Hw1+Solutions - f x(125 64 = 160 The marginal productivity...

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Solutions to Homework 1 7 . 1 . 2 g (1 , 1) = 3, g (0 , - 1) = 2, g ( a, b ) = a 2 + 2 b 2 . 7 . 1 . 8 C ( x, y, z ) = 3 xy + 5 xz + 10 yz . 7 . 1 . 12 C ( x, y ) = 100 x + 200 y . 7 . 1 . 13 a) f (2 . 5 , 200000 , 5000) = 1875. b) f (3 , 200000 , 5000) = 2250. The percentage increase in taxes is the difference in taxes divided by the old taxes then multiplied by 100. 2250 - 1875 1875 × 100 = 37500 1875 = 20% . 7 . 1 . 16 (Full disclosure: I don’t know how to create graphs on the computer so what follows is a description of the graphs. If you have any questions about this solution please contact me.) You end up with the graph of 3 parabolas. When c = 0 you graph the parabola y = x 2 / 2. When c = 1 you graph the parabola y = x 2 / 2 + 1 / 2. When c = 2 you graph the parabola y = x 2 / 2 + 1. 7 . 2 . 2 f x = 2 x and f y = - 2 y . 7 . 2 . 5 f x = 1 y - y x 2 and f y = 1 x - x y 2 . 7 . 2 . 6 f x = - 1 ( x + y ) 2 and f y = - 1 ( x + y ) 2 . 7 . 2 . 17 f x = ze yz , f y = xz 2 e yz , and f z = xe yz + xyze yz . 7 . 2 . 23 The first derivatives are f x = 3 x 2 y + 2 y 2 and f y = x 3 + 4 xy . From these we obtain the 4 second derivatives: f xx = 6 xy , f yy = 4 x , f yx = 3 x 2 +4 y and f xy = 3 x 2 + 4 y . 1
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7 . 2 . 26 a) The first order partials are f x = 200 x - 1 / 3 y 1 / 3 and f y = 200 x 2 / 3 y - 2
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Unformatted text preview: f x (125 , 64) = 160. The marginal productivity of capital is given by f y (125 , 64) = 156 . 25. b) f (125 , 66) ≈ f (125 , 64) + 2 · f y (125 , 64) = 30 , 312 . 5. c) f (124 , 64) ≈ f (125 , 64)-1 · f y (125 , 64) = 29 , 840. 7 . 2 . 31 ∂V ∂P =-. 08 T P 2 which tell us ∂V ∂P (20 , 300) =-. 06. This means that if pressure is changed from 20 by h units and temperature remains fixed at 300 units then the volume decreases by approximately . 06 h units. ∂V ∂T = . 08 P which tell us ∂V ∂T (20 , 300) = . 004. This means that if temperature is changed from 300 by k units and pressure remains fixed at 20 units then the volume increases by approximately . 004 k units. 2...
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