|Module 24 - Power Series|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 24.3: Taylor Series|
In Lesson 24.2 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.
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 24.2, is a Taylor series centered at zero.
The Taylor Polynomial of ex Centered at 1
The second-order Taylor polynomial centered at 1 for the function f(x) = ex can be found by using a procedure similar to the procedure given in Lesson 24.2.
The coefficient of the term (x - 1)k in the Taylor polynomial is given by . This formula is very similar to the formula for finding the coefficient of xk in a Maclaurin polynomial where the derivative is evaluated at 0. In this Taylor polynomial, the derivative is evaluated at 1, the center of the series.
The coefficients of the second-order Taylor polynomial centered at 1 for ex are
f(1) = e
f '(1) = e
So the second-order Taylor polynomial for ex centered at 1 is , and near x = 1, ex P2(x).
The Taylor series for ex centered at 1 is similar to the Maclaurin series for ex found in Lesson 24.2. However, the terms in the Taylor series have powers of (x - 1) rather than powers of x and the coefficients contain the values of the derivatives evaluated at x = 1 rather than evaluated at x = 0.
Graphing the function and the polynomial illustrate that the polynomial is a good approximation near
The second-order Maclaurin polynomial you found in Lesson 24.2, , is tangent to f(x) = ex at x = 0 and has the same concavity as f(x) = ex at that point. The polynomial , which is centered at x = 1, is tangent to f(x) = ex at x = 1 and has the same concavity as f(x) = ex at that point.
24.3.1 Find the second-order Taylor polynomial centered at 1 for the function f(x) = ln x. Graph this polynomial together with f(x) = ln x. Click here for the answer.
Other Taylor Series
The Taylor series for one function can be used to find a Taylor series for a related function.
The third-order Taylor polynomial centered at 1 for f(x) = ln x is .
The derivative of f(x) = ln x is . The derivative of p(x) gives the second-order Taylor polynomial for centered at 1.
24.3.2 Find the second-order Taylor polynomial for centered at 1 using the derivative of p(x) and graph it with . Click here for the answer.
The Taylor Series for cos x2
Other modifications of a Taylor series yield other Taylor-series. For example, replacing each x with x2 in the Taylor series for f(x) = cos(x) gives the Taylor series for g(x) = f(x2) = cos(x2).
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