Module 22  Power Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 22.3: Taylor Series  
In Lesson 22.2 you found Maclaurin series that approximate functions near x = 0. This lesson investigates how to find a series that approximates a function near x = a, where a is any real number. Defining a Taylor Series Given a function f that has all its higher order derivatives, the series
where
is called the Taylor series for f centered at a. The Taylor series is a power series that approximates the function f near x = a. The partial sum
is called the nthorder Taylor polynomial for f centered at a. Every Maclaurin series, including those studied in Lesson 22.2, is a Taylor series centered at zero. Finding Taylor Polynomials The TI89 taylor( command can be used to find partial sums of Taylor series.
Find the secondorder Taylor polynomial centered at 1 for the function f(x) = e^{x}.
Near The series above is similar to the Maclaurin series for y = e^{x} found in Lesson 22.2. However, the terms in this series have powers of (x  1) rather than powers of x and the coefficients contain the values of the derivatives evaluated at x = 1. The coefficient of the term (x  1)^{k} is
More Taylor Polynomials
The polynomial you found in Lesson 22.2,
, was tangent to y = e^{x} at x = 0 and has the same concavity as y = e^{x} at that point. The polynomial
, which is centered at 22.3.1 Find the fifthdegree Taylor polynomial centered at 1 for y = ln x and interpret the result. Click here for the answer. 22.3.2 Find the fourthdegree Taylor polynomial centered at 1 for . Click here for the answer. 22.3.3 Describe the relationship between the two polynomials found in 22.3.1 and 22.3.2. Click here for the answer. 22.3.4 Predict the relationship between the fifthdegree Maclaurin polynomial for y = sin x and the fourthdegree Maclaurin polynomial for y = cos x. Click here for the answer.
Verify your prediction of the relationship between the Maclaurin polynomials for
Find the fourthdegree Maclaurin polynomial for y = cos x.
The results are the same as predicted. 

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