MATH 152 Fall 2019 Section 06---Prof. Dean MIDTERM EXAM 3 Tues. December 3, 2019 NAME (please print legibly): ______________________________________________ Student ID Number: _____________________________________________________ Discussion Section (please circle): 07: Gerson (9:00) 08: Abhishek (9:00) 09: Karan (10:00) 10: Abhishek (10:00) No calculators or notes are allowed on this exam. Please show all your work. You may use the backs of pages if necessary. You may not receive full credit for a correct answer if there is no work shown. QUESTION VALUE SCORE 1 20 2 20 3 20 4 10 5 20 6 20 TOTAL 1001) (20 points) Find the radius of convergence and interval of convergence of the following power series. (a) (b) 2) (20 points) (a) Find a power series representation for the function What is the radius of convergence of this series? (b) Use the series you found in part (a) to find a power series representation for the function What is the radius of convergence of this series?3) (20 points) (a) Use the definition of Taylor series to find the Taylor series of centered at . (You do not need to find the radius of convergence.) (b) Evaluate the following integral as an infinite series. (You do not need to find the radius of convergence.) 4) (10 points) Let (a) Approximate by a Taylor polynomial with degree 2 centered at . (b) when lies in the interval . (You do not need to simplify your answer.)5) (20 points) (a) Find the center, foci, and vertices of the ellipse and sketch the graph. (b) Find an equation of the parabola with focus and directrix . Sketch its graph.6) (20 points) (a) Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. (Simplify your answer.) (b) Find the arc length of the curve.
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