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ProblemSet8 Solutions

# ProblemSet8 Solutions - Department of Economics Ryerson...

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Department of Economics Ryerson University ECN230-Mathematics For Economics Problem Set #8 - Solutions EXERCISE 7.2 Problem 1: Variable-cost function: V C = Q 3 5 Q 2 + 12 Q . The derivative: d ( V C ) dQ = 3 Q 2 10 Q + 12 is the marginal-cost ( MC ) function. Problem 2: First, we can obtain the total-cost function as follows: AC = C Q C = AC · Q = Q 3 4 Q 2 + 174 Q . Thus, MC = dC dQ = 3 Q 2 9 Q + 174. Since the total-cost function has no fi xed cost, the situation depicted is the long-run. Problem 3: (a) 3 ¡ 27 x 2 + 6 x 2 ¢ . (b) 54 x 2 + 78 x 70. (c) 12 x ( x + 1). (d) cx (3 ax 2 b ). 1

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(e) x (9 x + 14). (f) 2 ³ x 2 +3 x 2 ´ = x 2 3 x 2 . Problem 5: Let the average curve represented by A = a + bx . Then, the total curve will be T = A · x = ax + bx 2 , and the marginal curve will be M = dT/dx = a +2 bx . For x = 0, A = M = a . Therefore, the average and marginal curves have the same vertical intercepts. In addition, dM/dx = 2 b and dA/dx = b . Therefore, the marginal curve is twice as steep as the average curve. Problem 7: (a) x 2 3 x 2 . (b) 9 x 2 . (c) 30 ( x + 5) 2 . (d) acx 2 + 2 adx bc ( cx + d ) 2 . Problem 8: Problem 9: (a) True. The continuity of f ( x ) is a necessary condition for f ( x ) to be di ff erentiable. (b) False. A continuous function may not have a continuous derivative function (See for example, Figure 7.1(c)).
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ProblemSet8 Solutions - Department of Economics Ryerson...

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