AP Calculus BC Name: Module Three Lesson Five Activity 1) Let f(x) be a function with the following properties: i. f(0) = 1 ii. For all integers n 2 0, the nth derivative, f)(x) = (-1)" a"f(x), where a > 0 & a# 0 (a) Write the first four non-zero terms and the general term of the power series of f(x) centered at zero, in terms of a. (b) Write f(x) as a familiar function in terms of a. (c) How many terms of the power series are necessary to approximate f (0.2) with an error less than 0.001 with a = 2? Justify your answer. 2) The fifth-degree Maclaurin polynomial for tan(x) is M(x) = x + 5x3 + 3x5 . (a) Use M(x) to find the 4th degree Maclaurin polynomial for sect(x). (b) Use M(x) to find the 4th degree Maclaurin polynomial for tan(x) (c) Use the polynomial found in (b) to find lim tan(x) (d) Is the limit found in (c) exact or an approximation? Explain your reasoning