2033 - Math 202 Part I Final Exam Fall,2008 Do all parts of...

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Math 202 Final Exam Fall,2008 Part I Do all parts of the following six problems. (1) Compute the derivative dy dx for each of the following (15 points) : (a) y = ln( 1 x + 1 ) . Ans: y 0 = - 1 x + 1 . (b) y = arcsin( x ); Ans: y 0 = 1 2 p x (1 - x ) (c) y = x cos( x ) + cos( x ); Ans: y 0 = x cos( x ) [ cos( x ) x - sin( x )ln( x )] - sin( x ) . (2) Compute each of the following integrals(30 points): (a) Z 2 x 2 - 4 x 3 - 2 x 2 dx ; Ans: = Z 1 x + 2 x 2 + 1 x - 2 dx = ln | x | - 2 x + ln | x - 2 | + C. (b) Z x ln( x ) dx ; Ans: (let u = ln( x ) and dv = xdx ) = 1 2 x 2 ln( x ) - 1 4 x 2 + C. (c) Z p 4 - x 2 dx ; Ans: (let cos( t ) = x ) = - 4 Z sin 2 ( t ) dt = - 2 t + sin(2 t ) + C = - 2arccos( x 2 ) + x 4 - x 2 2 + C. 1
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(d) Z 1 0 arctan( x ) dx ; Ans: (let u = arctan( x ) and dv = dx ) = x arctan( x ) - 1 2 ln(1 + x 2 ) | 1 0 = π 4 - ln(2) 2 (e) Z (sin( x ) + 1) 2 dx. Ans: = 3 x 2 - 2cos( x ) + sin(2 x ) 4 + C. (3) Compute each of the following limits (10 points):
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2033 - Math 202 Part I Final Exam Fall,2008 Do all parts of...

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