Taylor and Maclaurin Series

Preliminary Remarks and examples: If a function is times continuously differentiable at a point compute its Taylor polynomial of degree at as follows: where a remainder term which satisfies with we can the maximum value of is for all such that . The Taylor polynomial is an approximation of the function; the remainder term measures the error in this approximation. For example, the Taylor polynomial of degree at for the function is computed as follows: Putting these terms together we get The class software can do the entire computation for us, by using Compute+Power Series and choosing 5 for the number of terms and for the power. The term represents the remainder, and signifies that the remainder is a fifth order term. In the figure below both and its 4th degree Taylor polynomial about zero are plotted. (When you do this you should make sure that you do not include the term in the plot.) We see that close to 0 the two curves are identical, further away they no longer match. If we want a better measure of the actual error we plot We see that in the interval The Taylor series of at is the Maclaurin series of . Problems 1. a. this error is less than 0.05. . The special case of is called i.Use the class software to find the Taylor polynomials up to degree 6 for centered at . Graph polynomials on a common screen. ii.Evaluate and these polynomials at and these and . iii.Comment on how the Taylor polynomials converge to b. . i.Use the class software to find the Taylor polynomials up to degree 3 for centered at on a common screen. ii.Evaluate . Graph and these polynomials at and these polynomials and . iii.Comment on how the Taylor polynomials converge to . More Problems Use the class software to find the Taylor polynomial of degree 7 for the given functions around the indicated points. Plot the function together with the Taylor polynomial in a single graph. Then plot the actual error: your units carefully and estimate the size of the error. . Choose c. d. e. f. Would the Taylor polynomials of degree 5 be better or worse approximations to the functions? Explain. What about the degree 9 Taylor polynomials? Explain. g. Find the Maclaurin series for which is equal to . . Then use it to find a series