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final-02

# final-02 - f using the information from this question 5...

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Faculty of Arts and Science University of Toronto April/May 2002 Examinations MAT 137Y, Calculus! Time Alloted: 3 hours 1. Evaluate the following expressions. (6%) (i) lim t t 2 t 3 sin 1 t . (6%) (ii) 1 0 xe x dx . (6%) (iii) 3 x 4 x 2 3 x 2 dx . (6%) (iv) dx 1 x 1 3 . (6%) 2. Find the volume of the solid generated by revolving the region bounded by the curves y e x 2 x 0 x 2 y 0 about the y -axis. (10%) 3. A solid consists of a hemisphere of radius r that is glued onto the top face of a right circular cylinder of height h and base radius r . If the surface area (including the bottom face of the cylinder) of the solid is a constant, show that the volume is maximized when h r . (It may help to remember that the surface area of a sphere with radius r is 4 π r 2 . 4. Consider the function f x x 1 3 x 4 3 . (2%) (a) Find f x and f x . (4%) (b) Find all vertical and horizontal asymptotes and find all x and y -intercepts. (4%) (c) By locating all critical points of f , find the intervals for which f is increasing and decreasing, and classify all critical points. (4%) (d) Find the intervals of concavity of f and locate all inflection points. (4%) (e) Sketch the graph of

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Unformatted text preview: f , using the information from this question. 5. Consider the function f x sin x . (3%) (a) Give the Taylor polynomial P 2 n 1 x of sin x of degree 2 n 1 (in powers of x ). (5%) (b) Use an estimate on the error R 2 n 1 x sin x P 2 n 1 x to ±nd a value of n such that P 2 n 1 1 5 approximates sin 1 5 with error less than 10 4 . 6. Determine, with justi±cation, whether the following series converges or diverges. (5%) (a) ∞ ∑ k 1 ln k k . 1 (5%) (b) ∞ ∑ k 1 2 k k ! k k . 7. Consider the function f x e x . (3%) (a) Write down the Taylor series (or the power series expansion in powers of x ) for f x . Include the k-th (non-zero) term in your answer. (3%) (b) Express 1 e x 2 dx as a series; include the k-th non-zero term in your answer. (3%) (c) Find the radius of convergence of the power series found in part (a). Justify your answer. (10%) 8. Use the ε-δ de±nition of limit to prove that lim x 3 x 2 9. 2...
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