Exam2007Post - [20 tionally or diverges Justify your...

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CARLETON UNIVERSITY FINAL EXAMINATION December 2007 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 0 e x - x - 1 x 2 . [4] 2. Evaluate the following integrals. [25] (a) ± 4 x 1 - x 2 dx (b) ± 4 x cos(2 x ) dx (c) ± cos 3 ( x ) sin( x ) dx (d) ± 1 0 x 1 - x 2 dx (e) ± 4 x 2 - 4 dx 3. Evaluate the improper integral ± 1 - 1 2 x 3 dx or show that it diverges. [4] 4. Find the equation of the tangent line to the parametric curve x = t 3 ,y = t 6 at ( x, y )=(1 , 1). [5] 5. Find the Cartesian coordinates of the point(s) on the polar curve r = 2cos θ, - π/ 2 θ π/ 2, [6] where the tangent is horizontal. 6. Find the sum of the series ² n =1 2 · 3 n +1 5 n . [4]
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2 7. For each of the following series, determine whether it converges absolutely, converges condi-
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Unformatted text preview: [20] tionally, or diverges. Justify your answer. (a) ∞ ± n =0 (-1) n ( n 2 + 1) √ n 4 + 2 (b) ∞ ± n =2 (-1) n n 3 ln( n ) (c) ∞ ± n =1 (-1) n √ n (d) ∞ ± n =0 (-1) n 2 n √ 2 n + 1 3 n √ n + 2 8. Find the radius and interval of convergence of the power series ∞ ± n =1 1 n (2 x-3) n . [7] 9. Find the MacLaurin series (Taylor series about 0) of f ( x ) = xe 2 x . [6] 10. Find the Taylor series of f ( x ) = 2 4-x about a = 2. [6] 11. Find the general solution of the following differential equations. [8] (a) y ± = 3 x 2 2 y (b) y ± + 2 xy = e-x 2 12. Solve the initial-value problem xy ±-xy = e x , y (1) = 2 e . [5]...
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Exam2007Post - [20 tionally or diverges Justify your...

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