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SECTION 9.7 15 y ϭ (x Ϫ 1)2(x ϩ 1) 16 f (x) ϭ (5x ϩ 1)(x ϩ 5x)2 (x Ϫ 4)3 18 17 y ϭ x ϩ1 19 p ϭ [(q ϩ 1)(q3 Ϫ 3)]3 20 s ϭ [(4 Ϫ t 2)(t ϩ 5t)]4 21 R(x) ϭ [x 2(x ϩ 3x)]4 22 2x Ϫ 23 y ϭ 24 x ϩx 25 g (x) ϭ (8x ϩ 3)2(x Ϫ 4x)3 26 y ϭ (3x Ϫ 4x)3(4x Ϫ 8)2 x ϩ5 27 f (x) ϭ 28 Ϫ x2 30 29 y ϭ x 4x Ϫ 31 c(x) ϭ 2x x ϩ 32 Using Derivative Formulas 601 37 Revenue Suppose that the revenue in dollars from the sale of x campers is given by yϭ (x Ϫ 3)4 x c(x) ϭ [x 3(x ϩ 1)]Ϫ3 Ϫ x2 yϭ x4 R(x) ϭ 60,000x ϩ 40,000(10 ϩ x)Ϫ1 Ϫ 4000 (a) Find the marginal revenue when 10 units are sold (b) How is revenue changing when 10 units are sold? 38 Production Suppose that the production of x items of a new line of products is given by x ϭ 200[(t ϩ 10) Ϫ 400(t ϩ 40)Ϫ1] 2x Ϫ 2x ϩ y ϭ 3x 4x ϩ R(x) ϭ x 3x ϩ g (x) ϭ In Problems 33 and 34, find the derivative of each function 3(x ϩ 1)5 33 (a) F1(x) ϭ (b) F2(x) ϭ 5(x ϩ 1)5 (3x ϩ 1)5 (c) F3(x) ϭ (d) F4(x) ϭ (5x ϩ 1)5 2(x Ϫ 5)3 34 (a) G1(x) ϭ (2x Ϫ 5)3 (b) G2(x) ϭ (c) G3(x) ϭ 3(x Ϫ 5)3 (d) G4(x) ϭ (3x Ϫ 5)3 APPLICATIONS 35 Physical output The total physical output P of workers is a function of the number of workers, x The function P ϭ f (x) is called the physical productivity function Suppose that the physical productivity of x construction workers is given by P ϭ 10(3x ϩ 1)3 Ϫ 10 Find the marginal physical productivity, dP dx 36 Revenue Suppose that the revenue function for a certain product is given by R(x) ϭ 15(2x ϩ 1)Ϫ1 ϩ 30x Ϫ 15 where x is in thousands of units and R is in thousands of dollars (a) Find the marginal revenue when 2000 units are sold (b) How is revenue changing when 2000 units are sold? where t is the number of weeks the line has been in production Find the rate of production, dx dt 39 National consumption If the national consumption function is given by C(y) ϭ 2(y ϩ 1)1 ϩ 0.4y ϩ find the marginal propensity to consume, dC dy 40 Demand Suppose that the demand function for q units of an appliance priced at $p per unit is given by pϭ 400(q ϩ 1) (q ϩ 2)2 Find the rate of change of price with respect to the number of appliances 41 Volume When squares of side x inches are cut from the corners of a 12-inch-square piece of cardboard, an open-top box can be formed by folding up the sides The volume of this box is given by V ϭ x(12 Ϫ 2x)2 Find the rate of change of volume with respect to the size of the squares 42 Advertising and sales Suppose that sales (in thousands of dollars) are directly related to an advertising campaign according to Sϭ1ϩ 3t Ϫ (t ϩ 3)2 where t is the number of weeks of the campaign (a) Find the rate of change of sales after weeks (b) Interpret the result in part (a) 43 Advertising and sales An inferior product with an extensive advertising campaign does well when it is released, but sales decline as people discontinue use of the product If the sales S (in thousands of dollars) after t weeks are given by S(t) ϭ 200t , (t ϩ 1)2 tՆ0 what is the rate of change of sales when t ϭ 9? Interpret your result 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 602 CHAPTER Derivatives (a) Find the instantaneous rate of change of per capita health care expenditures in 2005 and 2015 (b) Interpret the rate of change for 2015 found in part (a) (c) Use the data to find the average rate of change of per capita health care expenditures from 2004 to 2006 How well does this approximate the instantaneous rate of change in 2005? 44 Advertising and sales An excellent film with a very small advertising budget must depend largely on word-of-mouth advertising If attendance at the film after t weeks is given by Aϭ 100t (t ϩ 10)2 what is the rate of change in attendance and what does it mean when (a) t ϭ 10? (b) t ϭ 20? 45 Per capita expenditures for U.S health care The dollars spent per person per year for health care (projected to 2018) are shown in the table These data can be modeled by yϭ 4.38(x Ϫ 10)2 ϩ 78(x Ϫ 10) ϩ 1430 0.0029x ϩ 0.25 where x is the number of years past 1990 and y is the per capita expenditures for health care OBJECTIVE To find second derivatives and higher-order derivatives of certain functions $ per Person Year $ per Person 2000 2002 2004 2006 2008 4789 5563 6331 7091 7826 2010 2012 2014 2016 2018 8465 9275 10,289 11,520 12,994 Source: U.S Medicare and Medicaid Services 9.8 Higher-Order Derivatives | APPLICATION PREVIEW | Since cell phones were introduced, their popularity has increased enormously Figure 9.32(a) shows a graph of the billions of worldwide cellular subscribers (actual and projected) as a function of the number of years past 1990 (Source: International Telecommunications Union and Key Global Telecom Indicators) Note that the number of subscribers is always increasing and that the rate of change of that number (as seen from tangent lines to the graph) is always positive However, the tangent lines shown in Figure 9.32(b) indicate that the rate of change of the number of subscribers is greater at B than at either A or C y y 10 Billions of Subscribers Billions of Subscribers • Year C 10 B A x Figure 9.32 x 2000 2005 2010 2015 2020 2000 2005 2010 2015 2020 Year (a) Year (b) Furthermore, the rate of change of the number of subscribers (the slopes of tangents) increases from A to B and then decreases from B to C To learn how the rate of change of the number of subscribers is changing, we are interested in finding the derivative of the rate of change of the number of subscribers—that is, the derivative of the derivative of the number of subscribers (See Example 4.) This is called the second derivative In this section we will discuss second and higher-order derivatives 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 ... Advertising and sales An excellent film with a very small advertising budget must depend largely on word-of-mouth advertising If attendance at the film after t weeks is given by Aϭ 100t (t ϩ 10)2 what... y is the per capita expenditures for health care OBJECTIVE To find second derivatives and higher-order derivatives of certain functions $ per Person Year $ per Person 2000 2002 2004 2006 2008... 2014 2016 2018 8465 9275 10,289 11,520 12,994 Source: U.S Medicare and Medicaid Services 9.8 Higher-Order Derivatives | APPLICATION PREVIEW | Since cell phones were introduced, their popularity has
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