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This educational resource, created by Vickie Kelly in 2003 at Hanford High School, explores the Maclaurin series, highlighting their significance in calculus and their common occurrences in mathematics. The piece delves into the derivation of these series from functions and their derivatives, explains how to evaluate them at specific points, and discusses their relationship to geometric series. By demonstrating how to use these series to evaluate transcendental functions like sine and cosine, this document also emphasizes their practical applications.
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9.2 day 2 Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Finding Common Maclaurin Series Liberty Bell, Philadelphia, PA
Maclaurin Series: (generated by f at ) There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet, but today we are going to look at where they come from.
List the function and its derivatives.
List the function and its derivatives. Evaluate column one for x = 0. This is a geometric series with a = 1 and r = x.
We could generate this same series for with polynomial long division:
This is a geometric series with a = 1 and r = -x.
We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly. They do help to explain where the formula for the sum of an infinite geometric comes from. We will find other uses for these series, as well. A more impressive use of Taylor series is to evaluate transcendental functions.
Both sides are even functions. Cos (0) = 1 for both sides.
Both sides are odd functions. Sin (0) = 0 for both sides.
and substitute for , we get: If we start with this function: This is a geometric series with a = 1 and r = -x2. If we integrate both sides: This looks the same as the series for sin (x), but without the factorials.
