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.             