test3-03 - x y 2 2 y and the y-axis is rotated about the...

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Department of Mathematics, University of Toronto Term Test 3 – March 12, 2003 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: B. Begun, K. Consani, J. Colliander, A. del Junco, J. Korman, R. Rotman, D. Slepcev, R. Sreekantan, S. Uppal 1. Evaluate the following expressions. (6%) (i) d dx ln x 1 . (6%) (ii) d dx x 1 x . (6%) (iii) f e , where f x tan 1 ln x 1 e t 2 dt . Simplify your answer. 2. Evaluate the following integrals. (8%) (i) e x x dx . (8%) (ii) e 2 x cos x dx . (8%) (iii) π 0 cos 4 θ d θ . (8%) (iv) 9 0 1 u 1 1 2 du . (10%) 3. Find the enclosed area between the curves y e x 2 x and y e x x 3 x . (12%) 4. The enclosed area bounded by the curves
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Unformatted text preview: x y 2 2 y and the y-axis is rotated about the line y 1. Find the volume of the resulting solid. 5. Consider the function f x 1 e x 1 . (5%) (a) State the domain. Find all x-intercepts and all horizontal and vertical asymptotes, if any. (5%) (b) On what intervals is f x increasing? (5%) (c) On what intervals is f x concave up? (5%) (d) Sketch the graph of f x , using the information in the previous parts. (8%) 6. Prove that log 2 3 is irrational. 1...
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