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: VC = Q 3 5 Q 2 +12 Q . The derivative: d ( ) dQ =3 Q 2 10 Q 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, = dC dQ Q 2 9 Q + 174. Since the total-cost function has no f 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 .F o r 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 adx bc ( cx + d ) 2 . Problem 8: (a) d dx ( ax + b )= a (b) d dx x ( ax + b )=2 ax + b (c) d dx ³ 1 ax + b ´ = a ( ax + b ) 2 (d) d dx ¡ ax + b x ¢ = b x 2 Problem 9: (a) True. The continuity of f ( x ) is a necessary condition for f ( x )tobe di f erentiable. (b) False. A continuous function may not have a continuous derivative function (See for example, Figure 7.1(c)). 2
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Problem 10: (a) MC = dC dQ =6 Q +7 AC = C Q =3 Q +7+ 12 Q (b) MR = dR dQ =10 2 Q AR = R Q Q (c) MP = dQ dL = a +2 bL cL 2 AP = Q L = a + bL cL 2 EXERCISE 7.3 Problem 1: dy
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This homework help was uploaded on 04/18/2008 for the course ECONOMICS ECN 230 taught by Professor Angrydis during the Winter '08 term at Ryerson.

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ProblemSet8 Solutions - Department of Economics Ryerson...

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