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614 CHAPTER Derivatives EXAMPLE Profit and Marginal Profit In Example 5, we found that the profit (in thousands of dollars) for a company’s products is given by P (x) ϭ 20 x ϩ Ϫ 2x Ϫ 22 and its marginal profit is given by 10 Ϫ PЈ(x) ϭ xϩ1 (a) Use the graphs of P(x) and PЈ(x) to determine the relationship between the two functions (b) When is the marginal profit 0? What is happening to profit at this level of production? Solution (a) By comparing the graphs of the two functions (shown in Figure 9.39), we see that for x Ͼ 0, profit P(x) is increasing over the interval where the marginal profit PЈ(x) is positive, and profit is decreasing over the interval where the marginal profit PЈ(x) is negative (b) By using ZERO or SOLVER, or by using algebra, we see that PЈ(x) ϭ when x ϭ 24 This level of production (x ϭ 24) is where profit is maximized, at 30 (thousand dollars) ■ 35 P (x) P(x) 20 x 2x Figure 9.39 ✓ CHECKPOINT ANSWERS | EXERCISES | 10 x 22 50 5 MC ϭ CЈ(x) ϭ 0.03x Ϫ 1.8x ϩ 33 CЈ(50) ϭ 18 The 51st unit will cost approximately $18 to produce C(51) Ϫ C(50) ϭ 18.61 True Yes Yes, because PЉ(x) Ͻ for x Ն 9.9 MARGINAL REVENUE, COST, AND PROFIT In Problems 1–8, total revenue is in dollars and x is the number of units (a) If the total revenue function for a product is R(x) ϭ 4x, what is the marginal revenue function for that product? (b) What does this marginal revenue function tell us? If the total revenue function for a product is R(x) ϭ 32x, what is the marginal revenue for the product? What does this mean? Suppose that the total revenue function for a commodity is R ϭ 36x Ϫ 0.01x (a) Find R(100) and tell what it represents (b) Find the marginal revenue function (c) Find the marginal revenue at x ϭ 100, and tell what it predicts about the sale of the next unit and the next units (d) Find R(101) Ϫ R(100) and explain what this value represents Suppose that the total revenue function for a commodity is R(x) ϭ 25x Ϫ 0.05x (a) Find R(50) and tell what it represents (b) Find the marginal revenue function (c) Find the marginal revenue at x ϭ 50, and tell what it predicts about the sale of the next unit and the next units (d) Find R(51) Ϫ R(50) and explain what this value represents Suppose that demand for local cable TV service is given by p ϭ 80 Ϫ 0.4x where p is the monthly price in dollars and x is the number of subscribers (in hundreds) 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 9.9 where x is the number of units and p is the price in dollars (a) Find the total revenue from the sale of 500 units (b) Find and interpret the marginal revenue at 500 units (c) Is more revenue expected from the 501st unit sold or from the 701st? Explain (a) Graph the marginal revenue function from Problem (b) At what value of x will total revenue be maximized for Problem (c) What is the maximum revenue? (a) Graph the marginal revenue function from Problem (b) Determine the number of units that must be sold to maximize total revenue (c) What is the maximum revenue? In Problems 9–16, cost is in dollars and x is the number of units Find the marginal cost functions for the given cost functions C(x) ϭ 40 ϩ 8x 10 C(x) ϭ 200 ϩ 16x 11 C(x) ϭ 500 ϩ 13x ϩ x 12 C(x) ϭ 300 ϩ 10x ϩ 100 x2 13 C ϭ x Ϫ 6x ϩ 24x ϩ 10 14 C ϭ 0.1x Ϫ 1.5x ϩ 9x ϩ 15 15 C ϭ 400 ϩ 27x ϩ x 16 C(x) ϭ 50 ϩ 48x ϩ x 17 Suppose that the cost function for a commodity is C(x) ϭ 40 ϩ x dollars (a) Find the marginal cost at x ϭ units and tell what this predicts about the cost of producing additional unit (b) Calculate C(6) Ϫ C(5) to find the actual cost of producing additional unit 18 Suppose that the cost function for a commodity is C(x) ϭ x Ϫ 4x ϩ 30x ϩ 20 dollars find the marginal cost at x ϭ units and tell what this predicts about the cost of producing additional unit and additional units 20 If the cost function for a commodity is C(x) ϭ 90 x ϩ 4x ϩ 4x ϩ 10 dollars find the marginal cost at x ϭ units and tell what this predicts about the cost of producing additional unit and additional units 21 If the cost function for a commodity is C(x) ϭ 300 ϩ 4x ϩ x graph the marginal cost function 22 If the cost function for a commodity is C(x) ϭ x Ϫ 12x ϩ 63x ϩ 15 graph the marginal cost function In each of Problems 23 and 24, the graph of a company’s total cost function is shown For each problem, use the graph to answer the following questions (a) Will the 101st item or the 501st item cost more to produce? Explain (b) Does this total cost function represent a manufacturing process that is getting more efficient or less efficient? Explain C(x) 23 Dollars p ϭ 160 Ϫ 0.1x x Units 24 C(x) C(x) ϭ 300 ϩ 6x ϩ 20 x dollars (a) Find the marginal cost at x ϭ units and tell what this predicts about the cost of producing 1 additional unit 615 (b) Calculate C(9) Ϫ C(8) to find the actual cost of producing additional unit 19 If the cost function for a commodity is Dollars (a) Find the total revenue as a function of the number of subscribers (b) Find the number of subscribers when the company charges $50 per month for cable service Then find the total revenue for p ϭ $50 (c) How could the company attract more subscribers? (d) Find and interpret the marginal revenue when the price is $50 per month What does this suggest about the monthly charge to subscribers? Suppose that in a monopoly market, the demand function for a product is given by Applications: