1b-2009-final_exam_sample_2_solutions

1b-2009-final_exam_sample_2_solutions - Math 1B Sample...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 1B Sample Final #2 Solutions Fall Program for Freshmen 2009 Fred Bourgoin 1. Evaluate each integral. (a) Z cos x 4- sin 2 x dx Solution. Let u = sin x . Then du = cos x dx and Z cos x 4- sin 2 x dx = Z 1 4- u 2 du = 1 4 Z 1 2- u- 1 2 + u du = 1 4 ln 2- u 2 + u + C = 1 4 ln 2- sin x 2 + sin x + C (b) Z / 2 cos 3 sin d Solution. Let u = cos . Then du =- sin d and Z / 2 cos 3 sin d =- Z 1 u 3 du = 1 4 u 4 1 = 1 4 . 2. Determine whether the series X n =2 1 n ln n converges or diverges. Solution. Let f ( x ) = 1 x ln x . Then f is positive, continuous, and decreasing on [2 , ), so we can use the integral test. Z 2 1 x ln x dx = lim t h 2 ln x i t 2 = lim t (2 ln t- 2 ln 2) = . The integral diverges, so the series diverges as well. 3. Evaluate Z 4 ln x x dx or show that it is divergent. Solution. For the integration, proceed by parts with u = ln x and dv = 1 x dx . Z ln x x dx = 2 x ln x- Z 2 x dx = 2 x ln x- 4 x + C = 2 x (ln x- 2) + C. Hence, Z 4 ln x x dx = lim t + Z 4 t ln x x dx = lim t + h 2 x (ln x- 2) i 4 t = lim t + 8(ln 2- 1)- 2 t (ln t- 2) = . (LH opitals rule was used there.) Therefore, the integral diverges. 1 4. Consider the integral Z 4 f ( x ) dx , where f ( x ) = x 2 + 1. [Note that f 00 ( x ) = ( x 2 + 1)- 3 / 2 .] (a) If we use n = 4, which is more likely to give a better approximation: the midpoint rule or the trapezoidal rule? Why?...
View Full Document

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.

Page1 / 6

1b-2009-final_exam_sample_2_solutions - Math 1B Sample...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online