4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4

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1 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward on (, and concave upward on ( 2 2) 2,) and on,. The inflection point is ( 2, e 2 ). To sketch the graph of f we first draw the horizontal asmptote (as a dashed line), together with the parts of the curve near the asmptotes in a preinar sketch [Figure 3(a)]. These parts reflect the information concerning its and the fact that f is decreasing on both, and,. Notice that we have indicated that f l as l even though f does not eist. In Figure 3(b) we finish the sketch b incorporating the information concerning concavit and the inflection point. In Figure 3(c) we check our work with a graphing device. = 4 = inflection point = FIGURE 3 _3 3 (a) Preinar sketch (b) Finished sketch (c) Computer confirmation 4.3 Eercises 2 Use the given graph of f to find the following. (a) The open intervals on which f is increasing. (b) The open intervals on which f is decreasing. (c) The open intervals on which f is concave upward. (d) The open intervals on which f is concave downward. (e) The coordinates of the points of inflection.. 2. (b) How do ou determine where the graph of f is concave upward or concave downward? (c) How do ou locate inflection points? 4. (a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circum - stances is it inconclusive? What do ou do if it fails? 5 6 The graph of the derivative f of a function f is shown. (a) On what intervals is f increasing or decreasing? (b) At what values of does f have a local maimum or minimum? =fª() =fª() 3. Suppose ou are given a formula for a function f. (a) How do ou determine where f is increasing or decreasing? ; Graphing calculator or computer required CAS Computer algebra sstem required. Homework Hints available at stewartcalculus.com

2 298 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION 7. In each part state the -coordinates of the inflection points 23. Suppose f is continuous on,. of f. Give reasons for our answers. (a) If f 2 and f 2 5, what can ou sa about f? (a) The curve is the graph of f. (b) If f 6 and f 6, what can ou sa about f? (b) The curve is the graph of f. (c) The curve is the graph of f Sketch the graph of a function that satisfies all of the given conditions Vertical asmptote, f if 2, f if 2, f if, f if 8. The graph of the first derivative f of a function f is shown. (a) On what intervals is f increasing? Eplain. (b) At what values of does f have a local maimum or minimum? Eplain. (c) On what intervals is f concave upward or concave downward? Eplain. (d) What are the -coordinates of the inflection points of f? Wh? =fª() 9 8 (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maimum and minimum values of f. (c) Find the intervals of concavit and the inflection points. 9.. f f f 2 3. f sin cos, 2 4. f cos 2 2sin, 2 5. f e 2 e 6. f 2 ln 7. f 2 ln 8. f 4 e 9 2 Find the local maimum and minimum values of f using both the First and Second Derivative Tests. Which method do ou prefer? 9. f f 2 2. f f s s f f 2 f 4, f if or 2 4, f if 2 or 4, f if 3, f if or f f, f if, f if 2, f if 2, f if 2, inflection point, f 2 f 2 f l if, if, f 2,, f if 2 f 2 f if, if, f 2, f, f f, l f if 3, f if f and f for all 3. Suppose f 3 2, f 3 2, and f and f for all. (a) Sketch a possible graph for f. (b) How man solutions does the equation f have? Wh? (c) Is it possible that f 2 3? Wh? 3 32 The graph of the derivative f of a continuous function f is shown. (a) On what intervals is f increasing? Decreasing? (b) At what values of does f have a local maimum? Local minimum? (c) On what intervals is f concave upward? Concave downward? (d) State the -coordinate(s) of the point(s) of inflection. (e) Assuming that f, sketch a graph of f. 3. =fª() (a) Find the critical numbers of f 4 3. (b) What does the Second Derivative Test tell ou about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell ou? _2

3 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH _2 =fª() ; (a) Use a graph of f to estimate the maimum and minimum values. Then find the eact values. (b) Estimate the value of at which f increases most rapidl. Then find the eact value. 55. f 56. f 2 e s (a) Find the intervals of increase or decrease. (b) Find the local maimum and minimum values. (c) Find the intervals of concavit and the inflection points. (d) Use the information from parts (a) (c) to sketch the graph. Check our work with a graphing device if ou have one. 33. f f f t h h F s6 4. G C f ln f 2cos cos 2, 44. S sin, (a) Find the vertical and horizontal asmptotes. (b) Find the intervals of increase or decrease. (c) Find the local maimum and minimum values. (d) Find the intervals of concavit and the inflection points. (e) Use the information from parts (a) (d) to sketch the graph of f. 45. f f s f f e e 49. f e 2 5. f ln 5. f ln ln 52. f e arctan 53. Suppose the derivative of a function f is f On what interval is f increasing? 54. Use the methods of this section to sketch the curve 3 3a 2 2a 3, where a is a positive constant. What do the members of this famil of curves have in common? How do the differ from each other? ; (a) Use a graph of f to give a rough estimate of the intervals of concavit and the coordinates of the points of inflection. (b) Use a graph of f to give better estimates. 57. f cos 2 cos 2, 58. f CAS 59 6 Estimate the intervals of concavit to one decimal place b using a computer algebra sstem to compute and graph f. 59. f s 2 6. A graph of a population of east cells in a new laborator culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point. Number of east cells f 2 tan Time (in hours) 62. Let f t be the temperature at time t where ou live and suppose that at time t 3 ou feel uncomfortabl hot. How do ou feel about the given data in each case? (a) f 3 2, f 3 4 (b) f 3 2, f 3 4 (c) f 3 2, f 3 4 (d) f 3 2, f Let K t be a measure of the knowledge ou gain b studing for a test for t hours. Which do ou think is larger, K 8 K 7 or K 3 K 2? Is the graph of K concave upward or concave downward? Wh?

