Review Exercises Elderly in the Workforce, 1970–2040 Men Women 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% 26.8 19.0 9.7 1970 17.7 16.3 1980 9.4 8.6 8.1 1990 2000 21.0 19.6 21.0 19.5 17.3 12.6 11.7 12.5 11.1 10.1 2010 2015 2020 2030 621 where x is the number of $30 rent increases (and also the resulting number of unrented apartments) Find the marginal revenue when x ϭ 10 Does this tell you that the rent should be raised (causing more vacancies) or lowered? Explain 95 Productivity Suppose the productivity of a worker (in units per hour) after x hours of training and time on the job is given by 2040 P(x) ϭ ϩ Source: Bureau of the Census, U.S Department of Commerce 70x x ϩ 1000 (a) Find and interpret P(20) (b) Find and interpret PЈ(20) Section 9.4 91 Demand Suppose that the demand for x units of a product is given by x ϭ (100 p) Ϫ 1, where p is the price per unit of the product Find and interpret the rate of change of demand with respect to price if the price is (a) $10 (b) $20 92 Severe weather ice makers Thunderstorms severe enough to produce hail develop when an upper-level low (a pool of cold air high in the atmosphere) moves through a region where there is warm, moist air at the surface These storms create an updraft that draws the moist air into subfreezing air above 10,000 feet Data from the National Weather Service indicates that the strength of the updraft, as measured by its speed s in mph, affects the size of the hail according to h ϭ 0.000595s1.922 where h is the diameter of the hail (in inches) Find and interpret h(100) and hЈ(100) 93 Revenue The graph shows the revenue function for a commodity Will the (A ϩ 1)st item sold or the (B ϩ 1)st item sold produce more revenue? Explain R(x) Section 9.6 96 Demand The demand q for a product at price p is given by q ϭ 10,000 Ϫ 50 0.02p2 ϩ 500 Find the rate of change of demand with respect to price 97 Supply The number of units x of a product that is supplied at price p is given by xϭ pϪ1, pՆ1 If the price p is $10, what is the rate of change of the supply with respect to the price, and what does it tell us? Section 9.8 98 Acceleration Suppose an object moves so that its distance to a sensor, in feet, is given by s(t) ϭ 16 ϩ 140t ϩ t where t is the time in seconds Find the acceleration at time t ϭ seconds 99 Profit Suppose a company’s profit (in dollars) is given by Dollars P(x) ϭ 70x Ϫ 0.1x Ϫ 5500 where x is the number of units Find and interpret PЈ(300) and PЉ(300) Section 9.9 x A Units B Section 9.5 94 Revenue In a 100-unit apartment building, when the price charged per apartment rental is (830 ϩ 30x) dollars, then the number of apartments rented is 100 Ϫ x and the total revenue for the building is R(x) ϭ (830 ϩ 30x)(100 Ϫ x) In Problems 100–107, cost, revenue, and profit are in dollars and x is the number of units 100 Cost If the cost function for a particular good is C(x) ϭ 3x ϩ 6x ϩ 600, what is the (a) marginal cost function? (b) marginal cost if 30 units are produced? (c) interpretation of your answer in part (b)? 101 Cost If the total cost function for a commodity is C(x) ϭ 400 ϩ 5x ϩ x 3, what is the marginal cost when units are produced, and what does it mean? 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 622 CHAPTER Derivatives 102 Revenue The total revenue function for a commodity is R ϭ 40x Ϫ 0.02x 2, with x representing the number of units (a) Find the marginal revenue function (b) At what level of production will marginal revenue be 0? 103 Profit If the total revenue function for a product is given by R(x) ϭ 60x and the total cost function is given by C ϭ 200 ϩ 10x ϩ 0.1x 2, what is the marginal profit at x ϭ 10? What does the marginal profit at x ϭ 10 predict? 104 Revenue The total revenue function for a commodity is given by R ϭ 80x Ϫ 0.04x (a) Find the marginal revenue function (b) What is the marginal revenue at x ϭ 100? (c) Interpret your answer in part (b) 105 Revenue If the revenue function for a product is R(x) ϭ 107 Profit A small business has weekly costs of C ϭ 100 ϩ 30x ϩ x2 10 where x is the number of units produced each week The competitive market price for this business’s product is $46 per unit Find the marginal profit 108 Cost, revenue, and profit The graph shows the total revenue and total cost functions for a company Use the graph to decide (and justify) at which of points A, B, and C (a) the revenue from the next item will be least (b) the profit will be greatest (c) the profit from the sale of the next item will be greatest (d) the next item sold will reduce the profit 60x 2x ϩ R(x) find the marginal revenue 106 Profit A firm has monthly costs given by C(x) C ϭ 45,000 ϩ 100x ϩ x where x is the number of units produced per month The firm can sell its product in a competitive market for $4600 per unit Find the marginal profit A B Units C x C h a p t e r TEST Evaluate the following limits, if they exist Use algebraic methods 4x Ϫ x (a) lim x Ϫ2 4x Ϫ 8x Ϫ 4x ϩ (b) lim x ϱ ϩ x Ϫ 5x x Ϫ 5x Ϫ 14 (c) lim x x Ϫ 6x Ϫ 5x Ϫ 25 (d) lim x Ϫ5 x ϩ (a) Write the limit definition for f Ј(x) (b) Use the definition from (a) to find f Ј(x) for f (x) ϭ 3x Ϫ x ϩ 4x Identify all x-values where f (x) is Let f (x) ϭ x Ϫ 8x not continuous Use derivative formulas to find the derivative of each of the following Simplify, except for part (d) (a) B ϭ 0.523W Ϫ 5176 (b) p ϭ 9t 10 Ϫ 6t Ϫ 17t ϩ 23 (c) y ϭ 3x 2x ϩ 11 (d) f (x) ϭ (3x Ϫ 2x ϩ 3)(4x 10 ϩ 10x Ϫ 17) (e) g (x) ϭ 34(2x ϩ 7x Ϫ 5)12 (f) y ϭ (x ϩ 3)(2x ϩ 5)6 10 (g) f (x) ϭ 12 x Ϫ ϩ 17 x Find d 3y dx for y ϭ x Ϫ x Ϫ3 Let f (x) ϭ x Ϫ 3x Ϫ 24x Ϫ 10 (a) Write the equation of the line tangent to the graph of y ϭ f (x) at x ϭ Ϫ1 (b) Find all points (both x- and y-coordinates) where f Ј(x) ϭ Find the average rate of change of f (x) ϭ Ϫ x Ϫ 2x over [1, 6] Use the given tables to evaluate the following limits, if they exist (b) lim g (x) (a) lim f (x) (c) lim g (x) x x x 5Ϫ 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 ... equation of the line tangent to the graph of y ϭ f (x) at x ϭ Ϫ1 (b) Find all points (both x- and y-coordinates) where f Ј(x) ϭ Find the average rate of change of f (x) ϭ Ϫ x Ϫ 2x over [1, 6]... for f Ј(x) (b) Use the definition from (a) to find f Ј(x) for f (x) ϭ 3x Ϫ x ϩ 4x Identify all x-values where f (x) is Let f (x) ϭ x Ϫ 8x not continuous Use derivative formulas to find the derivative