1053714_cop3.qxd 11/4/08 9:03 AM Page 163 Applications 3 of Differentiation This chapter discusses several applications of the derivative of a function. These applications fall into three basic categories—curve sketching, optimization, and approximation techniques. In this chapter, you should learn the following. ■ How to use a derivative to locate the minimum and maximum values of a function on a closed interval. (3.1) ■ How numerous results in this chapter depend on two important theorems called Rolle’s Theorem and the Mean Value Theorem. (3.2) ■ How to use the first derivative to deter- mine whether a function is increasing or decreasing. (3.3) ■ How to use the second derivative to determine whether the graph of a ■ function is concave upward or concave downward. (3.4) ■ How to find horizontal asymptotes of the graph of a function. (3.5) ■ How to graph a function using the techniques from Chapters P–3. (3.6) © E.J. Baumeister Jr./Alamy ■ How to solve optimization problems. A small aircraft starts its descent from an altitude of 1 mile, 4 miles west of the (3.7) ■ runway. Given a function that models the glide path of the plane, when would the ■ How to use approximation techniques plane be descending at the greatest rate? (See Section 3.4, Exercise 75.) to solve problems. (3.8 and 3.9) In Chapter 3, you will use calculus to analyze graphs of functions. For example, you can use the derivative of a function to determine the function’s maximum and minimum values. You can use limits to identify any asymptotes of the function’s graph. In Section 3.6, you will combine these techniques to sketch the graph of a function. 163 1053714_0301.qxp 11/4/08 9:04 AM Page 164 164 Chapter 3 Applications of Differentiation 3.1 Extrema on an Interval ■ Understand the definition of extrema of a function on an interval. ■ Understand the definition of relative extrema of a function on an open interval. ■ Find extrema on a closed interval. Extrema of a Function In calculus, much effort is devoted to determining the behavior of a function f on an interval I. Doesf have a maximum value on I? Does it have a minimum value? Where is the function increasing? Where is it decreasing? In this chapter you will learn how derivatives can be used to answer these questions. You will also see why these y questions are important in real-life applications. Maximum 5 (2, 5) DEFINITION OF EXTREMA 4 f(x) = x2 + 1 Let f be defined on an interval I containing c. 3 1. f ͑c͒ is the minimum of f on I if f ͑c͒ Յ f ͑x͒ for all x in I. 2 2. f ͑c͒ is the maximum of f on I if f ͑c͒ Ն f ͑x͒ for all x in I. (0, 1) Minimum The minimum and maximum of a function on an interval are the extreme x values, or extrema (the singular form of extrema is extremum), of the −121 3 function on the interval. The minimum and maximum of a function on an (a) f is continuous,[Ϫ1, 2͔ is closed. interval are also called the absolute minimum and absolute maximum, or the global minimum and global maximum, on the interval. y Not a 5 maximum A function need not have a minimum or a maximum on an interval. For instance, 2 4 f(x) = x2 + 1 in Figure 3.1(a) and (b), you can see that the function f ͑x͒ ϭ x ϩ 1 has both a minimum and a maximum on the closed interval ͓Ϫ1, 2͔, but does not have a maxi- 3 mum on the open interval ͑Ϫ1, 2͒. Moreover, in Figure 3.1(c), you can see that 2 continuity (or the lack of it) can affect the existence of an extremum on the interval. This suggests the theorem below. (Although the Extreme Value Theorem is intuitively (0, 1) Minimum plausible, a proof of this theorem is not within the scope of this text.) x −121 3 (b) f is continuous,͑Ϫ1, 2͒ is open. THEOREM 3.1 THE EXTREME VALUE THEOREM If f is continuous on a closed interval ͓a, b͔, then f has both a minimum and a y maximum on the interval. Maximum 5 (2, 5) 4 2 + ≠ g(x) = x 1, x 0 3 2, x = 0 EXPLORATION 2 Finding Minimum and Maximum Values The Extreme Value Theorem (like the Intermediate Value Theorem) is an existence theorem because it tells of the Not