Compute the 6th derivative of at a = 0. f(6) (0) = Hint: Use the MacLaurin series for f(x). -1 f(x) = tan- -() Taylor's Remainder Theorem: Consider the function f(z) = 3 cos (1x). (a) Compute f(8)(a), the 8th derivative of f(x). f)(x) = - (b) What is the maximum value of f(8)(z) on the interval [-0.5, 0.5]? (c) Recall that the remainder R(x) = f(x) - P(z) on the interval [-0.5, 0.5] satisfies R(x)= where c is some point between 0 and z. Use this fact and your computations above determine how closely P(x), the Maclaurin polynomial of degree 7, approximates f(z) on the interval [-0.5, 0.5]. |R7(a)| f(n+1)(c) (n+1)!