594 CHAPTER Derivatives When p ϭ 24, the rate of change is dx dp ϭ Ϫ98(48 ϩ 1)Ϫ3 ϭ Ϫ98 pϭ24 ϭϪ 493 This means that when the price is $24, demand is changing at the rate of Ϫ2 hundred units ■ per dollar, or if the price changes by $1, demand will change by about Ϫ200 units ✓ CHECKPOINT ANSWERS | EXERCISES | (a) No, f Ј(x) ϭ 10(3x ϩ 1)9 (12x 3) (b) Yes Ϫ48x x2 (b) f (a) f Ј(x) ϭ Ј(x) ϭ (2x Ϫ 1)2 x3 Ϫ 9.6 dy dy du , , and du dx dx y ϭ u3 and u ϭ x ϩ y ϭ u4 and u ϭ x ϩ 4x y ϭ u4 and u ϭ 4x Ϫ x ϩ y ϭ u10 and u ϭ x ϩ 5x In Problems 1–4, find (c) Yes Differentiate the functions in Problems 5–22 f (x) ϭ (3x Ϫ 2)20 g (x) ϭ (3 Ϫ 2x)10 h(x) ϭ 43 (x Ϫ 2x ϩ 5)8 k(x) ϭ 75 (2x Ϫ x ϩ 6)14 s(t) ϭ 5t Ϫ 3(2t ϩ 7)3 10 p(q) ϭ 4(3q2 Ϫ 1)4 Ϫ 13q 11 g (x) ϭ (x Ϫ 5x)Ϫ2 12 p ϭ (q3 ϩ 1)Ϫ5 14 g (t) ϭ 13 f (s) ϭ (2s ϩ 1)4 4t ϩ 1 15 g (x) ϭ (2x ϩ 3x ϩ 5)3 16 y ϭ (3x ϩ 4x ϩ 1)3 18 y ϭ x ϩ 3x 17 y ϭ 3x ϩ 4x ϩ 11(x Ϫ 7)6 Ϫ x3 19 y ϭ 20 y ϭ (3w ϩ 1)5 Ϫ 3w 21 z ϭ 2x Ϫ Ϫ x 22 y ϭ At the indicated point, for each function in Problems 23–26, find (a) the slope of the tangent line, and (b) the instantaneous rate of change of the function You may use the numerical derivative feature on a graphing calculator to check your work 23 y ϭ (x ϩ 2x)4 at x ϭ 24 y ϭ 5x ϩ 2x at x ϭ 25 y ϭ x ϩ at (2, 3) 26 y ϭ (4x Ϫ 5x ϩ 1)3 at (1, 0) In Problems 27–30, write the equation of the line tangent to the graph of each function at the indicated point As a check, graph both the function and the tangent line you found to see whether it looks correct 27 y ϭ (x Ϫ 3x ϩ 3)3 at (2, 1) 28 y ϭ (x ϩ 1)3 at (2, 125) 29 y ϭ 3x Ϫ at x ϭ at x ϭ 30 y ϭ (x Ϫ x)3 In Problems 31 and 32, complete the following for each function (a) Find fЈ(x) (b) Check your result in part (a) by graphing both it and the numerical derivative of the function (c) Find x-values for which the slope of the tangent is (d) Find points (x, y) where the slope of the tangent is (e) Use a graphing utility to graph the function and locate the points found in part (d) 31 f (x) ϭ (x Ϫ 4)3 ϩ 12 32 f (x) ϭ 10 Ϫ (x Ϫ 2x Ϫ 8)2 In Problems 33 and 34, the following for each function f (x) (a) Find fЈ(x) (b) Graph both f (x) and fЈ(x) with a graphing utility (c) Determine x-values where fЈ(x) ؍0, fЈ(x) 0, f(x) (d) Determine x-values for which f (x) has a maximum or minimum point, where the graph is increasing, and where it is decreasing 33 f (x) ϭ 12 Ϫ 3(1 Ϫ x 2)4 34 f (x) ϭ ϩ (x Ϫ 4x)4 16 In Problems 35 and 36, find the derivative of each function 2x 35 (a) y ϭ (b) y ϭ 3 3x (2x)3 (d) y ϭ (c) y ϭ (3x)3 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 9.6 3x 5 (3x)5 (d) y ϭ (5x)5 (c) y ϭ 5x 36 (a) y ϭ (b) y ϭ APPLICATIONS 37 Ballistics Ballistics experts are able to identify the weapon that fired a certain bullet by studying the markings on the bullet Tests are conducted by firing into a bale of paper If the distance s, in inches, that the bullet travels into the paper is given by s ϭ 27 Ϫ (3 Ϫ 10t)3 for Յ t Յ 0.3 second, find the velocity of the bullet one-tenth of a second after it hits the paper 38 Population of microorganisms Suppose that the population of a certain microorganism at time t (in minutes) is given by P ϭ 1000 Ϫ 1000(t ϩ 10)Ϫ1 Find the rate of change of population 39 Revenue The revenue from the sale of a product is R ϭ 1500x ϩ 3000(2x ϩ 3)Ϫ1 Ϫ 1000 dollars where x is the number of units sold Find the marginal revenue when 100 units are sold Interpret