FINAL - dy dx = 3 x 2 2 y 5. Find an equation of the...

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MAC 2312-01,02,03 Calculus II FINAL EXAM December 11, 2008 Dr. Jungmin Choi Name Instructions. 1. This test has 10 pages including this page. There are 12 problems. 2. Show all your work. You may not receive any credit from a correct answer, if there is no relevant work leading to the answer. 3. Do not separate the pages of the test. If any pages do become separated, write your name on them and point them out to your instructor when you turn in your test. 4. You are not allowed to use a calculator. 5. Please turn oF all cell phones and pagers and remove all headphones. Problem Points Score 1 40 2 10 3 15 4 10 5 10 6 15 7 10 8 40 9 15 10 10 11 10 12 15 Total 200 You may need the following formulas. sin 2 θ = 1 2 (1 - cos 2 θ ) cos 2 θ = 1 2 (1 + cos 2 θ ) 1
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1. Evaluate the following integrals. (10 points each) (a) ± x sin xdx (b) ² 1 0 1 4 - x 2 dx 2
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(c) ± 1 x 2 - 4 dx (d) ² x 2 e x 3 dx 3
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2. Determine whether the following integral is convergent or divergent. Evaluate if it is convergent. (10 points) ± 0 -∞ xe x dx 3. Find the area of the surface obtained by rotating the curve y = x 2 from (0 , 0) to (1 , 1) about the y - axis. (15 points) 4
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4. Solve the differential equation. (10 points)
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Unformatted text preview: dy dx = 3 x 2 2 y 5. Find an equation of the tangent line to the curve x = t 2 + 3 t , y = 2 t-1 at t = 0 . (10 points) 5 6. Find the shaded region enclosed by r = 1 + cos θ and the horizontal axis. (15 points) 6 7. Find the exact length of the polar curve. (10 points) r = e θ , ≤ θ ≤ π 8. Test for convergence or divergence. (10 points each) (a) ∞ ± n =0 1 (-10) n 7 (b) ∞ ± n =1 n-1 2 n (c) ∞ ± n =1 n n 4 n (d) ∞ ± n =1 1 + sin n 5 n 8 9. Find the radius and interval of convergence. (15 points) ∞ ± n =1 (3 x-2) n n · 3 n 10. Find a power series representation for the given function and determine the interval of convergence. (10 points) f ( x ) = x x 2 + 1 9 11. Find the Taylor series for f ( x ) = x-x 2 centered at x = 1 . (10 points) 12. Find the Maclaurin series for f ( x ) = e x . Use your answer to evaluate ∞ ± n =0 1 n ! . (15 points) 10...
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This note was uploaded on 02/25/2010 for the course MAC 2312 taught by Professor Zhang during the Spring '07 term at FSU.

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FINAL - dy dx = 3 x 2 2 y 5. Find an equation of the...

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