1b-2009-final_exam_sample_1

# 1b-2009-final_exam_sample_1 - y (0) = 1. 9. Find the length...

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Math 1B — Sample Final #1 Fall Program for Freshmen 2009 Fred Bourgoin 1. Compute the indeﬁnite integrals. (a) Z tan 3 x sec 3 x dx (b) Z dx x 3 + x 2. Evaluate each deﬁnite integral, or show that it diverges. (a) Z 1 0 1 2 - 3 x dx (b) Z e 1 dx x [1 + (ln x ) 2 ] 3. Determine whether each inﬁnite series converges absolutely, converges condition- ally, or diverges. (a) X n =1 ( - 1) n n 3 5 n (b) X n =2 1 n ln n 4. Solve both second-order diﬀerential equations. Use the method of undetermined coeﬃcients for one and the method of variation of parameters for the other. (a) y 00 - 3 y 0 + 2 y = e 5 x (b) 4 y 00 + 9 y = 2 x 2 - 3 5. Compute the indeﬁnite integrals. (a) Z dx x 2 - 4 x (b) Z e x cos x dx 6. Evaluate the deﬁnite integral Z 1 1 (2 x + 1) 3 dx , or show that it diverges. 7. Determine whether each inﬁnite series converges absolutely, converges condition- ally, or diverges. (a) X n =1 sin n 1 + n 2 1

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(b) X n =1 ( - 1) n - 1 n n + 1 8. Use power series to solve the initial-value problem y 00 + xy 0 + y = 0, y (0) = 0,
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Unformatted text preview: y (0) = 1. 9. Find the length of the curve y = x 2 2-ln x 4 on the interval 2 ≤ x ≤ 4. 10. Strontium-90 ( 90 Sr) has a half-life of 25 years. Suppose we start with a sample of 18 mg. (a) How much 90 Sr is left after t years? (b) When will there remain only 2 mg? 11. Use series to approximate Z 1 sin( x 2 ) dx to correct three decimal places. 12. Solve the initial-value problem xyy = ln x , y (1) = 2. 13. A 2-kg mass is attached at the end of a spring with damping constant 16. A force of 12.8 N is required to stretch the spring 0.2 m beyond its natural length. If the spring starts at equilibrium with velocity 2.4 m/s, what is the position function of the spring? 14. Find the Maclaurin series of f ( x ) = x 2 1 + x and its interval of convergence. 2...
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## This note was uploaded on 02/19/2012 for the course MATH 1 taught by Professor Wilkening during the Spring '08 term at University of California, Berkeley.

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1b-2009-final_exam_sample_1 - y (0) = 1. 9. Find the length...

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