MATH141: Calculus II Exam #4 7/21/2017 Page 1 Write legibly and show all work. No partial credit can be given for an unjustified, incorrect answer. Put your name in the top right corner and sign the honor pledge at the end of the exam. If you need more room than what s given, please continue onto the back. 1. (a) (Four points) Let R be the region in the plane between the x-axis and the graph of y = e x, bounded on the left by x = 0 and on the right by x = 2. Write down an integral that equals the volume of the solid obtained by rotating R around the y-axis. [Don t do anything more than write it down, have you seen part (b) yet?] (b) (Eight points) Evaluate the integral from part (a). (If you are not able to answer part (a), or if you re unsure, you may use the integral π x sin(x)dx as though it 0 were the correct answer, even though it s not.)
MATH141: Calculus II Exam #4 7/21/2017 Page 2 2. (Three points each) Decide whether the following items are true or false. No work is required for these problems: I ll give you full credit if you circle the correct choice. If you want to explain yourself, that may be worth partial credit if your answer is incorrect. T F (a) Suppose {a n } n=0 is a sequence and lim n a n = 0. The series a n must converge. n=0 T F (b) d [ ] x x = x x ln x. dx T F (c) Suppose f(x) is a function whose Taylor series centered at 0 begins This function f is decreasing at x = 0. 2 3x + 4x 2 5x 3 + 10x 8 +. T F (d) The function f(x) = x 3 + 3x has an inverse on the domain (, ). T F (e) The value of the series ( ) n 1 2 is 1 5 4. n=1 T F (f) Let R be the region above y = x 2 and below y = 4. (That region is pictured below.) The y-coordinate of the center of gravity of R is larger than 2. 4 y 2 3 2 1 1 2 3 x
MATH141: Calculus II Exam #4 7/21/2017 Page 3 3. (Four points each) For the five integrals below, determine what method you would use to perform the antidifferentiation. Your choices are: integration by parts, u- substitution, trig sub, and partial fractions. The correct method is worth two points. For the other two points, provide the following information, depending on the method you ve selected. If you d integrate by parts, pick u and dv, and then calculate du and v. If you d do a u-substitution, say which one, and then convert the integral into one that only involves u. If you d do a trig sub, show me specifically what substitution you d use. If you d use partial fractions, write down the template you d use to break down the integrand. You don t have to solve for the constants. Do not integrate any of these. Three of them appear on the back of this sheet. (a) x 2 (x 2 4) 3 2 dx. (b) x arctan(x) dx.
MATH141: Calculus II Exam #4 7/21/2017 Page 4 2x 3 + x + 7 (c) dx. x 4 + 4x 2 (d) tan 2 (x) sec 4 (x) dx. (e) dx x 2 + 2x + 3.
MATH141: Calculus II Exam #4 7/21/2017 Page 5 4. (Eleven points) Does the series n=2 ( 1) n n ln(n) n converge conditionally, converge absolutely, or diverge? two-word answer is not sufficient. Give some explanation a
MATH141: Calculus II Exam #4 7/21/2017 Page 6 5. (Seven points) Let f(x) = cos(2x 2 ). Find the Taylor series for f(x), centered at x = 0. Your answer should be in summation notation. 6. (Seven points) Let g(x) = e x. Use the Taylor series for g(x), centered at x = 0, to find an approximation of 1 which is accurate to within 0.01. (That s 1.) (Your answer e 100 should be an appropriate partial sum of an appropriate Taylor series.) The Lagrange remainder formula says error max f (n+1) (x) a x b (b a) n+1 (n + 1)! where a is the center of the series and b is the number being approximated.
MATH141: Calculus II Exam #4 7/21/2017 Page 7 7. (a) (Nine points) On the axes below, draw the graphs of the circle r = 2 cos θ and the cardioid r = 2 + 2 cos θ. π/2 2π/3 π/3 5π/6 π/6 π 1 2 3 0 7π/6 11π/6 4π/3 5π/3 3π/2 (b) (Six points) Set up, but do not evaluate, an integral or integrals that would determine the amount of area that s inside the circle and outside the cardioid.
MATH141: Calculus II Exam #4 7/21/2017 Page 8 8. (Ten points) Calculate the three cube roots of the complex number 27i. Report them in the form re iθ. Draw a rough sketch of where they lie in the complex plane. (Your picture doesn t have to be a work of art.)
MATH141: Calculus II Exam #4 7/21/2017 Page 9 9. (a) (Thirteen points) Find the interval of convergence of the power series n=2 4 n 1 n 5 n xn. (b) (Five points) Let f be the function defined by the power series from the previous part. Find the value of f ( 1 4 ). Please copy the honor pledge below and sign your name next to it. I pledge on my honor that I have not given or received any unauthorized assistance on this examination.