Marginals and Derivatives x Units 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 Derivatives In Problems 25–28, cost, revenue, and profit are in dollars and x is the number of units 25 If the total profit function is P(x) ϭ 5x Ϫ 25, find the marginal profit What does this mean? 26 If the total profit function is P(x) ϭ 16x Ϫ 32, find the marginal profit What does this mean? 27 Suppose that the total revenue function for a product is R(x) ϭ 50x and that the total cost function is C (x) ϭ 1900 ϩ 30x ϩ 0.01x (a) Find the profit from the production and sale of 500 units (b) Find the marginal profit function (c) Find MP at x ϭ 500 and explain what it predicts (d) Find P (501) Ϫ P (500) and explain what this value represents 28 Suppose that the total revenue function is given by R (x) ϭ 46x and that the total cost function is given by x C(x) ϭ 100 ϩ 30x ϩ 10 (a) (b) (c) (d) Find P(100) Find the marginal profit function Find MP at x ϭ 100 and explain what it predicts Find P(101) Ϫ P(100) and explain what this value represents R(x) Dollars $ R(x) C(x) x Units In each of Problems 31 and 32, the graph of a company’s profit function is shown For each problem, use the graph to answer the following questions about points A, B, and C (a) Rank from smallest to largest the amounts of profit received at these points Explain your choices, and note whether any point results in a loss (b) Rank from smallest to largest the marginal profit at these points Explain your choices, and note whether any marginal is negative and what this means P(x) 31 P(x) C B x A Units 32 P(x) B C Dollars In each of Problems 29 and 30, the graphs of a company’s total revenue function and total cost function are shown For each problem, use the graph to answer the following questions (a) From the sale of 100 items, 400 items, and 700 items, rank from smallest to largest the amount of profit received Explain your choices and note whether any of these scenarios results in a loss (b) From the sale of the 101st item, the 401st item, and the 701st item, rank from smallest to largest the amount of profit received Explain your choices, and note whether any of these scenarios results in a loss $ 29 30 Dollars CHAPTER Dollars 616 P(x) x A C(x) Units x Units 33 (a) Graph the marginal profit function for the profit function P(x) ϭ 30x Ϫ x Ϫ 200, where P(x) is in thousands of dollars and x is hundreds of units 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 Key Terms and Formulas (b) What level of production and sales will give a marginal profit? (c) At what level of production and sales will profit be at a maximum? (d) What is the maximum profit? 34 (a) Graph the marginal profit function for the profit function P(x) ϭ 16x Ϫ 0.1x Ϫ 100, where P(x) is in hundreds of dollars and x is hundreds of units (b) What level of production and sales will give a marginal profit? (c) At what level of production and sales will profit be at a maximum? (d) What is the maximum profit? 35 The price of a product in a competitive market is $300 If the cost per unit of producing the product is 160 ϩ 0.1x dollars, where x is the number of units produced per month, how many units should the firm produce and sell to maximize its profit? 36 The cost per unit of producing a product is 60 ϩ 0.2x dollars, where x represents the number of units produced per week If the equilibrium price determined by a competitive market is $220, how many units should the firm produce and sell each week to maximize its profit? 37 If the daily cost per unit of producing a product by the Ace Company is 10 ϩ 0.1x dollars, and if the price on the competitive market is $70, what is the maximum daily profit the Ace Company can expect on this product? 38 The Mary Ellen Candy Company produces chocolate Easter bunnies at a cost per unit of 0.40 ϩ 0.005x dollars, where x is the number produced If the price on the competitive market for a bunny this size is $10.00, how many should the company produce to maximize its profit? C h a p t e r Summary & Review KEY TERMS AND FORMULAS Section 9.1 Limit (p 536) One-sided limits (p 537) Properties of limits (p 539) Polynomial functions (p 539) Limits of rational functions (p 540) 0 indeterminate form Limits of piecewise defined functions (p 543) Section 9.2 Continuous function (p 549) Limit at infinity (p 553) Section 9.3 Average rate of change of f over [a, b] (p 560) f (b) Ϫ f (a) bϪa Average velocity (p 561) Instantaneous rate of change (p 561) Velocity (p 562) Derivative of f (x) (p 562) f (x ϩ h) Ϫ f (x) f Ј(x) ϭ lim h h Section 9.4 Powers of x Rule (p 573) d(x n ) ϭ nx nϪ1 dx Constant Function Rule (p 575) d(c) ϭ for constant c dx Coefficient Rule (p 575) d [c f (x)] ϭ c f Ј(x) dx 617 Vertical asymptote (p 553) Horizontal asymptote (p 554) Derivative notation (p 563) dy df (x) , yЈ, f Ј(x), dx dx Marginal revenue (p 563) MR ϭ RЈ(x) Slope of a tangent (p 565) Tangent line (p 566) Interpretations of the derivative (p 566) Differentiability and continuity (p 567) Sum Rule (p 576) d du dv (u ϩ v) ϭ ϩ dx dx dx Difference Rule (p 577) du dv d (u Ϫ v) ϭ Ϫ dx dx dx 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 ... its profit? C h a p t e r Summary & Review KEY TERMS AND FORMULAS Section 9.1 Limit (p 536) One-sided limits (p 537) Properties of limits (p 539) Polynomial functions (p 539) Limits of rational
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