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Applications of Differentiation

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Applications of Differentiation Not for Sale Applications of 2 Differentiation What You’ll Learn Why It’s Important 2.1 Using First Derivatives to Classify Maximum In this chapter, we explore many applications and Minimum Values and Sketch Graphs of differentiation. We learn to find maximum 2.2 Using Second Derivatives to Classify and minimum values of functions, and that skill Maximum and Minimum Values and Sketch allows us to solve many kinds of problems in Graphs which we need to find the largest and/or smallest 2.3 Graph Sketching: Asymptotes and Rational value in a real-world situation. Our differentiation Functions skills will be applied to graphing, and we will use 2.4 Using Derivatives to Find Absolute differentials to approximate function values. We Maximum and Minimum Values will also use differentiation to explore elasticity of 2.5 Maximum–Minimum Problems; Business, demand and related rates. Economics, and General Applications 2.6 Marginals and Differentials 2.7 Elasticity of Demand 2.8 Implicit Differentiation and Related Rates Where It’s Used y C(r) Minimizing Cost: Minimizing cost is a common goal in 0.32 manufacturing. For example, cylindrical food cans come in a 3 (4.21, 0.2673) variety of sizes. Suppose a soup can is to have a volume of 250 cm . 0.24 2 The cost of material for the two circular ends is $0.0008 cm , and 0.16 the cost of material for the side is $0.0015 cm2. What dimensions > Cost (in dollars) 0.08 minimize the cost of material for the soup can?(This problem > appears as Example 3 in Section 2.5.) 20 4 6 8 r Radius (in centimeters) 188 Copyright 2016 Pearson. All rights reserved. M02_BITT9391_11_SE_C02.1.indd 188 29/10/14 4:39 PM Not for Sale 2.1 ● Using First Derivatives to Classify Maximum and Minimum Values and Sketch Graphs 189 Using First Derivatives to Classify Maximum and Minimum Values 2.1 and Sketch Graphs The graph below shows a typical life cycle of a retail product and is similar to graphs ● Find relative extrema of a continuous function using the we will consider in this chapter. Note that the number of items sold varies with re- First-Derivative Test. spect to time. Sales begin at a small level and increase to a point of maximum sales, after which they taper off to a low level, possibly because of new competitive prod- ● Sketch graphs of continuous ucts. The company then rejuvenates the product by making improvements. Think functions. about versions of certain products: televisions can be traditional, flat-screen, or high- definition; music recordings have been produced as phonograph (vinyl) records, au- diotapes, compact discs, and MP3 files. Where might each of these products be in a typical product life cycle? Is the curve below appropriate for each product? S PRODUCT LIFE CYCLE Sales of product t Introduction Growth Maturity Decline Rejuvenation Time Finding the largest and smallest values of a function—that is, the maximum and minimum values—has extensive applications. The first and second derivatives of a function can provide information for graphing functions and finding maximum and minimum values. Increasing and Decreasing Functions If the graph of a function rises from left to right over an interval I, the function is said to be increasing on, or over, I. If the graph drops from left to right, the function is said to be decreasing on, or over, I. y y f Decreasing f(b) g(a) f(a) g(b) Increasing g a b x a b x If the input a is less than the input b, If the input a is less than the input b, then the output for a is less than the then the output for a is greater than the output for b. output for b. Copyright 2016 Pearson. All rights reserved. M02_BITT9391_11_SE_C02.1.indd 189 29/10/14 4:39 PM Not for Sale 190 CHAPTER 2 ● Applications of Differentiation TECHNOLOGY CONNECTION We can define these concepts as follows. Exploratory DEFINITIONS Graph the function A function f is increasing over I if, for every a and b in I, 1 3 2 f x = - 3x + 6x - 11x - 50 if a 6 b, then f a 6 f b . and1 2 its derivative A function f is decreasing over I if, for every a and b in I, 2 1 2 1 2 f′ x = -x + 12x - 11 if a 6 b, then