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Unformatted text preview: Name: TA: Math 20B. Final Exam March 16, 2009 Sec. No: PID: Sec. Time: Turn oﬀ and put away your cell phone. You may use one page of notes, but no books, calculators, or other assistance during this exam. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clariﬁcation. # 1 2 3 4 5 6 7 8 Σ Points 8 4 4 6 8 4 4 4 42 Score 1. Evaluate the indeﬁnite integral: (a) (4 points) ex sin x dx (b) (4 points) x2 dx − 5x − 6 √ 2. (4 points) Find the volume of the solid whose base is the region bounded by y = x, y = 0, and x = 4 and whose crosssections perpendicular to the x−axis are squares. 3. (4 points) Find 1 1 −√ + i√ 2 2 32 . Please write your answer in the form a + ib. 4. (a) (2 points) Write down (but do not evaluate) a deﬁnite integral that equals the area of one leaf of the threepetaled rose r = sin 3θ.
0.5 0.5 0.5 0.5 1.0 (b) (4 points) Evaluate the integral from part (a). ∞ 5. (a) (4 points) Show that the improper integral
1 dx diverges. Please be sure to x1/2 show all of your work. (b) (4 points) Determine whether 6 − sin2 (2x) dx converges or diverges. You x1/2 1 must justify your answer to receive credit.
∞ 6. (4 points) The MacLaurin series of f (x) = ln(1 + x) is
∞ n=1 (−1)n−1 xn . n Find all values of x for which this series converges. Please be sure to justify your answer. 7. (4 points) Find the Taylor series centered at 5 of the function f (x) = 1 . 1−x 8. (4 points) Solve the initial value problem: dy 2 = y 2 tet , dt y (0) = 3. ...
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This note was uploaded on 01/26/2010 for the course MATH 20B taught by Professor Justin during the Winter '08 term at UCSD.
 Winter '08
 Justin
 Math

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