02Final9C Sample

02Final9C Sample - P + 1 n =1 ( & 1) n x n n ....

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University of California, Riverside Department of Mathematics Final Exam Mathematics 9C - First Year of Calculus Sample 2 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 ::: . of each convergent sequence. (a) (5 points) a n = ln( n ) ln( n +1) (b) (5 points) a n = ( n n +1 ) n . diverges, explain why. (a) (5 points) 4 2 + 1 1 2 + 1 4 1 8 + ±±± (b) (5 points) P + 1 n =1 1 (2 n 1)(2 n +1) . 3. Determine if the following series converges or diverges. Please give your reason(s). (a) (5 points) P + 1 n =1 n ! (2 n )! . (b) (5 points) P + 1 n =1 ( 1) n 1 n +1 . 4. 1
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(a) (5 points) Find the radius of convergence for the power series
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Unformatted text preview: P + 1 n =1 ( & 1) n x n n . (b) (5 points) Find the interval of convergence for the above series. 5. (10 points) Find the Taylor Polynomials of orders , 1 , 2 , 3 generated by f ( x ) = cos( x ) at x = & 4 . 6. (a) (8 points) Express the inde&nite integral R sin( x 2 ) dx as a power series. (b) (2 points) Express the de&nite integral R 1 sin( x 2 ) dx as a number series. 7. (a) (5 points) Consider the function f ( x ) = (1 & 1 2 x ) & 2 . Find the &rst 3 terms of its Binomial Series. (b) (5 points) Find its radius of convergence. 8. (10 points) Solve the following initial value problem: dy dx = x 2 e & y y (0) = 0 : 9. (10 points) Find the general solution of (1 + x 2 ) y + 2 xy = e x 10. (10 points). Find the length of the curve y 2 = ( x + 1) 3 from (0 ; 1) to (3 ; 8) . 2...
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02Final9C Sample - P + 1 n =1 ( & 1) n x n n ....

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