fx166s2010 - Hint ∞ X n =0 3 n x n is a geometric series...

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Math 166 Final Exam May 4, 2010 Full Name: Instructor & Section: Instructions: Problems 1-3 have several parts totalling 50 points. Problems 4-8, which are more involved, are worth 10 points each. Calculators cannot be used during the exam. You must justify your answers to get full credit. 1. (5 points each) Evaluate each of the limits or state why it does not exist. (a) lim x →∞ ln x x . (b) lim x 0 sec x - 1 x 2 . (c) lim x 0 + ( 1 x - 1 sin x ) .
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Math 166 Final Exam Page 2 2. (5 points each) Evaluate each improper integral or show that it diverges. (a) 0 3 x ( x 2 + 9) 2 dx . (b) 2 0 x 4 - x 2 dx . (c) 0 xe - 5 x dx .
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Math 166 Final Exam Page 3 3. (5 points each) Use the integral, limit comparison, alternating series, or n -th term test to determine whether each series converges or diverges. (a) n =1 n n + 100 . (b) n =1 1 n ln n . (c) n =1 5 n 2 11 n 3 + 4 n 2 + 6 n + 3 . (d) n =1 ( - 1) n +1 n + 1 .
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Math 166 Final Exam Page 4 4. Find the convergence set of the power series n =1 x n n . 5. Find a power series for the function f ( x ) = 1 (1 - 3
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Unformatted text preview: . Hint: ∞ X n =0 3 n x n is a geometric series with ratio r = 3 x . What does it converge to? Math 166 Final Exam Page 5 6. Taylor’s Formula with remainder applied to f ( x ) = ln(2 + 3 x ), at the centering point a = 0, shows that ln(2 + 3 x ) = P 2 ( x ) + R 2 ( x ), where P 2 ( x ) is the Maclaurin polynomial of degree 2 and R 2 ( x ) is the remainder term. Determine P 2 ( x ) and R 2 ( x ) in this case. Math 166 Final Exam Page 6 7. Find the tangent line to the parametric curve x = t 3 + t , y = t 2-1, 0 < t < 2, at the point (2 , 0). Hint: The curve passes through the point (2 , 0) when t = 1. 8. Find the area of one leaf of the 3-leaved rose r = 5 sin 3 θ . (Recall the trigonometric identities: cos 2 x = 1 2 (1 + cos 2 x ), sin 2 = 1 2 (1-cos 2 x ), and sin 2 x = 2 sin x cos x .)...
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