final-03 - University of Toronto MAT 137Y1Y Calculus...

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University of Toronto MAT 137Y1Y Calculus! April/May Examinations; April 29, 2003 Time Alloted: 3 hours Instructors: B. Begun, J. Colliander, K. Consani, A. del Junco, J. Korman, R. Rotman, D. Slepcev, R. Sreekantan, S. Uppal 1. (a) Find the derivatives of the following functions. (4%) (i) f ( x ) = ( sin x ) 2 x . (4%) (ii) f ( x ) = Z x 2 0 ( 1 + t 2 ) 1 / 3 dt . (5%) (b) Find the equation for the tangent line to the curve y 3 - x 3 y + x - 1 = 0 at the point ( 1 , 1 ) . 2. Evaluate the following integrals. (7%) (i) Z e 2 x sin3 x dx . (7%) (ii) Z 4 0 u 1 / 2 1 + u 1 / 2 du . (7%) (iii) Z 2 x 2 - 2 x + 1 x 3 - 2 x 2 + x dx . (10%) 3. One end of a rope 20 metres long is attached to a box resting on the floor. The other end is passed over a pulley directly above the box, 5 metres above the floor, and attached to the back of a truck at a point one metre above the ground. The truck then drives in a straight line away from the pulley at a speed of 0.5 metres per second. At what speed is the box rising when the top of the box is 2 metres above the ground?
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