f a 7 f b . using1 2 the window -10, 25, -100, 150 , with Xscl = 5 and 1 2 1 2 3 Yscl = 25. Then TRACE from left The above definitions can be restated in terms of secant lines as shown below. to right along4 each graph. Mov- ing the cursor from left to right, f b - f a f b - f a Increasing: 7 0. Decreasing: 6 0. note that the x-coordinate always b a b a increases. If the function is 1 2 - 1 2 1 2 - 1 2 increasing, the y-coordinate will y y increase as well. If a function is decreasing, the y-coordinate will f f(b) f(a) decrease. Slope of secant Slope of secant Over what intervals is f line is positive. line is negative. increasing? Over what intervals is f f(a) decreasing? Over what intervals is f′ f positive? Over what intervals is f′ f(b) negative? What rules might relate the sign of f′ to the behavior of f? a b x a b x The following theorem shows how we can use the derivative (the slope of a tan- gent line) to determine whether a function is increasing or decreasing. THEOREM 1 Let f be differentiable over an open interval I. If f′ x 7 0 for all x in I, then f is increasing over I. If f′ x 6 0 for all x in I, then f is decreasing over I. 1 2 1 2 1 3 2 Theorem 1 is illustrated in the following graph of f x = 3x - x + 3. 1 2 4± ±1, 3 y 1 f(x)= ± x3 ±x + 2± 3 3 1 ±2 ±1 (1, 0) x Increasing Decreasing Increasing f9(x) > 0 f9(x) < 0 f9(x) > 0 f is increasing over the intervals (±, ∞ ±1) and (1, ∞); slopes of tangent lines are positive. f is decreasing over the interval (±1, 1); slopes of tangent lines are negative. Note in the graph above that x = -1 and x = 1 are not included in any interval over which the function is increasing or decreasing. These values are examples of critical values. Copyright 2016 Pearson. All rights reserved. M02_BITT9391_11_SE_C02.1.indd 190 29/10/14 4:39 PM Not for Sale 2.1 ● Using First Derivatives to Classify Maximum and Minimum Values and Sketch Graphs 191 Critical Values Consider the following graph of a continuous function f. y f x c1 c2 c3 c4 c5 c6 c7 c8 Note the following: 1. f′ x = 0 at x = c1, c2, c4, c7, and c8. That is, the tangent line to the graph is horizontal for these values. 1 2 2. f′ x does not exist at x = c3, c5, and c6. The tangent line is vertical at c3, and there are corners at both c5 and c6. (See also the discussion at the end of Section1 2 1.4.) DEFINITION A critical value of a function f is any number c in the domain of f for which the tangent line at c, f c is horizontal or for which the derivative does not exist. That is, c is a critical value if f c exists and 1 1 22 f′ c = 0 or f′ c does1 2 not exist. 1 2 1 2 Thus, in the graph of f above: ● c1, c2, c4, c7, and c8 are critical values because f′ c = 0 for each value. ● c , c , and c are critical values because f′ c does not exist for each value. 3 5 6 1 2 A continuous function can change from increasing1 2 to decreasing or from decreas- ing to increasing only at a critical value. In the above graph, c1, c2, c4, c5, c6, and c7 separate intervals over which the function f increases from those in which it decreases or intervals in which the function decreases from those in which it in- creases. Although c3 and c8 are critical values, they do not separate intervals over which the function changes from increasing to decreasing or from decreasing to increasing. Copyright 2016 Pearson. All rights reserved. M02_BITT9391_11_SE_C02.1.indd 191 29/10/14 4:39 PM Not for Sale 192 CHAPTER 2 ● Applications of Differentiation Finding Relative Maximum and Minimum Values Now consider a graph with “peaks” and “valleys” at x = c1, c2, and c3. y f(b) Relative Maxima maximum f f(c2) f(c3) Relative Minima f(a) minimum f(c1) abc1 c2 c3 x Here, f c2 is an example of a relative maximum (plural: maxima). Each of f c1 and f c3 is called a relative minimum (plural: minima). Collectively, maximum and minimum1 2values are called extrema (singular: extremum). 1 2 1 2 DEFINITIONS Let I be the domain of f. f c is a relative minimum if there exists within I an open interval I1 containing c such that f c … f x , for all x in I1; 1 2 and 1 2 1 2 f c is a relative maximum if there exists within I an open interval I2 containing c such that f c Ú f x , for all x in I2.
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