Analyzing the Graph of a Function


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1 SECTION A Summar of Curve Sketching Section Different viewing windows for the graph of f Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the Graph of a Function It would be difficult to overstate the importance of using graphs in mathematics Descartes s introduction of analtic geometr contributed significantl to the rapid advances in calculus that began during the midseventeenth centur In the words of Lagrange, As long as algebra and geometr traveled separate paths their advance was slow and their applications limited But when these two sciences joined compan, the drew from each other fresh vitalit and thenceforth marched on at a rapid pace toward perfection So far, ou have studied several concepts that are useful in analzing the graph of a function intercepts and  intercepts (Section P) Smmetr (Section P) Domain and range (Section P) Continuit (Section ) Vertical asmptotes (Section 5) Differentiabilit (Section ) Relative etrema (Section ) Concavit (Section ) Points of (Section ) Horizontal asmptotes (Section 5) Infinite limits at infinit (Section 5) When ou are sketching the graph of a function, either b hand or with a graphing utilit, remember that normall ou cannot show the entire graph The decision as to which part of the graph ou choose to show is often crucial For instance, which of the viewing windows in Figure better represents the graph of f 5 7 0? B seeing both views, it is clear that the second viewing window gives a more complete representation of the graph But would a third viewing window reveal other interesting portions of the graph? To answer this, ou need to use calculus to interpret the first and second derivatives Here are some guidelines for determining a good viewing window for the graph of a function Guidelines for Analzing the Graph of a Function Determine the domain and range of the function Determine the intercepts, asmptotes, and smmetr of the graph Locate the values for which f and f either are zero or do not eist Use the results to determine relative etrema and points of NOTE In these guidelines, note the importance of algebra (as well as calculus) for solving the equations f 0, f 0, and f 0
2 0 CHAPTER Applications of Differentiation EXAMPLE Sketching the Graph of a Rational Function f() = ( 9) Vertical asmptote: = Horizontal asmptote: = Vertical asmptote: = ( ) 0, (, 0) (, 0) Using calculus, ou can be certain that ou have determined all characteristics of the graph of f Figure 5 Relative minimum Analze and sketch the graph of f 9 First derivative: f 0 Second derivative: f 0 intercepts:, 0,, 0 intercept: 0, 9 Vertical asmptotes:, Horizontal asmptote: Critical number: 0 Possible points of : None Domain: All real numbers ecept ± Smmetr: With respect to ais Test intervals:,,, 0, 0,,, The table shows how the test intervals are used to determine several characteristics of the graph The graph of f is shown in Figure 5 FOR FURTHER INFORMATION For more information on the use of technolog to graph rational functions, see the article Graphs of Rational Functions for Computer Assisted Calculus b Stan Brd and Terr Walters in The College Mathematics Journal MathArticle < < < < < < < < f Decreasing, concave downward Undef Undef Undef Vertical asmptote 9 f 0 Relative minimum f Increasing, concave upward Undef Undef Undef Vertical asmptote Increasing, concave downward Tr It Eploration A Video Video Video Video 8 B not using calculus ou ma overlook important characteristics of the graph of g Figure Be sure ou understand all of the implications of creating a table such as that shown in Eample Because of the use of calculus, ou can be sure that the graph has no relative etrema or points of other than those shown in Figure 5 TECHNOLOGY PITFALL Without using the tpe of analsis outlined in Eample, it is eas to obtain an incomplete view of a graph s basic characteristics For instance, Figure shows a view of the graph of g 9 0 From this view, it appears that the graph of g is about the same as the graph of f shown in Figure 5 The graphs of these two functions, however, differ significantl Tr enlarging the viewing window to see the differences
3 SECTION A Summar of Curve Sketching EXAMPLE Sketching the Graph of a Rational Function Analze and sketch the graph of f 8 (0, ) Figure 7 Vertical asmptote: = (, ) Relative minimum Relative maimum f() = + First derivative: f 8 Second derivative: f intercepts: None intercept: 0, Vertical asmptote: Horizontal asmptotes: None End behavior: lim f, lim f Critical numbers: 0, Possible points of : None Domain: All real numbers ecept Test intervals:, 0, 0,,,,, The analsis of the graph of f is shown in the table, and the graph is shown in Figure 7 f f f < < 0 Increasing, concave downward 0 0 Relative maimum 0 < < Decreasing, concave downward Undef Undef Undef Vertical asmptote < < 0 Relative minimum < < Increasing, concave upward 8 Vertical asmptote: = A slant asmptote Figure 8 Slant asmptote: = f() = + Tr It Although the graph of the function in Eample has no horizontal asmptote, it does have a slant asmptote The graph of a rational function (having no common factors and whose denominator is of degree or greater) has a slant asmptote if the degree of the numerator eceeds the degree of the denominator b eactl To find the slant asmptote, use long division to rewrite the rational function as the sum of a firstdegree polnomial and another rational function f Eploration A Write original equation Rewrite using long division In Figure 8, note that the graph of f approaches the slant asmptote as approaches or
