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The Taylor series for f (ac ) = ac at -3 is Cn ( 2 + 3 ) " . n=0 Find the first few coefficients. CO = C1 = C2 = C3 C4The Maclaurin series of the function x) 2 5:3 tan1 (91:2) 00 can be written as f(:c) = 2 Cum\" n20 where a few of the coefficients are: 32 d m2+4 m 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? dw (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function 32 $2+4 . Then, integrate it from 0 to 2, and call the result S. 5 should be an infinite series. What are the first few terms of S? (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 7r by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.) l 1 The Taylor series for f(x) = In(sec(x) ) at a = 0 is E Cn (2 )n n=0 Find the first few coefficients. Co C1 = C2 = C3 - C4 Find the exact error in approximationg In (sec(0.4) ) by its fourth degree Taylor polynomial at a = 0. The error isFind the Taylor polynomial of degree 3, centered at a = 4 for the function f (a) = Va + 1.Let T4(:c): be the Taylor polynomial of degree 4 of the function f(m) = 1n(1 + 21:) at a = 0. Suppose you approximate f(a:) by T403), 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(:c): Taylor polynomial of degree 5 of the function at) 2 cos(a:) at a = 0. w 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 |93| S 1. msS Assume that sin(:L-) equals its Maclaurin series for all x. Use the Maclaurin series for sin(7m2) to evaluate the integral 0.73 / sin(7m2) d9: 0 Your answer will be an infinite series. Use the first two terms to estimate its value. Let F() = sin (7t2 ) dt. 0 Find the MacLaurin polynomial of degree 7 for F(x). 0.77 Use this polynomial to estimate the value of sin (7x2) dx. 0