spring04final

# Thomas' Calculus: Early Transcendentals

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Mathematics 104 Spring Term 2004 Final Examination May 12, 2004 1. Evaluate ( θ 2 + 1) cos θ dθ . 2. Evaluate 4 xe x 2 e 2 x 2 + 2 e x 2 + 2 dx . 3. Evaluate x 2 - 1 x 2 dx . Hint : you may at some point want to use sin 2 θ = 1 - cos 2 θ . 4. Does 0 sin 2 x x 2 dx converge or diverge? Give your reasons. 5. For each of the following three series, state whether it converges or diverges and give your reasons. a) n =0 7 n - 2 n (2 n )! . b) n =1 n n 2 + n . c) n =1 ( - 1) n 2 n 2 n + n 2 . 6. For what values of x does each of the following two series converge? Give your reasons. a) n =1 ( x + 3) n n 3 . b) n =1 (2 x - 1) n n . 7. Find the second order Taylor polynomial of tan - 1 x about the center a = 1 2 . 8. Find 3 1 . 01 with an error at most 0.0001. Hint : 3 1 . 01 = (1 + 0 . 01) 1 / 3 . 9. a) Draw the graph of the first two turns of the spiral given in polar coordinates by

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• Fall '07
• Nelson
• Math, Taylor Series, Boundary value problem, Polar coordinate system, reasons. x2

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