math20C_W07_Practice_mid2

math20C_W07_Practice_mid2 - vector and the initial position...

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Math 20C, Practice Midterm 2 February 20, 2007 Name: Section: This exam consists of 6 pages including this front page. Ground Rules 1. No calculator is allowed. 2. Show your work for every problem. A correct answer without any justiﬁcation will receive no credit. 3. You may use one 4-by-6 index card, both sides. Score 1 10 2 10 3 10 4 10 5 10 Total 60 1

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1. (a) Let f ( x, y ) = x e y . Find f x . (b) Consider x - z = arctan( yz ). Find ∂z/∂y by using implicit diﬀerentia- tion. 2
2. (a) Let r ( t ) = h t, 3 t 2 , e t - 1 i . Find the unit tangent vector at t = 1. (b) Find the position vector of a particle with the following the velocity

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Unformatted text preview: vector and the initial position. v ( t ) = t i + sin t j + t 2 k , x (0) = i-k 3 3. Show that the following limit does not exist. lim ( x,y ) → (0 , 0) x cos y x + y 2 4 4. Let f ( x, y ) = sin xy . Find all the numbers a so that the maximum directional derivative at (0 , a ) is 9. 5 5. Let f ( x, y ) = 9-2 x + 4 y-x 2-4 y 2 . Find all the local maximum and minimum values and saddle points of f , if there is any. 6...
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math20C_W07_Practice_mid2 - vector and the initial position...

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