In or Zoom Out and press ENTER . A cursor will appear in the graph, which will determine the center of the new viewing window. Move the cursor to the desired center and press ENTER . The graph will be redrawn. The viewing window Xmin=-10, Xmax=10, Xscl=1, Ymin=-10, Ymax=10, Yscl=1 is the default set at the factory. You can restore this window by selecting ZStandard . A square viewing area is sometimes necessary, ZSquare sets the dimensions of the viewing window so that a circle will look like a circle, not like an ellipse. When plotting statistical data points ZoomStat sets the viewing window so all data points are visible in the window. ZoomFit resizes the window, changing only the Y values in such a way that the graph is displayed within the prespecified values of X . The other items in the Zoom menu are discussed in the guidebook that came with your calculator. 2.4.6 TRACE The TRACE key allows you to move the cursor along the graph of a function as the calculator displays the values of the coordinates of the points on the graph. Press the TRACE key, and you will see your graph displayed and the trace cursor will appear on the graph. Use the left and right arrow keys to move the cursor along the graph. You can also move the cursor to a specific point by entering the x - value of the point and pressing the ENTER key. If the values of x and y are within the viewing window, the cursor will immediately move to the point on the graph that has the given x -coordinate and the calculator will display both coordinates. Figure 12 shows the cursor on the graph of the function y 2 = x 2 − y 1 ( x ) = x 2 − 5 x + 2 and the coordinates of the point where the cursor is positioned. Use the up and down arrows to move from function to function. Figure 12: TRACE 2.4.7 TABLE If you have entered a function into Y1 (or any other dependent variable), the table feature will allow you to compute values for this function for many values of the independent variable. First, press 2nd [TBLSET] to set the starting value of X , TblStart=-1 , and the increment of X , ∆ Tbl=.5 . Set both Indpnt and Depend to AUTO , press ENTER to save the values (Figure 13). Press 2nd [TABLE] to view a table in which the values for Y1 are computed automatically. Figure 14 displays a table of values for the function y 1 = 5 x − 2. You can scroll through the table of values using the up and down arrow keys. When setting the options for the table, you can also set Indpnt to ASK and Depend to AUTO . Press ENTER to save these options, then press 2nd [TABLE] . Enter a value for X , press ENTER and the corresponding value for Y1 will be computed. For more information on tables, see the guidebook that 7 came with your calculator. Figure 13: TBLSET Figure 14: TABLE 2.4.8 Finding Zeros of Functions This section contains methods for finding zeros of functions, that is, points where the graph of the function crosses the x -axis. Your calculator has built-in algorithms, that make use of graphs and tables, for finding zeros of functions. The values obtained with these methods may be very rough approximations, depending on your calculator. (See Section 2.3 for other methods of finding zeros of functions.) Trace. Enter and graph the function y 1 = x 3 + 2 . 55 x 2 − 2 . 655 x − 5 . 13 in the viewing window Xmin=-3, Xmax=3, Xscl=1, Ymin=-4, Ymax=2, Yscl=1 . Press the TRACE key and use the arrow keys to move the cursor to the point where the graph meets the x -axis. Once you establish an x -value that gives you a y -value close to zero, you can experiment with the graph and zoom in to reach other x -values that may give a y -value closer to zero (Figure 15). In many cases, you may not be able to arrive at an x -value that lies exactly on the x -axis. Figure 15: Finding zeros of a function with TRACE Table. Enter the function y 1 = x 3 + 2 . 55 x 2 − 2 . 655 x − 5 . 