# 1232 - Math 202 Part I Final Exam Spring,2008 Do all parts...

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Math 202 Final Exam Spring,2008 Part I Do all parts of the following six problems. (1) Compute the derivative dy dx for each of the following (18 points) : (a) y = 2 arctan(3 x ) ; Ans: y 0 = 3 ln(2)2 arctan(3 x ) (1 + 9 x 2 ) . (b) y = x 2 + x x ; Ans: y 0 = 2 x + x x (ln( x ) + 1) . (c) y = ln( q x 4 + 3 x ) . Ans: y 0 = 4 x 3 + 3 2( x 4 + 3) (2) Compute each of the following integrals(24 points): (a) Z x 3 p 4 + x 2 dx ; Ans: with u = 4 + x 2 Z u - 4 2 u du = 1 3 (4 + x 2 ) 3 / 2 - 4(4 + x 2 ) 1 / 2 + C Alternatively, use x = 2 tan( t ) , q 4 + x 2 = 2 sec( t ) . (b) Z x 3 - 1 x 3 + x dx Ans: Z 1 + - 1 x + x - 1 x 2 + 1 dx = x - ln( x ) + 1 2 ln( x 2 + 1) - arctan( x ) + C. 1

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(c) Z cos 3 ( x ) dx ; Ans: let u = sin( x ) , Z (1 - u 2 ) du = sin( x ) - 1 3 sin 3 ( x ) + C. (d) Z 1 0 x ln( x + 1) dx. Ans: let z = x + 1 , u = ln( z ) , dv = ( z - 1) dz Z 2 1 ( z - 1) ln( z ) dz = [ 1 2 z 2 - z ] ln( z ) - [ 1 4 z 2 - z ] | 2 1 = 1 4 . (3) Compute each of the following limits (10 points): (a) Lim x →∞ x 2 + e x x 3 + e x ; Ans = 1 . (b) Lim x →∞ x 1 /x ; Ans = e 0 = 1 . (4) The region R in first quadrant of the xy plane is bounded by the curves y = 9 - x 2 , y = 0 and x = 0. Set up two integrals (method of washers and method of shells) for the volume of the solid obtained by rotating R
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