|Module 22 - Power Series|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|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
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 nth-order 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 TI-89 taylor( command can be used to find partial sums of Taylor series.
Find the second-order Taylor polynomial centered at 1 for the function f(x) = ex.
The series above is similar to the Maclaurin series for y = ex 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 = ex at x = 0 and has the same concavity as y = ex at that point. The polynomial
, which is centered at
22.3.1 Find the fifth-degree Taylor polynomial centered at 1 for y = ln x and interpret the result. Click here for the answer.
22.3.2 Find the fourth-degree 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 fifth-degree Maclaurin polynomial for y = sin x and the fourth-degree Maclaurin polynomial for y = cos x. Click here for the answer.
Verify your prediction of the relationship between the Maclaurin polynomials for
Find the fourth-degree Maclaurin polynomial for y = cos x.
The results are the same as predicted.
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