13 and construct a table of values for the function (see Section 2.4.7). You might want to take a peek at the graph to see if there is a zero between 1 and 2. If this is the case, it’s a good idea to set TblStart=1 and ∆ Tbl=0.1 , and both Indpnt and Depend to AUTO . Scroll through the values in the table to find values of the dependent variable close to zero. Once you establish an x -value that gives you a y -value close to zero, you can experiment with 8 other values of TblStart and ∆ Tbl to see if you can achieve a y -value of closer to zero (Figure 16). In many cases, you may not be able to arrive at an x -value that yields a y -value of exactly zero. Figure 16: Finding zeros of a function using TABLE Zero. Enter the function y 1 = x 3 + 2 . 55 x 2 − 2 . 655 x − 5 . 13 and graph it using the viewing window Xmin=-3, Xmax=3, Xscl=1, Ymin=-8, Ymax=4, Yscl=2 . Press 2nd [CALC] to access the CALCULATE menu, select zero by pressing 2 . Use the arrow keys to move the cursor to select the left bound, the right bound, and a guess, as prompted by the calculator. Press ENTER to save each of your selections. The cursor will move to the zero of the function, and the calculator will display the values of x and y at that point (Figures 17–19). Figure 17: Left bound Figure 18: Right bound Figure 19: The zero Intersection. Suppose you want to solve e 3 x − 5 x − 7 = 0 for x . This problem is equivalent to finding the x -value of the point where the graphs of y 1 = e 3 x and y 2 = 5 x + 7 meet. Enter both functions into memory and graph them. Use the viewing window Xmin= -5, Xmax=5, Xscl=1, Ymin=-3, Ymax=15, Yscl=1 . Press 2nd [CALC] to access the [CALCULATE] menu, select intersect by pressing 5 . Use the arrow keys to move the cursor to select the first curve, the second curve, and a guess, as prompted by 9 the calculator. Press ENTER to save each one. The cursor will move to the point of intersection of the curves, and the calculator will display the values of x and y at that point (Figures 20–22). Figure 20: First curve Figure 21: Second curve Figure 22: The intersection 2.4.9 Composition of Functions Functions defined in the TI-83+/84+ can be combined to form new functions, one such combination is the composition of two functions. Enter the functions y 1 = 1 − x and y 2 = e x into your calculator. Both functions have domain equal to the set of real numbers, therefore the compositions y 1 ( y 2 ( x )), and y 2 ( y 1 ( x )) can both be formed without restrictions. Enter Y 3 = Y 1 ( Y 2 ( x )) as shown in Figure 23. This is the function y 3 = 1 − e x ; its graph is shown in Figure 24, using the viewing window Xmin= -5, Xmax=5, Xscl=1, Ymin=-5, Ymax=5, Yscl=1 . Enter Y 4 = Y 2 ( Y 1 ( x )), this is the function y 4 = e 1 − x ; its graph is shown in Figure 25. (Recall that the symbol Y 1 ( Y 2 , respectively) is obtained by means of the keystroke sequence VARS Y-VARS Function Y1 ( VARS Y-VARS Function Y2 , respectively). Figure 23: The function y 3 = y 1 ( y 2 ( x )) entered 10 Figure 24: The graph of y 3 = 1 − e x Figure 25: The graph of y 4 = e 1 − x 2.4.10 Piecewise-defined functions In many applications, functions cannot be given by one unique formula. Instead, functions related to applications are given in parts. Such functions are called piecewise-defined functions . The TI- 83+/84+ allows you to enter and graph piecewise-defined functions. Consider the function f ( x ) = { e x + 1 − 2 ≤ x ≤ 0 x 2 − 2 x + 2 0 < x ≤ 3 2 . In order to avoid any vertical lines, you must first change the Graph Style to Dot , (see Section 2.4.2) then enter the function as shown in Figure 26. The symbols ‘ ≤ ’, ‘ ≥ ’ and ‘ > ’ are found in the 2nd [TEST] menu. Set the viewing window to Xmin=-2.5, Xmax=2, Xscl=1, Ymin=-1, Ymax=4, Yscl=1 , and press GRAPH . The graph of the piece-wise defined function is shown in Figure 27. Notice that the graph is limited to the interval [ − 2 , 3 2 ]. Figure 26: Entering a piecewise-defined function Figure 27: The graph of a piecewise-defined function 11