BC 3 Taylor Quiz Name: Calculator Allowed Clearly show all appropriate work for full credit. NO magically calculator "leaps of faith," please. 1 (1 pt each) The function y f ( x) is approximated near x = 0 by the third degree Taylor Polynomial 1 P3 ( x) 3 x x 2 x 3 . 3 Find. f (0) f (0) f (0) f (0) 2 (5 pts) Determine all values of x (exact) for which x 2 x3 x 4 x5 x6 L L L 1 . 3(3 pts each) Find the value of each series by recognizing the function and the point at which it is evaluated. [Exact answer please.] Indicate clearly the function and the value of x you use. 32 34 36 38 2! 4! 6! 8! 1 1 1 1 1 3 5 7 9 IMSA = = 4(6 pts) Suppose g x xe x2 . a. Find the Maclaurin series for g x . b. Evaluate g (100) 0 and g (101) 0 , the 100th and 101st derivatives of g at 5(8 pts) Let f x be a function such that f n 0 (1)n n2 x = 0. for all n ≥ 0. Thus, f 0 0, f 1, f 2 4 , and so on. a. Find the MacLaurin series for f x . Write out the first 4 non-zero terms along with the general term. b. Given that the MacLaurin series converges at x 1 , how many non-zero terms are needed to approximate f 1 with an error of at most 0.001? Clearly show analysis. IMSA 6. (2 pts) Evaluate the series n0 IMSA 4 (4n)! .