Winter 2007 - Takeda's Class - Practice Exam 2

# Winter 2007 - Takeda's Class - Practice Exam 2 - Math 20C...

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Math 20C, Practice Midterm 2 Solutions February 24, 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 50 1

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1. (a) Let f ( x, y ) = x e y . Find f x . f x = e y x e y - 1 (b) Consider x - z = arctan( yz ). Find ∂z/∂y by using implicit diﬀerentia- tion. By diﬀerentiating both sides with respect to y , we have 0 - ∂z ∂y = 1 ( yz ) 2 + 1 · ( z + y ∂z ∂y ) . By solving this for ∂z/∂y , we get ∂z ∂y = - z ( yz ) 2 + y + 1 . 2
2. (a) Let r ( t ) = h t, 3 t 2 , e t - 1 i . Find the unit tangent vector at t = 1. r 0 ( t ) = h 1 , 6 t, e t - 1 i . So r 0 (1) = h 1 , 6 , 1 i . Thus the unit tangent vector is r 0 (1) | r 0 (1) | = h 1 38 , 6 38 , 1 38 i (b) Find the position vector of a particle with the following velocity vector

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## This note was uploaded on 04/29/2008 for the course MATH 20C taught by Professor Helton during the Winter '08 term at UCSD.

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Winter 2007 - Takeda's Class - Practice Exam 2 - Math 20C...

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