4 CHAPTER Applications of Differentiation EXAMPLE Sketching the Graph of a Radical Function Analze and sketch the graph of f f() = Horizontal asmptote: = Figure 9 + Horizontal asmptote: = (0, 0) f f 5 The graph has onl one intercept, 0, 0 It has no vertical asmptotes, but it has two horizontal asmptotes: (to the right) and (to the left) The function has no critical numbers and one possible point of (at 0) The domain of the function is all real numbers, and the graph is smmetric with respect to the origin The analsis of the graph of f is shown in the table, and the graph is shown in Figure 9 < < < < f f Increasing, concave upward 0 0 f Increasing, concave downward Tr It Eploration A EXAMPLE Sketching the Graph of a Radical Function Analze and sketch the graph of f 5 5 Relative f() = 5/ 5 / maimum (0, 0) 8 (, ) (8, ) Relative minimum Figure 50 ( ) 5 8, 0 f 0 f 0 9 The function has two intercepts: 0, 0 and 8, 0 There are no horizontal or vertical asmptotes The function has two critical numbers ( 0 and 8) and two possible points of ( 0 and ) The domain is all real numbers The analsis of the graph of f is shown in the table, and the graph is shown in Figure 50 < < < < < < < < f 5 Increasing, concave downward 0 0 Undef Relative maimum f f Decreasing, concave downward 0 0 Relative minimum Increasing, concave upward Tr It Eploration A
5 SECTION A Summar of Curve Sketching EXAMPLE 5 Sketching the Graph of a Polnomial Function Analze and sketch the graph of f 8 5 (a) (0, 0) 5 (, 0) (, ) (, 7) Relative minimum f() = Generated b Derive (b) A polnomial function of even degree must have at least one relative etremum Figure 5 Begin b factoring to obtain f 8 Then, using the factored form of f, ou can perform the following analsis First derivative: f Second derivative: f intercepts: 0, 0,, 0 intercept: 0, 0 Vertical asmptotes: None Horizontal asmptotes: None End behavior: lim f, lim f Critical numbers:, Possible points of :, Domain: All real numbers Test intervals:,,,,,,, The analsis of the graph of f is shown in the table, and the graph is shown in Figure 5(a) Using a computer algebra sstem such as Derive [see Figure 5(b)] can help ou verif our analsis < < < < < < < < f 7 f 0 Relative minimum f Increasing, concave upward 0 Increasing, concave downward Increasing, concave upward Tr It Eploration A Eploration B Open Eploration The fourthdegree polnomial function in Eample 5 has one relative minimum and no relative maima In general, a polnomial function of degree n can have at most n relative etrema, and at most n points of Moreover, polnomial functions of even degree must have at least one relative etremum Remember from the Leading Coefficient Test described in Section P that the end behavior of the graph of a polnomial function is determined b its leading coefficient and its degree For instance, because the polnomial in Eample 5 has a positive leading coefficient, the graph rises to the right Moreover, because the degree is even, the graph also rises to the left
6 CHAPTER Applications of Differentiation EXAMPLE Sketching the Graph of a Trigonometric Function (a) π Vertical asmptote: = π (b) Figure 5 π π π ( π, 0) π π π π π (0, ) f() = cos + sin π Vertical asmptote: = Generated b Derive Analze and sketch the graph of f cos sin Because the function has a period of, ou can restrict the analsis of the graph to an interval of length For convenience, choose, First derivative: Second derivative: Period: intercept: intercept: Vertical asmptotes: Horizontal asmptotes: Critical numbers: f sin f, 0 0,, None None See Note below Possible points of : n Domain: All real numbers ecept Test intervals:,,, The analsis of the graph of f on the interval, is shown in the table, and the graph is shown in Figure 5(a) Compare this with the graph generated b the computer algebra sstem Derive in Figure 5(b) < < < < f f cos sin Undef Undef Undef Vertical asmptote 0 0 f Decreasing, concave downward Undef Undef Undef Vertical asmptote Tr It Eploration A NOTE B substituting or into the function, ou obtain the form 00 This is called an indeterminate form and ou will stud this in Section 87 To determine that the function has vertical asmptotes at these two values, ou can rewrite the function as follows f cos cos sin cos sin sin sin sin sin cos cos In this form, it is clear that the graph of f has vertical asmptotes when and
7 SECTION A Summar of Curve Sketching 5 Eercises for Section The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, match the graph of f in the left column with that of its derivative in the right column Graph of f Graph of f (a) (b) (c) (d) 5 Graphical Reasoning The graph of f is shown in the figure (a) For which values of is f zero? Positive? Negative? (b) For which values of is f zero? Positive? Negative? (c) On what interval is f an increasing function?,,,, (d) For which value of is f minimum? For this value of, how does the rate of change of f compare with the rate of change of f for other values of? Eplain Graphical Reasoning Identif the real numbers 0 and in the figure such that each of the following is true (a) f 0 (b) f 0 (c) f does not eist (d) f has a relative maimum (e) f has a point of Figure for 5 Figure for In Eercises 7, analze and sketch a graph of the function Label an intercepts, relative etrema, points of, and asmptotes Use a graphing utilit to verif our results f g f 5 f f g 9 h f f In Eercises 5 8, use a computer algebra sstem to analze and graph the function Identif an relative etrema, points of, and asmptotes 5 f 0 7 f 8 f 7 In Eercises 9, sketch a graph of the function over the given interval Use a graphing utilit to verif our graph 9 sin 8 sin, 0 0 cos cos, f 5 5 f
