psCalculus1Ans - Fall 2007 Problem Set #09- Answer key...

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Fall 2007 ARE211 Problem Set #09- Answer key First Calculus Problem Set Please note: There are parts of one question in this assignment to which you may want to reply: “I beg your pardon?” or something possibly ruder than this. That kind of response is entirely appropriate, but you should add a sentence explaining why you are led to make such a remark. (1) Consider the function f ( x ) = 20 x 3 - 120 x 2 - 5 x + 36 a) What is the derivative of this function? Ans: f p ( x ) = 60 * x 2 - 240 * x - 5 b) What is the derivative of this function evaluated at x 0 = 4? Ans: f p ( x 0 = 4) = 60 * 4 2 - 240 * 4 - 5 = 240 * 4 - 240 * 4 - 5 = - 5 c) What is the diFerential of this function at x 0 = 4? Ans: df = - 5 dx d) Approximate the change in the function when moving from x 0 = 4 to x 1 = 5. Ans: df | x 0 =4 = - 5 * 1 = - 5 e) What is the actual change in the function when moving from x 0 = 4 to x 1 = 5? Ans: Δ f = f ( x 1 = 5) - f ( x 0 = 4) = - 489 - ( - 624) = 135 . Not a very good approximation, huh! Any ideas why? Try plotting the function. (2) Multivariate calculus drill: Simon and Blume, question 14.1, parts (a), (d) and (f); Ans: a) ∂f ∂x = 8 xy - 3 y 3 + 6 , ∂f ∂y = 4 x 2 - 9 xy 2
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2 d) ∂f ∂x = 2 e 2 x +3 Y , ∂f ∂y = 3 e 2 x +3 y . f) ∂f ∂x = 6 xy - 7 y, ∂f ∂y = 3 x 2 - 7 x 2 y (3) Gradients: Simon and Blume, question 14.19 Ans: Δ y 2 e 3 x = (3 y 2 e 3 x , 2 ye 3 x ) .At (0 , 3) , this vector is proportional to (9 , 2) . Normalizing to length 1 , the vector is (9 / 85 , 2 / 85) . (4) More diFerential approximations: Simon and Blume, question 14.4. Ans: a) Q = 5400 b) Q (998 , 216) = 5398 . 798 . The approximation gives Q 5392 . 8 , which is in error by - 0 . 002 . c) Q (1000 , 217 . 5) = 5412 . 471 . The approximation gives Q 5412 . 5 , which is in error by - 0 . 029 .
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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psCalculus1Ans - Fall 2007 Problem Set #09- Answer key...

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