SECTION 12.4 EXAMPLE Applications of the Indefinite Integral in Business and Economics 769 Consumption and Savings | APPLICATION PREVIEW | If the consumption is $9 trillion when income is $0, and if the marginal propensity to save is 0.25, find the consumption function Solution If dS dy ϭ 0.25, then 0.25 ϭ Ϫ dC dy, or dC dy ϭ 0.75 Thus C ϭ 30.75 dy ϭ 0.75y ϩ K If C ϭ when y ϭ 0, then ϭ 0.75(0) ϩ K, or K ϭ Then the consumption function is ■ C ϭ 0.75y ϩ (trillions of dollars) ✓ CHECKPOINT If the marginal propensity to save is 0.4 dS ϭ 0.7 Ϫ dy y find the marginal propensity to consume Find the national consumption function if the marginal propensity to consume is dC ϭ dy ϩ 0.2 yϩ4 and national consumption is $6.8 trillion when disposable income is $0 ✓ CHECKPOINT ANSWERS (a) False C(0) equals the fixed costs It may or may not be the constant of integration (b) False We use R(0) ϭ to determine the constant of integration, but it may be nonzero C(x) ϭ 200 x ϩ ϩ 7800 dS 0.4 dC ϭ1Ϫ ϭ 0.3 ϩ dy dy y C( y) ϭ y ϩ ϩ 0.2y ϩ 2.8 | EXERCISES | 12.4 TOTAL COST AND PROFIT In Problems 1–12, cost, revenue, and profit are in dollars and x is the number of units If the daily marginal cost for a product is MC ϭ 2x ϩ 100, with fixed costs amounting to $200, find the total cost function for each day If the monthly marginal cost for a product is MC ϭ x ϩ 30 and the related fixed costs are $5000, find the total cost function for the month If the marginal cost for a product is MC ϭ 4x ϩ and the production of 10 units results in a total cost of $300, find the total cost function If the marginal cost for a product is MC ϭ 3x ϩ 50 and the total cost of producing 20 units is $2000, find the total cost function If the marginal cost for a product is MC ϭ 4x ϩ 40 and the total cost of producing 25 units is $3000, find the cost of producing 30 units If the marginal cost for producing a product is MC ϭ 5x ϩ 10, with a fixed cost of $800, find the cost of producing 20 units A firm knows that its marginal cost for a product is MC ϭ 3x ϩ 20, that its marginal revenue is MR ϭ 44 Ϫ 5x, and that the cost of production of 80 units is $11,400 (a) Find the optimal level of production (b) Find the profit function (c) Find the profit or loss at the optimal level A certain firm’s marginal cost for a product is MC ϭ 6x ϩ 60, its marginal revenue is MR ϭ 180 Ϫ 2x, and its total cost of production of 10 items is $1000 (a) Find the optimal level of production (b) Find the profit function (c) Find the profit or loss at the optimal level of production (d) Should production be continued for the short run? (e) Should production be continued for the long run? 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 770 CHAPTER 12 Indefinite Integrals Suppose that the marginal revenue for a product is MR ϭ 900 and the marginal cost is MC ϭ 30 x ϩ 4, with a fixed cost of $1000 (a) Find the profit or loss from the production and sale of units (b) How many units will result in a maximum profit? 10 Suppose that the marginal cost for a product is MC ϭ 60 x ϩ and its fixed cost is $340.00 If the marginal revenue for the product is MR ϭ 80x, find the profit or loss from production and sale of (a) units (b) units 11 The average cost of a product changes at the rate 16 If national consumption is $9 trillion when income is $0 and if the marginal propensity to consume is 0.30, what is consumption when disposable income is $20 trillion? 17 If consumption is $8 trillion when income is $0 and if the marginal propensity to consume is CЈ(x) ϭ Ϫ6x Ϫ2 ϩ dC 0.3 ϭ 0.4 ϩ dy y and the average cost of units is $10.00 (a) Find the average cost function (b) Find the average cost of 12 units 12 The average cost of a product changes at the rate CЈ(x) ϭ Ϫ10 ϩ x 10 0.2 dC ϭ 0.3 ϩ dy y find the national consumption function 18 If consumption is $5 trillion when disposable income is $0 and if the marginal propensity to consume is find the national consumption function 19 If consumption is $6 trillion when disposable income is $0 and if the marginal propensity to consume is dC ϭ dy ϩ 0.4 yϩ1 and the average cost of 10 units is $20.00 (a) Find the average cost function (b) Find the average cost of 20 units 13 Suppose that marginal cost for a certain product is given by MC ϭ 1.05(x ϩ 180)0.05 and marginal revenue is given by MR ϭ (1 0.5x ϩ 4) ϩ 2.8, where x is in thousands of units and both revenue and cost are in thousands of dollars Fixed costs are $200,000 and production is limited to at most 200 thousand units (a) Find C(x) and R(x) (b) Graph C(x) and R(x) to determine whether a profit can be made (c) Determine the level of production that yields maximum profit, and find the maximum profit (or minimum loss) 14 Suppose that the marginal cost for a certain product is given by MC ϭ 1.02(x ϩ 200)0.02 and marginal revenue is given by MR ϭ (2 4x ϩ 1) ϩ 1.75, where x is in thousands of units and revenue and cost are in thousands of dollars Suppose further that fixed costs are $150,000 and production is limited to at most 200 thousand units (a) Find C(x) and R(x) (b) Graph C(x) and R(x) to determine whether a profit can be made (c) Determine what level of production yields maximum profit, and find the maximum profit (or minimum loss) find the national consumption function 20 If consumption is $5.8 trillion when disposable income is $0 and if the marginal propensity to consume is NATIONAL CONSUMPTION AND SAVINGS and that consumption is $8.6 trillion when disposable income is $0 Find the national consumption function 25 Suppose that the marginal propensity to save is 15 If consumption is $7 trillion when disposable income is $0 and if the marginal propensity to consume is 0.80, find the national consumption function (in trillions of dollars) dC ϭ dy ϩ 0.8 2y ϩ find the national consumption function 21 Suppose that the marginal propensity to consume is dC ϭ 0.7 Ϫ e Ϫ2y dy and that consumption is $5.65 trillion when disposable income is $0 Find the national consumption function 22 Suppose that the marginal propensity to consume is ln ( y ϩ 1) dC ϭ 0.04 ϩ dy yϩ1 and that consumption is $6.04 trillion when disposable income is $0 Find the national consumption function 23 Suppose that the marginal propensity to save is dS ϭ 0.15 dy and that consumption is $5.15 trillion when disposable income is $0 Find the national consumption function 24 Suppose that the marginal propensity to save is dS ϭ 0.22 dy dS ϭ 0.2 Ϫ dy 3y 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