|Module 22 - Power Series|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 22.2: Maclaurin Series|
In Lesson 22.1 you explored several power series and their relationships to the functions to which they converge. In this lesson you will start with a function and find the power series that converges to that function for values of x near zero. This type of series is called a Maclaurin series.
Finding a Quadratic Polynomial that Approximates a Function Near x = 0
It turns out that finding a second-degree polynomial p(x) = ax2 + bx + c that satisfies p(0) = f(0),
The procedure below illustrates the method by finding a quadratic polynomial that satisfies these conditions for the function f(x) = ex.
Find the value of c when the two function values are equal at x = 0.
If c = 1, the function and the polynomial have the same value at x = 0, so p(x) = ax2 + bx + 1
Set the first derivatives equal and solve for b when x = 0.
If b = 1, the function and the polynomial will have the same slope at x = 0, so p(x) = ax2 + x + 1.
Set the second derivatives equal and solve for a when x = 0.
If a = 1/2, the function and the polynomial will have the same concavity at x = 0. Therefore,
Displaying Graphs of the Function and the Approximating Quadratic Polynomial
The parabola has the same value, the same slope and the same concavity as y = ex at x = 0. The quadratic polynomial is a good approximation for y = ex at x = 0.
22.2.1 Create a table of values for y = ex and with tblStart = -1 and tbl = 0.1. Compare the values of y = ex and the quadratic polynomial when x = -0.5 and when x = 0.5.
Click here for the answer.
Outlining the General Procedure of Finding Maclaurin Series
The general idea for finding the coefficients of a power series
that approximates a function
y = f(x)
near x = 0 is first to set the value of the power series equal to the value of f(x) at x = 0 and solve for c0. Then, set all of the derivatives of the power series (beginning with the first) equal to the corresponding derivative of f(x) at x = 0 and solve for the remaining coefficients of the power series.
The series found is this fashion is called the Maclaurin series for f(x).
Letting f (k)(x) represent the kth derivative of f, the coefficient of the term that contains the kth power of x in a Maclaurin series is
This method produces the following power series for y = ex :
The Maclaurin series above is more than an approximation of ex, it is equal to ex on the interval of convergence ( , ).
Every Maclaurin series is centered at 0 and the interval of convergence is centered at 0. When the Maclaurin series approximates a function, the series values and the function values are very close near x = 0.
The partial sum
is the nth-order Maclaurin polynomial for f.
Generating Maclaurin Polynomials on a TI-89
The TI-89 can generate Maclaurin polynomials for a function f using the taylor( command.
Generate the second-order Maclaurin polynomial for ex using the taylor( command.
22.2.2 Use the taylor command to find the fifth-order Maclaurin polynomial for
22.2.3 Graph y = ln(1 + x) and the fifth-order polynomial from 22.2.2 together. Click here for the answer.
|< Back | Next >|
2007 All rights reserved. |