11.1: Taylor Series Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. Brook Taylor 1685 - 1731 Greg Kelly, Hanford High School, Richland, Washington
The Maclaurin and Taylor series can be used to approximate the value of a function. (Just as a linear approximation can, but even more accurately.)
The Maclaurin Series is used to approximate values of f(x) close to zero. The more terms you use, the more accurate your approximation.
The Maclaurin Series Maclaurin Series: (generated by f at )
Use the 5th degree Taylor polynomial of to estimate the value of . Now compare the graphs. See page 468.
If the value you are approximating it not close to zero, use a Taylor Polynomial. The derivatives are evaluated at x=c, the center. The Maclaurin polynomial is the same as Taylor polynomial centered at zero. Neither one is more accurate than the other. One is just used for approximations close to zero, the other is used for all other approximations.
Maclaurin Series: (generated by f at ) Taylor Series: (generated by f at ) If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:
Homework Page 472 #1-21 odd