1'1 dz In this question, we will investigate the number of intervals necessary to achieve a given level of accuracy using Simpson's method. To make the oroiect concrete, we focus on the integral I = . 1 I 1 (a) Let at) : E' Find the fourth derivative of f[1). Mm) : I (b) Which of the following best describes the behaviour of |f(4)(1)| on the interval (1,1.1)? Increasing u Decreasing Sometimes increasing, sometimes decreasing (c) What is the maximum value of |f(4)($)| on the interval (1,1.1)? m : 12355.1": (EM I (d) Let Ldenote the value found above. Taking a: = land b : 1.1, and assuming n > 0, isolate nto express the inequality 07405 If(+) (x) | on (a, b). Determine the number of intervals n for which S, approximates the exact value p1/5 I = 1/10 (23 In a) da 1 with an absolute error of at most 3 x 109 Remark: You can (and should) do this without a calculator. Give the smallest even integer n that works, according to the error formula. n :Recall that when S, denotes the n-interval approximation for I = / f(x) dx from Simpson's Rule, one has 1I - Sals- L (b - a)5 180 n4 whenever L > If(+) (x) | on (a, b). Determine the number of intervals n for which S, approximates the exact value 1= 2e"' da with |I - Sn|