8 CHAPTER Applications of Differentiation 5 tan, cot, csc sec, sec 8 tan 8, g tan, g cot, < < < < 0 < < 0 < < < < Writing About Concepts < < In Eercises 7 and 8, the graphs of f, f, and are shown on the same set of coordinate aes Which is which? Eplain our reasoning To print an enlarged cop of the graph, select the MathGraph button 7 8 In Eercises 9 5, use the graph of to sketch a graph of f and the graph of f To print an enlarged cop of the graph, select the MathGraph button f f 8 8 (Submitted b Bill Fo, Moberl Area Communit College, Moberl, MO) f 0 8 f f f Writing About Concepts (continued) 5 Suppose ft < 0 for all t in the interval, 8 Eplain wh f > f 5 5 Suppose f 0 and f for all in the interval 5, 5 Determine the greatest and least possible values of f In Eercises 55 58, use a graphing utilit to graph the function Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asmptote Do ou think it is possible for the graph of a function to cross its vertical asmptote? Wh or wh not? 55 f sin h 58 f Writing In Eercises 59 and 0, use a graphing utilit to graph the function Eplain wh there is no vertical asmptote when a superficial eamination of the function ma indicate that there should be one 59 0 g h Writing In Eercises, use a graphing utilit to graph the function and determine the slant asmptote of the graph Zoom out repeatedl and describe how the graph on the displa appears to change Wh does this occur? g 8 5 f 5 f 5 Graphical Reasoning Consider the function f cos, 0 < < (a) Use a computer algebra sstem to graph the function and use the graph to approimate the critical numbers visuall (b) Use a computer algebra sstem to find and approimate the critical numbers Are the results the same as the visual approimation in part (a)? Eplain Graphical Reasoning Consider the function f tansin g 5 cos h (a) Use a graphing utilit to graph the function (b) Identif an smmetr of the graph (c) Is the function periodic? If so, what is the period? (d) Identif an etrema on, (e) Use a graphing utilit to determine the concavit of the graph on 0, f
9 SECTION A Summar of Curve Sketching 7 Think About It In Eercises 7 70, create a function whose graph has the given characteristics (There is more than one correct answer) 7 Vertical asmptote: 5 8 Vertical asmptote: Horizontal asmptote: 0 Horizontal asmptote: None 9 Vertical asmptote: 5 70 Vertical asmptote: 0 Slant asmptote: Slant asmptote: 7 Graphical Reasoning Consider the function f (a) Determine the effect on the graph of f if b 0 and a is varied Consider cases where a is positive and a is negative (b) Determine the effect on the graph of f if a 0 and b is varied 7 Consider the function f a a, a 0 (a) Determine the changes (if an) in the intercepts, etrema, and concavit of the graph of f when a is varied (b) In the same viewing window, use a graphing utilit to graph the function for four different values of a 7 Investigation Consider the function f a b n for nonnegative integer values of n (a) Discuss the relationship between the value of n and the smmetr of the graph (b) For which values of n will the ais be the horizontal asmptote? (c) For which value of n will be the horizontal asmptote? (d) What is the asmptote of the graph when n 5? (e) Use a graphing utilit to graph f for the indicated values of n in the table Use the graph to determine the number of etrema M and the number of points N of the graph n 0 5 M N 7 Investigation Let P 0, 0 be an arbitrar point on the graph of f such that f 0 0, as shown in the figure Verif each statement (a) The  intercept of the tangent line is 0 f 0 f 0, 0 (b) The intercept of the tangent line is 0, f 0 0 f 0 (c) The intercept of the normal line is 0 f 0 f 0, 0 (e) BC f 0 f 0 (g) (h) (f) 75 Modeling Data The data in the table show the number N of bacteria in a culture at time t, where t is measured in das A model for these data is given b N AB f 0 f 0 AP f 0 f 0 P( 0, 0 ) O A B C PC f 0 f 0 f 0 t N ,70 5,5t,50t 00 9t 7t, t 8 (a) Use a graphing utilit to plot the data and graph the model (b) Use the model to estimate the number of bacteria when t 0 (c) Approimate the da when the number of bacteria is greatest (d) Use a computer algebra sstem to determine the time when the rate of increase in the number of bacteria is greatest (e) Find lim Nt t Slant Asmptotes In Eercises 7 and 77, the graph of the function has two slant asmptotes Identif each slant asmptote Then graph the function and its asmptotes 7 77 Putnam Eam Challenge 78 Let f be defined for a b Assuming appropriate properties of continuit and derivabilit, prove for a < < b that f f a f b f a a b a b f where is some number between a and b This problem was composed b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserved f (d) The  intercept of the normal line is 0, 0 0 f 0
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