final_wi06 - Name: TA: Math 20B. Final Examination March...

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Unformatted text preview: Name: TA: Math 20B. Final Examination March 22, 2006 Sec. No: PID: Sec. Time: Turn off and put away your cell phone. No calculators or any other devices are allowed on this exam. You may use one page of notes, but no books or other assistance on this exam. Read each question carefully, answer each question completely, and show all of your work. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification. 1. (6 points) Find a function y = y(t) that satisfies the following differential equation subject to the given initial condition. y = 4te-y y(1) = 0 # 1 2 3 4 5 6 7 8 Score 2. (12 points) The integral of sin2 (x) may be found using several different methods. (a) Find sin2 (x) dx using the double-angle identity for cos(2x). (b) Find sin2 (x) dx using integration by parts. 2. (c) Find sin2 (x) dx by first writing sin2 (x) in terms of the complex exponential. Leave the result in exponential form. 3. (6 points) Evaluate the integral 2x2 + 5x - 17 dx x2 + 2x - 15 4. (6 points) Find the volume common to two spheres, each with radius A, if the center of each sphere lies on the surface of the other. A 5. (6 points) The function (x - 1)2 ln(x) is represented as a Taylor series about x = 1 by (-1)n+1 (x - 1)n . n-2 n=3 Find the radius of convergence for this series. 6. (6 points) Find the area enclosed by the polar curve r = 4 + 3 sin(). 5 0 0 7. (6 points) The Taylor "series" representing a polynomial is a polynomial. For example, x2 is represented as a Taylor polynomial about x = 1 by 1 + 2(x - 1) + (x - 1)2 . Represent the polynomial x3 - 4x2 + 4x + 1 as a Taylor polynomial about x = 2. 8. (8 points) z is a complex number written in polar form as z = ei 3 . (a) Write z in rectangular form, a + bi. (b) Write z 11 in rectangular form, a + bi. (c) Find the 4th roots of z in polar form, r ei . ...
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