# UVA MATH 1320 - Final Exam (12 pages)

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## Final Exam

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- Pages:
- 12
- School:
- University Of Virginia
- Course:
- Math 1320 - Calculus II

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Math 132 Spring 2008 Final Exam Page 1 Name Instructor Instructions Write clearly You must show all work to receive credit Missed pg1 pg2 pg8 pg9 pg3 pg10 pg4 pg5 pg11 pg6 pg12 pg7 Total Score 300 1 10 points Set up the partial fraction decomposition of the following rational function Do not solve for the coefficients 5x2 6x 2008 x 1 5x 3 2 x2 4 x2 x 1 2 2 10 points each Evaluate the following definite integrals Z 2 a x2 sin x dx 0 Math 132 Spring 2008 Z b 0 3 x2 Final Exam 5x dx 1 2 3 3 10 points each Compute the following indefinite integrals Z 1 dx a x2 2x 2 Page 2 Math 132 Spring 2008 Z b Final Exam Page 3 x2 dx 1 x2 3 2 4 15 points Compute the arc length of the curve f x 14 x2 12 ln x on the interval 1 x 5 Math 132 Spring 2008 Final Exam Page 4 5 5 points each Complete the following definitions R a The improper integral a f x dx is convergent if b The improper integral R a f x dx is divergent if 6 10 points Set up but do not evaluate an integral to compute the surface area of the solid of revolution generated by revolving the curve f x cos x 0 x 2 about the y axis 7 5 points each Find polar coordinates r for the point with Cartesian coordinates x y 4 4 such that a r 0 b r 0 Math 132 Spring 2008 Final Exam Page 5 8 Consider the parametric curve defined by the equations x t cos3 t y t sin3 t 0 t 2 a 15 points Write the equation of the tangent line to the curve at the point where t 4 b 15 points Compute the length of the parametric curve Math 132 Spring 2008 Final Exam Page 6 9 15 points Find the area of the shaded region below inside the polar curve r 2 and outside the polar curve r 2 cos 2 Math 132 Spring 2008 Final Exam Page 7 10 15 points each Evaluate the following double integrals ZZ a xey dA where D is the region bounded by the curves y 4 x y 0 and x 0 D Z 1 Z b 0 0 ln 3 2 xyexy dx dy Math 132 Spring 2008 Final Exam Page 8 11 10 points each Compute the sums of the following infinite series a X e3 2n n 2 X 1 n 2n 1 b 62n 1 2n 1 n 0 12 10 points The letter k is an arbitrary real number that has been fixed ahead of time Show that X the infinite series nk 3 n converges no matter what value of k has been chosen n 1 Math 132 Spring 2008 Final Exam Page 9 13 10 points Determine whether the following infinite series are convergent or divergent State which test s you use to reach your conclusion Show all work a X n 2 b n3 n4 2n2 1 X arctan n n 1 n2 14 5 points each Complete the following definitions P a The infinite series n 1 an is convergent if b The infinite series P an is divergent if c The infinite series P an is absolutely convergent if d The infinite series P an is conditionally convergent if n 1 n 1 n 1 Math 132 Spring 2008 Final Exam Page 10 15 15 points Determine whether the following series is conditionally convergent absolutely convergent or divergent State which test s you use to reach your conclusion Show all work X 1 n 1 n ln n n 241 16 10 points Find the interval and radius of convergence of the following power series X n 12 en x 2 n Math 132 Spring 2008 Final Exam Page 11 17 10 points each Find Taylor series centered at a 0 for the following functions Simplify your answer State the radius of convergence a f x x 4 2x3 b f x 1 2x 2 Math 132 Spring 2008 Final Exam 18 15 points Find the degree three Taylor polynomial T3 x at a 4 for f x 19 Write out and sign the Honor Pledge Page 12 x

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