Taylor Series Section 9 2 b Do Now

Taylor Series Section 9. 2 b

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 (a) Write 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 derivative of f, at x = 0 and use it to approximate Second order Taylor polynomial for : , the .

Practice Problems: #18 on p. 478 The Maclaurin series for f(x) is (a) Find In this case, Similarly, and

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 (a) 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 (a) 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.