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11Breview1

# 11Breview1 - econ 11b ucsc ams 11b Review Questions 1...

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econ 11 b ucsc ams 11 b Review Questions 1 Partial derivatives and applications. 1. Compute the indicated partial derivatives of the functions below. a. z = 3 x 2 + 4 xy - 5 y 2 - 4 x + 7 y - 2 , z x = z y = b. F ( u, v, w ) = 60 u 2 / 3 v 1 / 6 w 1 / 2 ∂F ∂u = 2 F ∂w∂u = c. w = x 2 z ln( y 2 + z 3 ) w x = w y = w xx = w yz = w xyz = d. f ( x, y, z ) = 2 x 3 yz 2 - 3 xy 3 z + 5 x 2 y 2 - 7 yz 5 + 11 x - 1 f x = f y = f zx =. e. q ( u, v ) = u 2 v - 3 uv 3 2 u + 3 v ∂q ∂u = ∂q ∂v = 2. Suppose that z = 3 x 2 y 3 + 5 xy 2 - 3 x + y - 1, where x = 3 t - 2 and y = 2 t + 1. Use the chain rule to compute dz dt t =0 . 3. The production function for a firm is given by Q = F ( K, L ) , where Q is the firm’s monthly output, K is the firm’s monthly capital input and L is the firm’s monthly labor input. Furthermore, the labor input L is given by L = 2 m (60 h - h 2 ) ,

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where m is the number of the firm’s employees and h is the average number of hours each employee works per month. The firm’s profit function is given by P = p 0 Q - ( w 0 mh + K ) , where p 0 is the (fixed) price of the firm’s product and w 0 is the (average) hourly wage the firm pays its employees.
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