580 CHAPTER Derivatives Solution (a) The graphs of f (x) ϭ x Ϫ 3x ϩ and f Ј(x) ϭ 3x Ϫ are shown in Figure 9.31 (b) The values where f Ј(x) ϭ are the x-intercepts, x ϭ Ϫ1 and x ϭ The graph of y ϭ x Ϫ 3x ϩ appears to turn at both these values (c) f Ј(x) Ͻ where the graph of y ϭ f Ј(x) is below the x-axis, for Ϫ1 Ͻ x Ͻ The graph of y ϭ f (x) appears to be decreasing on this interval ■ (d) They appear to be the same intervals f (x) x3 3x f (x) Figure 9.31 ✓ CHECKPOINT ANSWERS | EXERCISES | 3x 3 True, by the Coefficient Rule True, by the Sum Rule True, by the Difference Rule Yes, f (x) ϭ x n c ϭ (1 c)x n, so the coefficient is (1 c) (a) f Ј(x) ϭ 10x Ϫ 10 ds Ϫ5 ϭ (b) dt t The slope of the tangent at x ϭ Ϫ1 is f Ј(Ϫ1) ϭ 11 9.4 Find the derivatives of the functions in Problems 1–14 y ϭ f (s) ϭ f (t) ϭ t s ϭ t 2 y ϭ Ϫ 8x ϩ 2x y ϭ 12 ϩ 2x Ϫ 7x f (x) ϭ 3x Ϫ x f (x) ϭ 3x Ϫ x y ϭ 10x Ϫ 3x ϩ 5x Ϫ 11 10 y ϭ 3x Ϫ 5x Ϫ 8x ϩ 11 w ϭ z Ϫ 3z ϩ 13 12 u ϭ 2t 10 Ϫ 5t Ϫ 13 g (x) ϭ 2x 12 Ϫ 5x ϩ 9x ϩ x Ϫ 14 h(x) ϭ 12x 20 ϩ 8x 10 Ϫ 2x ϩ 17x Ϫ In Problems 15–18, at the indicated points, find (a) the slope of the tangent to the curve, and (b) the instantaneous rate of change of the function 15 y ϭ 7x ϩ 2x ϩ 1, x ϭ 16 C(x) ϭ 3x Ϫ 5, (3, 22) 17 P(x) ϭ x Ϫ 6x, (2, Ϫ4) 18 R(x) ϭ 16x ϩ x 2, x ϭ In Problems 19–26, find the derivative of each function 19 y ϭ x Ϫ5 ϩ x Ϫ8 Ϫ 20 y ϭ x Ϫ1 Ϫ x Ϫ2 ϩ 13 21 z ϭ 3t 11 Ϫ 2t Ϫ t ϩ 22 w ϭ 5u8 Ϫ 3u5 ϩ u1 ϩ 23 f (x) ϭ 5x Ϫ4 ϩ 2x Ϫ4 24 f (x) ϭ 6x Ϫ8 Ϫ x Ϫ2 3 25 g (x) ϭ ϩ ϩ x x x 26 h(x) ϭ Ϫ ϩ x x x In Problems 27–30, write the equation of the tangent line to each curve at the indicated point As a check, graph both the function and the tangent line 27 y ϭ x Ϫ 5x ϩ at x ϭ 28 y ϭ x Ϫ 4x Ϫ at x ϭ 1 29 f (x) ϭ 4x Ϫ at x ϭ Ϫ x x3 30 f (x) ϭ Ϫ at x ϭ Ϫ1 x In Problems 31–34, find the coordinates of points where the graph of f (x) has horizontal tangents As a check, graph f (x) and see whether the points you found look as though they have horizontal tangents 31 f (x) ϭ Ϫx ϩ 9x Ϫ 15x ϩ 32 f (x) ϭ x Ϫ 3x Ϫ 16x ϩ 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 9.4 33 f (x) ϭ x Ϫ 4x ϩ 34 f (x) ϭ 3x Ϫ 5x ϩ In Problems 35 and 36, find each derivative at the given x-value (a) with the appropriate rule and (b) with the numerical derivative feature of a graphing calculator 35 y ϭ Ϫ x at x ϭ 36 y ϭ ϩ 3x at x ϭ Ϫ8 In Problems 37–40, complete the following (a) Calculate the derivative of each function with the appropriate formula (b) Check your result from part (a) by graphing your calculated derivative and the numerical derivative of the given function with respect to x evaluated at x 37 f (x) ϭ 2x ϩ 5x Ϫ p4 ϩ 38 f (x) ϭ 3x Ϫ 8x ϩ 25 Ϫ 20 10 10 39 h(x) ϭ Ϫ ϩ x ϩ x x 40 g (x) ϭ 10 ϩ ϩ x Ϫ x x The tangent line to a curve at a point closely approximates the curve near the point In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable Problems 41 and 42 explore this relationship Use each given function and the indicated point to complete the following (a) Write the equation of the tangent line to the curve at the indicated point (b) Use a graphing calculator to graph both the function and its tangent line Be sure your graph shows the point of tangency (c) Repeatedly zoom in on the point of tangency Do the function and the tangent line eventually become indistinguishable? 