finalsol_v1

Finalsol_v1 - Name PID TA Sec No Sec Time Math 20B Final Examination Turn off and put away your cell phone No calculators or any other devices are

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Name: ____________________________ PID: ________________ TA: ______________________ Sec. No: _____ Sec. Time: ______ Math 20B. Final Examination December 12, 2007 Turn off and put away your cell phone . No calculators or any other devices are allowed on this exam . You may use one page of notes, but no books or other assistance on this exam . Read each question carefully , and answer each question completely . Show all of your work; no credit will be given for unsupported answers . Write your solutions clearly and legibly ; no credit will be given for illegible solutions . If any question is not clear , ask for clarification . # Points Score 18 26 36 48 56 66 71 0 86 96 10 10 72 Σ
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1. Let 2 2 1 () (2 ) ( 1 ) xx fx +− = −+ . (a) (4 points) Find the partial fraction decomposition of f ( x ). 22 2 1( 1 ) ( ) ( 2 ) ) ( 1 ) 2 1 ) ( 1 ) A B x CA x B x C x x x + ++ + =+ = + So, we see that x 2 + x – 1 = A ( x 2 + 1) + ( Bx + C )( x – 2). Letting x = 2, we have: 5 = 5 A , i.e. A = 1. Letting x = 0, we have: -1 = A – 2 C , i.e. C = 1. Letting x = 1, we have: 1 = 2 A B C , i.e. B = 0 So, we end up with 2 111 ) ( 1 ) 2 1 x x + . (b) (4 points) Find f xdx . 2 ln 2 arctan( ) ) ( 1 ) 2 1 dx dx dx x x C x x = += + + + ∫∫
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2. (6 points) Find the area of the region that lines inside the circle r = 1 and outside the curve r = 1 – sin( q ). Notice that the two curves intersect at 0 and p . That is, we solve for when 1 = r = 1 – sin( q ) and the problem boils down to when sin( q ) = 0. (The picture helps to confirm our intuition.) The area will be given by () 22 2 00 2 0 0 0 11 1( 1 s i n ) 1 ( 12 s i n s i n 1 2sin sin 2 c o s 2 s i n 2 2cos 4 1 1 sin 2 1 sin 2(0) 2cos(0) (0) 4 2 4 20 2 0 dd d d ππ π θθ θ −− = + =− ⎛− ⎛⎞ ⎜⎟ ⎝⎠ ⎡⎤ + ⎢⎥ ⎣⎦ + + + + + ∫∫ 0 2 4
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3. (6 points) Using complex exponentials , compute 2 cos ( ) x ex d x . You need not simplify the result and may leave it in complex exponential form.
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This note was uploaded on 04/29/2010 for the course MATH MATH 20B taught by Professor Takeda during the Spring '07 term at UCSD.

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Finalsol_v1 - Name PID TA Sec No Sec Time Math 20B Final Examination Turn off and put away your cell phone No calculators or any other devices are

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