Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual
CHAPTER 7 Exercise 7.1 1.
2.
(a) dy/dx = 12x11
(b) dy/dx = 0
(c) dy/dx = 35x4
(d) dw/du = −3u−2
(e) dw/du = −2u−1/2
(f) dw/du = u−3/4
(a) 4x−5
(b) 3x−2/3
(c) 20w3
(d) 2cx
(e) abub−1
(f) abu−b−1
3. (a) f 0 (x) = 18; thus f 0 (1) = 18 and f 0 (2) = 18. (b) f 0 (x) = 3cx2 ; thus f 0 (1) = 3c and f 0 (2) = 12c. 1 (c) f 0 (x) = 10x−3 ; thus f 0 (1) = 10 and f 0 (2) = 10 8 = 44 √ √ (d) f 0 (x) = x1/3 = 3 x; thus f 0 (1) = 1 and f 0 (2) = 3 2
(e) f 0 (w) = 2w−2/3 ; thus f 0 (1) = 2 and f 0 (2) = 2 · 2−2/3 = 21/3 (f) f 0 (w) = 12 w−7/6 ; thus f 0 (1) =
1 2
and f 0 (2) = 12 (2−7/6 ) = 2−1 · 2−7/6
4. Refer to the following two graphs
Exercise 7.2 1. V C = Q3 − 5Q2 + 12Q. The derivative
d dQ V
C = 3Q2 − 10Q + 12 is the M C function.
2. C = AC · Q = Q3 − 4Q2 + 174Q. Thus M C = dC/dQ = 3Q2 − 9Q + 174. Since the total-cost function shows zero fixed cost, the situation depicted is the long run.
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