41 f (x) ϭ 3x ϩ 2x at x ϭ 42 f (x) ϭ 4x Ϫ x at x ϭ For each function in Problems 43–46, the following (a) Find f (x) (b) Graph both f (x) and f (x) with a graphing utility (c) Use the graph of f (x) to identify x-values where f (x) ؍0, f (x) 0, and f (x) (d) Use the graph of f (x) to identify x-values where f (x) has a maximum or minimum point, where the graph of f (x) is rising, and where the graph of f (x) is falling 43 f (x) ϭ Ϫ 2x Ϫ x 44 f (x) ϭ x ϩ 4x Ϫ 12 45 f (x) ϭ x Ϫ 12x Ϫ x3 46 f (x) ϭ Ϫ 3x Ϫ Derivative Formulas 581 APPLICATIONS 47 Revenue Suppose that a wholesaler expects that his monthly revenue, in dollars, for an electronic game will be R(x) ϭ 100x Ϫ 0.1x 2, Յ x Յ 800 where x is the number of units sold Find his marginal revenue and interpret it when the quantity sold is (a) x ϭ 300 (b) x ϭ 600 48 Revenue The total revenue, in dollars, for a commodity is described by the function R ϭ 300x Ϫ 0.02x (a) What is the marginal revenue when 40 units are sold? (b) Interpret your answer to part (a) 49 Metabolic rate According to Kleiber’s law the metabolic rate q of the vast majority of animals is related to the animal’s mass M according to q ϭ kM where k is a constant This means that a cat, with mass about 100 times that of a mouse, has a metabolism about 1003 32 times greater than that of a mouse Find the function that describes the rate of change of the metabolic rate with respect to mass 50 Capital investment and output The monthly output of a certain product is Q(x) ϭ 800x where x is the capital investment in millions of dollars Find dQ dx, which can be used to estimate the effect on the output if an additional capital investment of $1 million is made 51 Demand The demand for q units of a product depends on the price p (in dollars) according to qϭ 1000 Ϫ 1, p for p Ͼ Find and explain the meaning of the instantaneous rate of change of demand with respect to price when the price is (a) $25 (b) $100 52 Demand Suppose that the demand for a product depends on the price p according to D(p) ϭ 50,000 Ϫ , p2 pϾ0 where p is in dollars Find and explain the meaning of the instantaneous rate of change of demand with respect to price when (a) p ϭ 50 (b) p ϭ 100 53 Cost and average cost Suppose that the total cost function, in dollars, for the production of x units of a product is given by C(x) ϭ 4000 ϩ 55x ϩ 0.1x 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 582 CHAPTER Derivatives Then the average cost of producing x items is C(x) ϭ total cost 4000 ϭ ϩ 55 ϩ 0.1x x x (a) Find the instantaneous rate of change of average cost with respect to the number of units produced, at any level of production (b) Find the level of production at which this rate of change equals zero (c) At the value found in part (b), find the instantaneous rate of change of cost and find the average cost What you notice? 54 Cost and average cost Suppose that the total cost function, in dollars, for a certain commodity is given by C(x) ϭ 40,500 ϩ 190x ϩ 0.2x where x is the number of units produced (a) Find the instantaneous rate of change of the average cost Cϭ 40,500 ϩ 190 ϩ 0.2x x for any level of production (b) Find the level of production where this rate of change equals zero (c) At the value found in part (b), find the instantaneous rate of change of cost and find the average cost What you notice? 