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Unformatted text preview: econ/ams 11b winter 2010 Midterm 1 Review — Solutions 1. Compute the indicated partial derivatives. a. 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 = 6 x 2 yz 2 3 y 3 z + 10 xy 2 + 11 . f y = 2 x 3 z 2 9 xy 2 z + 10 x 2 y 7 z 5 . f zx = f xz = 12 x 2 yz 3 y 3 . b. q ( u,v ) = u 2 v 3 uv 3 2 u + 3 v ; ∂q ∂u = (2 uv 3 v 3 )(2 u + 3 v ) 2( u 2 v 3 uv 3 ) (2 u + 3 v ) 2 = 2 u 2 v + 6 uv 2 9 v 4 (2 u + 3 v ) 2 . ∂q ∂v = ( u 2 9 uv 2 )(2 u + 3 v ) 3( u 2 v 3 uv 3 ) (2 u + 3 v ) 2 = 2 u 3 18 u 2 v 2 18 uv 3 (2 u + 3 v ) 2 . c. w = r 2 ln(3 r + s 2 ); w r = 2 r ln(3 r + s 2 ) + 3 r 2 3 r + s 2 . w s = 2 r 2 s 3 r + s 2 . w rs = w sr = 4 rs (3 r + s 2 ) 6 r 2 s (3 r + s 2 ) 2 = 6 r 2 s + 4 rs 3 (3 r + s 2 ) 2 . 2. The demand function for a firm’s product is given by Q = 20 √ 6 Y + 5 p s 3 p + 4 , where • Q is the monthly demand for the firm’s product, measured in 1000’s of units, • Y is the average monthly disposable income in the market for the firm’s product, measured in 100’s of dollars, • p s is the average price of a substitute for the firm’s product, measured in dollars, • p is the price of the firm’s product, also measured in dollars. a. Find Q , Q Y , Q p s and Q p when the monthly income is $2500 and the prices are p s = 15 and p = 12. Round your (final) answers to two decimal places. 1 Q (25 , 15 , 12) = 20 √ 225 40 = 7 . 5 . Q Y = 60(6 Y + 5 p s ) 1 / 2 3 p + 4 ; Q Y (25 , 15 , 12) = 60 / 15 40 = 0 . 1. Q p s = 50(6 Y + 5 p s ) 1 / 2 3 p + 4 ; Q p s (25 , 15 , 12) = 50 / 15 40 = 1 12 ≈ . 08 . Q p = 60 √ 6 Y + 5 p s (3 p + 4) 2 ; Q p (25 , 15 , 12) = 60 · 15 1600 = 9 16 ≈  . 56 b. Compute the incomeelasticity of demand for the firm’s product at the point in part a....
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This note was uploaded on 03/15/2010 for the course ECON 11 taught by Professor Yk during the Spring '10 term at University of California, Santa Cruz.
 Spring '10
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