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## Taylor Series

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**Taylor Series**Section 9.2b**Do Now**Find the fourth order Taylor polynomial that approximates near Before finding a bunch of derivatives, remember that we can use a known power series to generate another… From the Table of Maclaurin Series on p.477: Therefore,**Do Now**Find the fourth order Taylor polynomial that approximates near The Taylor polynomial: Support by graphing both functions in [–3, 3] by [–2, 2]**Practice Problems: #16 on p.478**Let f be a function that has derivatives of all orders for all real numbers. Assume Write the third order Taylor polynomial for f at x = 0 and use it to approximate f(0.2).**Practice Problems: #16 on p.478**Let f be a function that has derivatives of all orders for all real numbers. Assume (b) Write the second order Taylor polynomial for , the derivative of f, at x = 0 and use it to approximate . Second order Taylor polynomial for :**Practice Problems: #18 on p.478**The Maclaurin series for f(x) is (a) Find and In this case, Similarly,**Practice Problems: #18 on p.478**The Maclaurin series for f(x) is (b) Let g(x) = x f(x). Write the Maclaurin series for g(x), showing the first three nonzero terms and the general term. Multiply each term of f(x) by x: (c) Write g(x) in terms of a familiar function without using series.**Practice Problems: #20 on p.478**Let and Find the first four terms and the general term for the Maclaurin series generated by f. From the Table of Maclaurin Series on p.477: So factor out 2 and substitute for ...**Practice Problems: #20 on p.478**Let and Find the first four terms and the general term for the Maclaurin series generated by f.**Practice Problems: #20 on p.478**Let and (b) Find the first four nonzero terms and the Maclaurin series for G. The first term: To find the other terms, integrate the terms of the series for f:**Practice Problems: #20 on p.478**Let and (b) Find the first four nonzero terms and the Maclaurin series for G.