Math 132 - Spring 2008 Final Exam Page 1Name: Instructor:Instructions: Write clearly. You must show all work to receive credit.Missed: pg1 pg2 pg3 pg4 pg5 pg6 pg7. pg8 pg9 pg10 pg11 pg12 Total Score /3001. (10 points) Set up the partial fraction decomposition of the following rational function. Do not solvefor the coefficients.5x2− 6x + 2008(x − 1)(5x + 3)2(x2+ 4)(x2+ x + 1)22. (10 points each) Evaluate the following definite integrals.(a)Zπ/20x2sin(x) dxMath 132 - Spring 2008 Final Exam Page 2(b)Z305x(x2− 1)2/3dx3. (10 points each) Compute the following indefinite integrals.(a)Z1x2+ 2x + 2dxMath 132 - Spring 2008 Final Exam Page 3(b)Zx2(1 − x2)3/2dx4. (15 points) Compute the arc length of the curve f(x) =14x2−12ln(x) on the interval 1 ≤ x ≤ 5.Math 132 - Spring 2008 Final Exam Page 45. (5 points each) Complete the following definitions:(a) The improper integralR∞af(x) dx is convergent if(b) The improper integralR∞af(x) dx is divergent if6. (10 points) Set up, but do not evaluate, an integral to compute the surface area of the solid ofrevolution generated by revolving the curve f(x) = cos(x), 0 ≤ x ≤ π/2, about the y-axis.7. (5 points each) Find polar coordinates (r, θ) 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 58. Consider the parametric curve defined by the equations x(t) = cos3(t), y(t) = sin3(t), 0 ≤ t ≤ π/2.(a) (15 points) Write the equation of the tangent line to the curve at the point where t = π/4.(b) (15 points) Compute the length of the parametric curve.Math 132 - Spring 2008 Final Exam Page 69. (15 points) Find the area of the shaded region below, inside the polar curve r = 2 and outside thepolar curve r = 2 cos(2θ).Math 132 - Spring 2008 Final Exam Page 710. (15 points each) Evaluate the following double integrals.(a)ZZDxeydA where D is the region bounded by the curves y = 4 − x, y = 0, and x = 0.(b)Z10Zln(3)0xyexy2dx dyMath 132 - Spring 2008 Final Exam Page 811. (10 points each) Compute the sums of the following infinite series.(a)∞Xn=2e3−2n(b)∞Xn=0(−1)nπ2n+162n+1(2n + 1)!12. (10 points) The letter k is an arbitrary real number that has been fixed ahead of time. Show thatthe infinite series∞Xn=1nk3−nconverges no matter what value of k has been chosen.Math 132 - Spring 2008 Final Exam Page 913. (10 points) Determine whether the following infinite series are convergent, or divergent. State whichtest(s) you use to reach your conclusion. Show all work.(a)∞Xn=2n3√n4− 2n2+ 1(b)∞Xn=1arctan(n)n214. (5 points each) Complete the following definitions.(a) The infinite seriesP∞n=1anis convergent if(b) The infinite seriesP∞n=1anis divergent if(c) The infinite seriesP∞n=1anis absolutely convergent if(d) The infinite seriesP∞n=1anis conditionally convergent ifMath 132 - Spring 2008 Final Exam Page 1015. (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.∞Xn=241(−1)n+1n ln(n)16. (10 points) Find the interval and radius of convergence of the following power series:∞Xn=12en(x − 2)nMath 132 - Spring 2008 Final Exam Page 1117. (10 points each) Find Taylor series centered at a = 0 for the following functions. Simplify youranswer. State the radius of convergence.(a) f(x) =x4 − 2x3(b) f(x) = (1 + 2x)−2Math 132 - Spring 2008 Final Exam Page 1218. (15 points) Find the degree three Taylor polynomial T3(x) at a = 4 for f(x) =√x.19. Write out and sign the Honor