1. Use Taylor series to evaluate the following limit lim x cos x-+005. Suppose f(x) = (1)+1,k K2 4 16 k =1 (a) Find the first three non-zero terms of the power series representing the function ac f ( a?) . (b) Use your results in part (a) to approximate the indefinite integral:2 2 3. (a) Use the Maclaurin series et = 1 + x+ 21 + 3! + 4! + ... to find the Maclaurin series of e(23 ) (b) Use the first three non-zero terms of the series you found in part (a) to approximate the definite integral and evaluate. ! 4. Using the Taylor series centered at x = 0 for xet, show that n! = 2e. n=02. A Ferris wheel has a radius 20 In and completes a revolution in the clockwise direction at constant speed in 3 min. Assume that :r and y measure the horizontal and vertical positions of a seat on the Ferris wheel relative to the coordinate system Whose origin is at the low point of the Wheel. Assume that the seat begins moving at the origin. (a) Find the parametric equations that describe the path of a seat. (b) What is the position of the rider when t = 1 min? 2. (a) Use the Maclaurin series of et to find a Maclaurin series of ex2 (b) Differentiate the Maclaurin series from (a) of et to find a Maclaurin series for 2xex2. (c) Do you get the same answer for a Maclaurin series for 2xe as in (b) if you multiply the Maclaurin series of e" by 2x