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Unformatted text preview: Math 115 -- Final Practice This exam is meant to suggest the kind, difficulty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 3 1/2 hours during which you can work on this exam without interruption. Refer to no books, notes, or calculators while working on the exam. This will give you a sense of your level of preparation for the real exam. 1. Evaluate these integrals (a) (b) (c) cos(x)esin(x) dx x2 sin(x)dx. dx x(ln x)2 1 + (ln x)2 dx 2. Determine if these integrals converge or diverge. cos2 (x) (a) dx. 2 - 1 + x
0 dx . sin(x)
2 3. Let R be the region bounded by y = ex , y = 0, x = 0, and x = 1. Find the volume of the solid obtained when R is rotated about the y-axis. 4. (a) Sketch the curves r = 1 and r = 1 + cos(). (b) Find the area inside r = 1 and outside r = 1 + cos(). 5. Determine if these series converge or diverge. Support your answer. (a)
n=1 n2 e-n 3 5 sin2 n . (b) n! n=1 (c) 1 + n ln n . n2 + 5 n=2 6. Is this series absolutely convergent, conditionally convergent, or divergent? (-1)n
n=1 n2 n +n+4 7. Find the radius and interval of convergence of the power series (-2)n (x - 3)n 2n + 3 n=1 8. Evaluate 3 dx as a power series. For what values of x is this power series 1 - x9 representation valid? 9. Find the sum of the series
1 ( 3 )n+1 . n+1 n=0 x2 ln x - , 10. Find the area of the surface of revolution formed by rotating the curve y = 4 2 1 x 2, about the x-axis. 11. Determine if these sequences converge or diverge. Find the limit of the converging sequences. n2 + 2n + 3 (a) an = 3n2 - 2n + 1 (b) an = n(e1/n - 1) ...
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