11hw5 - Homework Assignment#5 15.1 a Prove that the...

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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 non-zero, 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 profit-maximizing 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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11hw5 - Homework Assignment#5 15.1 a Prove that the...

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