4 3 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION 64. Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavit. What is the significance of the inflection point? 69. (a) If the function f 3 a 2 b has the local minimum value 2 9 s3 at s3, what are the values of a and b? (b) Which of the tangent lines to the curve in part (a) has the smallest slope? 7. For what values of a and b is 2, 2.5 an inflection point of the curve 2 a b? What additional inflection points does the curve have? 7. Show that the curve 2 has three points of inflection and the all lie on one straight line. 72. Show that the curves e and e touch the curve e sin at its inflection points. 65. A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S t At p e kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A., p 4, k.7, and t is measured in minutes, estimate the times corresponding to the inflection points and eplain their significance. If ou have a graphing device, use it to graph the drug response curve. 66. The famil of bell-shaped curves 2 s2 e 2 2 occurs in probabilit and statistics, where it is called the normal densit function. The constant is called the mean and the positive constant is called the standard deviation. For simplicit, let s scale the function so as to remove the factor ( s2 ) and let s analze the special case where. So we stud the function f e (a) Find the asmptote, maimum value, and inflection points of f. (b) What role does pla in the shape of the curve? ; (c) Illustrate b graphing four members of this famil on the same screen. 67. Find a cubic function f a 3 b 2 c d that has a local maimum value of 3 at 2 and a local minimum value of at. 68. For what values of the numbers a and b does the function f ae b2 have the maimum value f 2? 73. Show that the inflection points of the curve sin lie on the curve Assume that all of the functions are twice differentiable and the second derivatives are never. 74. (a) If f and t are concave upward on I, show that f t is concave upward on I. (b) If f is positive and concave upward on I, show that the function t f 2 is concave upward on I. 75. (a) If f and t are positive, increasing, concave upward functions on I, show that the product function ft is concave upward on I. (b) Show that part (a) remains true if f and t are both decreasing. (c) Suppose f is increasing and t is decreasing. Show, b giving three eamples, that ft ma be concave upward, concave downward, or linear. Wh doesn t the argument in parts (a) and (b) work in this case? 76. Suppose f and t are both concave upward on,. Under what condition on f will the composite function h f t be concave upward? 77. Show that tan for 2. [Hint: Show that f tan is increasing on, 2.] 78. (a) Show that e for. (b) Deduce that e 2 2 for. (c) Use mathematical induction to prove that for and an positive integer n, e 2 2! n n! 79. Show that a cubic function (a third-degree polnomial) alwas has eactl one point of inflection. If its graph has three -intercepts, 2, and 3, show that the -coordinate of the inflection point is

5 SECTION 4.4 INDETERMINATE FORMS AND L HOSPITAL S RULE 3 ; 8. For what values of c does the polnomial P 4 c 3 2 have two inflection points? One inflection point? None? Illustrate b graphing P for several values of c. How does the graph change as c decreases? 8. Prove that if c, f c is a point of inflection of the graph of f and f eists in an open interval that contains c, then f c. [Hint: Appl the First Derivative Test and Fermat s Theorem to the function t f.] 82. Show that if f 4, then f, but, is not an inflection point of the graph of f. t 83. Show that the function has an inflection point at, but t does not eist. 84. Suppose that f is continuous and f c f c, but f c. Does f have a local maimum or minimum at c? Does f have a point of inflection at c? 85. Suppose f is differentiable on an interval I and f for all numbers in I ecept for a single number c. Prove that f is increasing on the entire interval I. 86. For what values of c is the function f c 2 3 increasing on,? 87. The three cases in the First Derivative Test cover the situations one commonl encounters but do not ehaust all possibilities. Consider the functions f, t, andh whose values at are all and, for, f t 4 2 sin 4 sin h 4 2 sin (a) Show that is a critical number of all three functions but their derivatives change sign infinitel often on both sides of. (b) Show that f has neither a local maimum nor a local minimum at, t has a local minimum, and h has a local maimum. 4.4 Indeterminate Forms and l Hospital s Rule Suppose we are tring to analze the behavior of the function Although F is not defined when, we need to know how F behaves near. In particular, we would like to know the value of the it In computing this it we can t appl Law 5 of its (the it of a quotient is the quotient of the its, see Section 2.3) because the it of the denominator is. In fact, although the it in eists, its value is not obvious because both numerator and denominator approach and is not defined. In general, if we have a it of the form where both f l and t l as l a, then this it ma or ma not eist and is called an indeterminate form of tpe. We met some its of this tpe in Chapter 2. For rational functions, we can cancel common factors: l 2 2 l F l l a We used a geometric argument to show that ln f t l sin l ln 2