626 CHAPTER 8 INFINITE SEQUENCES AND SERIES CAS 9-10 Use a computer algebra system to find the Taylor poly- 26. Suppose you know that nomials 7, centered at a for n = 2, 3, 4. 5. Then graph these polynomials and f on the same screen. f" (4) = 3"(n + 1) 9. f(x) = cotx, a = 7/4 10. S (x) = Vitx. a=0 and the Taylor series of f centered at 4 converges to f(x) for all .x in the interval of convergence. Show that the fifth- degree Taylor polynomial approximates f(5) with error less than 0.0002. 11-18 (a) Approximate f by a Taylor polynomial with degree n at the 27. A car is moving with speed 20 m/s and acceleration 2 m/s' number a. at a given instant. Using a second-degree Taylor polynomial, (b) Use Taylor's Inequality to estimate the accuracy of the estimate how far the car moves in the next second. Would it approximation f(x) = T.(x) when .x lies in the given be reasonable to use this polynomial to estimate the distance interval. (c) Check your result in part (b) by graphing | R.(x) |. traveled during the next minute? 11. f(x) = Vx. a =4, n = 2, 45x $4.2 28. (a) Derive Equation 3 for Gaussian optics from Equation 1 by approximationg cos d in Equation 2 by its first-degree 12. f (x ) = x-? a = 1. n =2, 0.9