Exam2007Solutions

Exam2007Solutions - CARLETON UNIVERSITY FINAL EXAMINATION...

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Unformatted text preview: CARLETON UNIVERSITY FINAL EXAMINATION December 2007 SOLUTIONS DURATION: 3 HOURS Department Name and Course Number: Mathematics and Statistics, MATH 2007 Course Instructor(s): Dr. L. Haque, Dr. S. Melkonian AUTHORIZED MEMORANDA NON-PROGRAMMABLE CALCULATORS ONLY This examination may be released to the Library. This examination paper may not be taken from the examination room. [Marks] 1. Evaluate lim x → e x- x- 1 x 2 . [4] Solution: By L’Hospital’s rule, lim x → e x- x- 1 x 2 = lim x → e x- 1 2 x = lim x → e x 2 = 1 2 . 2. Evaluate the following integrals. [25] Solution: (a) 4 x √ 1- x 2 dx =- 2 √ 1- x 2 (- 2 x ) dx =- 2 u 1 / 2 du =- 4 3 u 3 / 2 + C =- 4 3 (1- x 2 ) 3 / 2 + C ( u = 1- x 2 , du dx =- 2 x ). (b) 4 x cos(2 x ) dx = 2 x sin(2 x )- 2 sin(2 x ) dx = 2 x sin(2 x ) + cos(2 x ) + C . (c) cos 3 ( x ) sin( x ) dx =- cos 3 ( x )[- sin( x )] dx =- u 3 du =- 1 4 u 4 + C =- 1 4 cos 4 ( x ) + C ( u = cos( x ) , du dx =- sin( x ))....
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This note was uploaded on 01/18/2012 for the course MATH 2007 taught by Professor S.melkonian during the Fall '11 term at Carleton CA.

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Exam2007Solutions - CARLETON UNIVERSITY FINAL EXAMINATION...

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