your result 40 Revenue The revenue from the sale of x units of a product is R ϭ 15(3x ϩ 1)Ϫ1 ϩ 50x Ϫ 15 dollars Find the marginal revenue when 40 units are sold Interpret your result 41 Pricing and sales Suppose that the weekly sales volume y (in thousands of units sold) depends on the price per unit (in dollars) of the product according to The Chain Rule and the Power Rule 595 (a) What is the rate of change of price with respect to the quantity demanded when q ϭ 49? (b) Interpret your answer to part (a) Stimulus-response The relation between the magnitude of a sensation y and the magnitude of the stimulus x is given by y ؍k (x ؊ x0)n where k is a constant, x0 is the threshold of effective stimulus, and n depends on the type of stimulus Find the rate of change of sensation with respect to the amount of stimulus for each of Problems 44–46 44 For the stimulus of visual brightness y ϭ k(x Ϫ x0)1 45 For the stimulus of warmth y ϭ k(x Ϫ x0)8 46 For the stimulus of electrical stimulation y ϭ k(x Ϫ x0)7 47 Demand If the demand for q units of a product priced at $p per unit is described by the equation pϭ 100 2q ϩ find the rate of change of p with respect to q 48 Advertising and sales The daily sales S (in thousands of dollars) attributed to an advertising campaign are given by Sϭ1ϩ 18 Ϫ t ϩ (t ϩ 3)2 where t is the number of weeks the campaign runs What is the rate of change of sales at (a) t ϭ 8? (b) t ϭ 10? (c) Should the campaign be continued after the 10th week? Explain 49 Body-heat loss The description of body-heat loss due to convection involves a coefficient of convection, Kc, which depends on wind velocity according to the following equation y ϭ 32(3p ϩ 1)Ϫ2 5, p Ͼ Kc ϭ 4v ϩ (a) What is the rate of change in sales volume when the price is $21? (b) Interpret your answer to part (a) 42 Pricing and sales A chain of auto service stations has found that its monthly sales volume S (in thousands of dollars) is related to the price p (in dollars) of an oil change according to Find the rate of change of the coefficient with respect to the wind velocity 50 Data entry speed The data entry speed (in entries per minute) of a data clerk trainee is Sϭ 98 , p Ͼ 10 pϩ5 (a) What is the rate of change of sales volume when the price is $44? (b) Interpret your answer to part (a) 43 Demand Suppose that the demand for q units of a product priced at $p per unit is described by pϭ 200,000 (q ϩ 1)2 S ϭ 10 0.8x ϩ , Յ x Յ 100 where x is the number of hours of training he has had What is the rate at which his speed is changing and what does this rate mean when he has had (a) 15 hours of training? (b) 40 hours of training? 51 Investments If an IRA is a variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of $1000 is S ϭ 1000 ϩ 0.01r 12 240 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... ϭ 10? (c) Should the campaign be continued after the 10th week? Explain 49 Body-heat loss The description of body-heat loss due to convection involves a coefficient of convection, Kc, which depends... with respect to the quantity demanded when q ϭ 49? (b) Interpret your answer to part (a) Stimulus-response The relation between the magnitude of a sensation y and the magnitude of the stimulus... paper is given by s ϭ 27 Ϫ (3 Ϫ 10t)3 for Յ t Յ 0.3 second, find the velocity of the bullet one-tenth of a second after it hits the paper 38 Population of microorganisms Suppose that the population