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Unformatted text preview: Homework Assignment #5 15.1 a ) Prove that the expression x 2 xy 3 + y 5 = 17 is an implicit function of y in terms of x in a neighborhood of ( x,y ) = (5 , 2) . b ) Then, estimate the y value which corresponds to x = 4 . 8. Answer: a ) Let f ( x,y ) = x 2 xy 3 + y 5 17. Then ∂f/∂y = 3 xy 2 + 5 y 4 . Evaluating at ( x,y ) = (5 , 2) we find ∂f/∂y (5 , 2) = 60 + 80 = 20. Since this is nonzero, the implicit function theorem tells us that we may write y as a function of x in a neighborhood of x = 5. b ) Now dy/dx = ∂f/∂x/∂f/∂y . Evaluating at ( x,y ) = (5 , 2), we obtain dy/dx = 2 / 20 = 1 / 10. Since Δ x = . 2, Δ y ≈ +0 . 02, yielding a value of approximately y = 2 . 02. 15.7 Consider the profitmaximizing firm in Example 15.5. If p increases by Δ p and w increases by Δ w , what will be the corresponding effect on the optimal input amount x ? Answer: Here pf ( x ) = w implicitly defines x as a function of ( p,w ). As long as f 00 6 = 0, the im plicit function theorem allows us to write...
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 Fall '08
 STAFF
 Derivative, implicit function theorem

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