2005a_x2b_sols

# 2005a_x2b_sols - Spring 2005 Math 152 Exam 2B Solutions c...

This preview shows pages 1–2. Sign up to view the full content.

Spring 2005 Math 152 Exam 2B: Solutions Mon, 28/Mar c ± 2005, Art Belmonte For speciﬁcity, lengths are in centimeters unless stated otherwise. 1. (c) For non-STEPS folks: We have ¯ x = 3 X k = 1 m k x k 3 X k = 1 m k = ( 3 )( 1 ) + ( 5 - 2 ) + ( 7 3 ) 3 + 5 + 7 = 14 15 . (Note that this only gives you the x -coordinate of the center of mass, not the entire center of mass.) For STEPS folks: Let p = [3 , 5 , 7] be the row vector of masses and r = 12 - 25 31 be a matrix whose rows are position vectors of the points. Then m = sum ( p ) = 3 + 5 + 7 = 15 is the total mass. The center of mass is [ ¯ x , ¯ y ] = 1 m ( pr ) = 1 15 [3 - 10 + 21 , 6 + 25 + 7] = ± 14 15 , 38 15 ² . So the x -coordinate of the center of mass is ¯ x = 14 15 . ( Remark: pr represents matrix multiplication, realized by taking the dot products of rows with columns. This immediately extends to 3-D center of mass problems in Calc 3.) 2. (c) Let f ( x ) = ln x . We’ll determine K = max 2 x 5 ³ ³ f 00 ( x ) ³ ³ , then employ the Trapezoidal error estimate. Now f 0 ( x ) = 1 x = x - 1 and f 00 ( x ) =- x - 2 . Thus K = max 2 x 5 ´ 1 x 2 µ = 1 4 . Therefore, | E T | ≤ K ( b - a ) 3 12 n 2 = 1 4 ( 5 - 2 ) 3 12 ( 4 ) 2 = 27 3 ( 4 ) 4 = 9 256 . 3. (d) The improper integral converges to π/ 4. Z 1 1 x 2 + 1 dx = lim t →∞ Z t 1 1 x 2 + 1 = lim t →∞ tan - 1 x · ³ ³ ³ t 1 = lim t →∞ tan - 1 t - tan - 1 1 · = π 2 - π 4 = π 4 4. (a) The step size is h = b - a n = 2 - 1 4 = 1 4 . Partition points are n 1 , 5 4 , 3 2 , 7 4 , 2 o . The Trapezoidal Rule gives T n = step size × (

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/25/2008 for the course MATH 152 taught by Professor Teitler during the Spring '08 term at Texas A&M.

### Page1 / 3

2005a_x2b_sols - Spring 2005 Math 152 Exam 2B Solutions c...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online