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Let T4(:n): be the Taylor polynomial of degree 4 of the function f(2:) = 1n(1 + cc) at a = 0. Suppose you approximate x) by T4(m), find all positive values of x for which this approximation is within 0.001 of the right answer. (Hint: use the alternating series approximation.) Find T5(2:): Taylor polynomial of degree 5 of the function f($) = cos(:n) at a = 0. Tabs) = :1 Using the Taylor Remainder Theorem, find all values of x for which this approximation is within 0.004414 of the right answer. Assume for simplicity that we limit ourselves to |m| S 1. msS Assume that sin(a:) equals its Maclaurin series for all x. Use the Maclaurin series for sin (7:132) to evaluate the integral 0.73 / sin (7:132) da: 0 Your answer will be an infinite series. Use the first two terms to estimate its value. 32 d $2+4 :1: 2 (a) Evaluate the integral: / 0 Your answer should be in the form krr, where k is an integer. What is the value of k? H' t d arctan( ) 1 : a: = m do: $2 + 1 (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function f0?) = 2 + 4 . Then, integrate it from 0 to 2, and call the result 5. 5 should be an infinite series. a; What are the first few terms of S? a1: ? x/d' a2: % s/d" a3: $ (3' a4: % 0' (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of 7r in terms of an infinite series. Approximate the value of 71' by the first 5 terms. 3.33968 ~/ 6' (d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.) E