55 Cost-benefit Suppose that for a certain city the cost C, in dollars, of obtaining drinking water that contains p percent impurities (by volume) is given by Cϭ 120,000 Ϫ 1200 p (a) Find the rate of change of cost with respect to p when impurities account for 10% (by volume) (b) Write a sentence that explains the meaning of your answer in part (a) 56 Cost-benefit Suppose that the cost C, in dollars, of processing the exhaust gases at an industrial site to ensure that only p percent of the particulate pollution escapes is given by C(p) ϭ 8100(100 Ϫ p) p (a) Find the rate of change of cost C with respect to the percent of particulate pollution that escapes when p ϭ (percent) (b) Write a sentence interpreting your answer to part (a) 57 Wind chill One form of the formula that meteorologists use to calculate wind chill temperature (WC) is WC ϭ 35.74 ϩ 0.6215t Ϫ 35.75s0.16 ϩ 0.4275t s0.16 where s is the wind speed in mph and t is the actual air temperature in degrees Fahrenheit Suppose temperature is constant at 15Њ (a) Express wind chill WC as a function of wind speed s (b) Find the rate of change of wind chill with respect to wind speed when the wind speed is 25 mph (c) Interpret your answer to part (b) 58 Allometric relationships—crabs For fiddler crabs, data gathered by Thompson* show that the allometric relationship between the weight C of the claw and the weight W of the body is given by C ϭ 0.11W 1.54 Find the function that gives the rate of change of claw weight with respect to body weight Recall that for all modeling problems, use the unrounded model for any calculations unless instructed otherwise 59 Modeling Consumer price index The table below gives the U.S consumer price index (CPI) for selected years from 2012 and projected to 2050 With the reference year as 2012, a 2020 CPI ϭ 120.56 means goods and services that cost $100.00 in 2012 are expected to cost $120.56 in 2020 (a) Find the quadratic function that is the best fit for the data, with x as the number of years past 2010 and y as the CPI in dollars Report the model as y ϭ f (x) with three significant digit coefficients (b) Use the data to find the average rate of change of the CPI from 2012 to 2020 (c) Find the derivative of the reported model found in part (a) (d) Find the instantaneous rate of change of the CPI for the year 2020 (e) Use the rate of change from part (d) to predict the CPI for 2022 Year CPI Year CPI 2012 2014 2016 2018 2020 2025 100.00 104.00 108.58 114.09 120.56 138.41 2030 2035 2040 2045 2050 158.90 182.43 209.44 240.45 276.05 Source: Social Security Administration 60 Modeling E-commerce The following table gives the online sales, in billions of dollars, from 2000 and projected to 2017 (a) Model these data with a power function E(t), where t is the number of years past 1990 Report the model with three significant digit coefficients *d’Arcy Thompson, On Growth and Form (Cambridge, England: Cambridge University Press, 1961) 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 ... graphing utility (c) Use the graph of f (x) to identify x-values where f (x) ؍0, f (x) 0, and f (x) (d) Use the graph of f (x) to identify x-values where f (x) has a maximum or minimum point,... tangent line to a curve at a point closely approximates the curve near the point In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable... f (x) ϭ x Ϫ 4x ϩ 34 f (x) ϭ 3x Ϫ 5x ϩ In Problems 35 and 36, find each derivative at the given x-value (a) with the appropriate rule and (b) with the numerical derivative feature of a graphing