a minimum existence of minimum and maximum values but does not show how to find x these values. Use the extreme-value capability of a graphing utility to find the −121 3 minimum and maximum values of each of the following functions. In each case, (c) g is not continuous,[Ϫ1, 2͔ is closed. do you think the x-values are exact or approximate? Explain your reasoning. Extrema can occur at interior points or end- ͑ ͒ ϭ 2 Ϫ ϩ ͓Ϫ ͔ points of an interval. Extrema that occur at a. f x x 4x 5 on the closed interval 1, 3 the endpoints are called endpoint extrema. b. f ͑x͒ ϭ x3 Ϫ 2x2 Ϫ 3x Ϫ 2 on the closed interval ͓Ϫ1, 3͔ Figure 3.1 1053714_0301.qxp 11/4/08 9:04 AM Page 165 3.1 Extrema on an Interval 165 Relative Extrema and Critical Numbers 3 2 Hill y f(x) = x3 − 3x2 In Figure 3.2, the graph of f ͑x͒ ϭ x Ϫ 3x has a relative maximum at the point (0, 0) ͑0, 0͒ and a relative minimum at the point ͑2, Ϫ4͒. Informally, for a continuous x −1 12 function, you can think of a relative maximum as occurring on a “hill” on the graph, and a relative minimum as occurring in a “valley” on the graph. Such a hill and valley can occur in two ways. If the hill (or valley) is smooth and rounded, the graph −2 has a horizontal tangent line at the high point (or low point). If the hill (or valley) is sharp and peaked, the graph represents a function that is not differentiable at the high −3 Valley point (or low point). (2, −4) −4 DEFINITION OF RELATIVE EXTREMA ͑ ͒ f has a relative maximum at 0, 0 and a 1. If there is an open interval containing c on which f ͑c͒ is a maximum, then relative minimum at ͑2, Ϫ4͒. f ͑c͒ is called a relative maximum of f, or you can say that f has a relative Figure 3.2 maximum at ͧc, fͧcͨͨ. 2. If there is an open interval containing c on which f ͑c͒ is a minimum, then f ͑c͒ is called a relative minimum of f, or you can say that f has a relative minimum at ͧc, fͧcͨͨ. y 9(x2 − 3) The plural of relative maximum is relative maxima, and the plural of relative Relative f(x) = 3 minimum is relative minima. Relative maximum and relative minimum are maximum x 2 sometimes called local maximum and local minimum, respectively. (3, 2) x 246 Example 1 examines the derivatives of functions at given relative extrema. (Much −2 more is said about finding the relative extrema of a function in Section 3.3.) −4 EXAMPLE 1 The Value of the Derivative at Relative Extrema (a) fЈ͑3͒ ϭ 0 Find the value of the derivative at each relative extremum shown in Figure 3.3. y Solution 9͑x2 Ϫ 3͒ f(x) = ⏐x⏐ 3 a. The derivative of f ͑x͒ ϭ is x3 2 x3͑18x͒ Ϫ ͑9͒͑x2 Ϫ 3͒͑3x2͒ fЈ͑x͒ ϭ Differentiate using Quotient Rule. 1 Relative ͑x3͒2 minimum x 9͑9 Ϫ x2͒ − − ϭ . Simplify. 2 1 1 2 x4 −1 (0, 0) At the point ͑3, 2͒, the value of the derivative is fЈ͑3͒ ϭ 0 [see Figure 3.3(a)]. (b) fЈ͑0) does not exist. b. At x ϭ 0, the derivative of f ͑x͒ ϭ ԽxԽ does not exist because the following y one-sided limits differ [see Figure 3.3(b)]. f(x) = sin x f͑x͒ Ϫ f ͑0͒ ԽxԽ 2 π lim ϭ lim ϭϪ1 Limit from the left , 1 Relative → Ϫ Ϫ → Ϫ ( 2 ( x 0 x 0 x 0 x 1 maximum f͑x͒ Ϫ f ͑0͒ ԽxԽ lim ϭ lim ϭ 1 Limit from the right x x→0ϩ x Ϫ 0 x→0ϩ x π 3π 2 2 −1 c. The derivative of f ͑x͒ ϭ sin x is Relative π 3 , −1 ( 2 ( −2 minimum fЈ͑x͒ ϭ cos x. ␲ 3␲ At the point ͑␲͞2, 1͒, the value of the derivative is fЈ͑␲͞2͒ ϭ cos͑␲͞2͒ ϭ 0. At the (c) fЈ΂ ΃ ϭ 0; fЈ΂ ΃ ϭ 0 2 2 point ͑3␲͞2, Ϫ1͒, the value of the derivative is fЈ͑3␲͞2͒ ϭ cos͑3␲͞2͒ ϭ 0 [see Figure 3.3 Figure 3.3(c)]. ■ 1053714_0301.qxp 11/4/08 9:04 AM Page 166 166 Chapter 3 Applications of Differentiation Note in Example 1 that at each relative extremum, the derivative either is zero or does not exist. The x-values at these special points are called critical numbers. Figure 3.4 illustrates the two types of critical numbers. Notice in the definition that the critical number c has to be in the domain of f, but c does not have to be in the domain of fЈ. DEFINITION OF A CRITICAL NUMBER Let f be defined at c. If fЈ͑c͒ ϭ 0 or if f is not differentiable at c, then c is a critical number of f.