Practice Final

# Practice Final - has a limit as n . (7) (a) What does the...

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Mathematics 115b Final Examination 5/10/08 Name: Section: Please answer all TWELVE questions; each question is worth TEN points. You may use the table of integrals supplied with the test. However, books, notes, calculators, computers, cellphones may NOT be used. 1 2 3 4 5 6 7 8 9 10 11 12 Total (1) (a) Find F 0 ( x ) where F ( x ) = Z cos x sin x e t 1 + t 2 dt . (b) For each of Z π/ 3 0 sec 2 x dx , Z 2 π/ 3 π/ 3 sec 2 x dx , Z π 2 π/ 3 sec 2 x dx , either evaluate the integral or explain why the Fundamental Theorem of Calculus does not apply.

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(2) Find the volume of the solid obtained by rotating the region between the x -axis and the curve y = 1 - x 2 about the line x = 2. (3) Either evaluate the integral Z 1 ln x (1 + x ) 2 dx , or explain why it is divergent.
(4) Find the arc length of the curve x = 1 3 ( y - 5) y - 2 , 3 y 11. (5) Find dy/dx and d 2 y/dx 2 for the curve with parametric equation x = t 2 + t, y = t 3 + t .

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(6) Determine whether the sequence ± (ln n ) 2 /n

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Unformatted text preview: has a limit as n . (7) (a) What does the Ratio Test say about the series X n =1 n n 2 + 1 ? (b) Does the above series converge or diverge? (8) Give an example of a series such that X n =1 a n converges but X n =1 | a n | diverges. Give reasons for convergence and divergence. (9) (a) Find a formula expressing (1-x )-2 as a power series centered at 0. For which values of x is your formula correct? (b) Evaluate the sum X n =1 n 3 n . (10) Evaluate the indenite integral Z x 1-x 4 dx as a power series, and state its radius of convergence. (11) Find the Maclaurin series of xe-x and its radius of convergence. (12) Find the power series of 1 4-x centered at 2 and its radius of convergence....
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## This note was uploaded on 01/11/2010 for the course MATH 115 at Yale.

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Practice Final - has a limit as n . (7) (a) What does the...

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