01Final9C Sample

# 01Final9C Sample - (b(2 points Determine the interval of...

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University of California, Riverside Department of Mathematics Final Exam Mathematics 9C - First Year of Calculus Sample 1 Instructions: This exam has a total of 100 points. You have 3 hours. You must show all your work to receive full credit. You may use any result done in class. The points attached to each problem are indicated beside the problem. You are not allowed to use books, notes, or calculators. Answers should be written as p 2 as opposed to 1 : 4142135 ::: . 1. Compute (a) (5 points) lim n !1 3 2 n 2 5 n 2 + n + 1 (b) (5 points) lim n !1 ln n ln 3 n 2. Find the sum of the following series: (a) (5 points) 1 X n =0 ( 2) n e n (b) (5 points) 1 X n =1 1 2 n 1 2 n +1 ± 1

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3. (10 points) Determine whether the following series converges or di- verges. 1 X n =1 ( 1) n n ! n n 4. (10 points) Find the interval of convergence of the following series. 1 X n =0 ( 1) n ( x + 2) n n 2 5. Let f ( x ) = 1 X n =1 nx n (a) (2 points) Find the radius of convergence of the power series.
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Unformatted text preview: (b) (2 points) Determine the interval of convergence of the power se-ries. (c) (4 points) Obtain an explicit formula for the function f ( x ) . (d) (2 points) Find the sum of the number series f & 1 5 ± = 1 X n =1 n 5 n : 6. (10 points) Find the Taylor polynomial of degree 4 of f ( x ) = cos 2 x at a = & 4 . 7. (10 points) Find the length of the curve y = ln (cos x ) ; ± x ± & 3 : 8. (10 points) The curve y = 1 & x 2 ; ± x ± 1 is rotated about the y-axis. Find the area of the resulting surface. 2 9. (10 points) Solve the initial value problem: dy dx = 3 x 2 e y ; y (0) = 1 10. (10 points) Solve the initial value problem: x dy dx + ( x & 2) y = 3 x 3 e & x ; y (1) = 0 3...
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01Final9C Sample - (b(2 points Determine the interval of...

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