MATH
fall03final

# Thomas' Calculus: Early Transcendentals

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Mathematics 104 Fall Term 2003-2004 Final Examination January 16, 2004 1. Evaluate 5 dx x 3 + 2 x 2 + 5 x . 2. For each of the following integrals, state whether it converges or diverges, and give your reasons carefully and clearly. a. -∞ cos 2 t dt . b. 1 x 3 dx 1 + x 4 . 3. For each of the following series, state whether it converges or diverges, and give your reasons carefully and clearly. a. n =1 e - n ln n . b. n =1 ( - 1) n 1 1 + 1 n . 4. Find the Taylor series, centered at - 1, of f ( x ) = 1 x . 5. Estimate 1 / 2 0 e - x 3 dx with an error no bigger than 1 / 100. Give your reasons. 6. Find lim x 0 ( x - sin x ) 2 x 6 . 7. Find the area between the origin and the curve given in polar coordinates by r = θe θ for 0 θ π . 8. Find all roots of x 6 - 3 x 3 + 9 = 0 in polar form:
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Unformatted text preview: x = re iθ . 9. Consider the region under the curve y = e-x and above the x-axis for 0 ≤ x < ∞ . a. Revolve it around the x-axis and ﬁnd the volume. b. Revolve it around the y-axis and ﬁnd the volume. 10. Find the arc length of the curve given by y = x 2 for 0 ≤ x ≤ √ 2. (You may ﬁnd the formula Z sec 3 θ dθ = sec θ tan θ 2 + 1 2 Z sec θ dθ useful.) 11. The mass m of a crystal in a solution grows at a rate proportional to m 2 / 3 . The original mass is 1 gram and the mass after 24 hours is 8 grams. Find the exact value of the mass as a function of time....
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