Taylor and Maclaurin Series Examples

Graph the function f(x) = e^x. Then add, on the same set of axes:

(a) the first-degree Taylor polynomial for f(x) at 0.

(b) the second-degree Taylor polynomial for f(x), centered at 0.

(c) the 3rd-degree Taylor polynomial for f(x) at 0.

(d) the 4th-degree Taylor polynomial for f(x) at 0.

(e) the 5th-degree Taylor polynomial for f(x) at 0.

We calculate the derivatives of f, evaluate them at 0, and plug them into the formula

with the appropriate value of n for each part of the problem.

Since the function we're looking at is f(x) = e^x, all the derivatives are also

e^x, so each derivative evaluated at 0 is

e⁰ = 1.

We get

then

and so on, up to

Another way to do this problem would be to find g₅(x) to start, and then cut the polynomial off at the appropriate degrees to get g₁ through g₄.

(a) The first-degree Taylor polynomial for f(x) at 0 is g₁(x) = 1 + x. This is our linear approximation:

(b) The second-degree Taylor polynomial for f(x), centered at 0, is . This function curves more like e^x but still gets away pretty quickly:

(c) The 3rd-degree Taylor polynomial for f(x) at 0 is . If we graph the 2nd and 3rd degree polynomials with f(x) we can see that g₃ sticks a little more closely to f(x) when x > 0:

(d) The 4th-degree Taylor polynomial for f(x) at 0 is .

This function gets even closer to f(x) = e^x for x > 0, and is starting to look a little more like e^x for x below zero, also.

(e) The 5th-degree Taylor polynomial for f(x) at 0 is . Getting even closer...

What is the Maclaurin series for f(x) = e^x?

To get the Maclaurin series, we look at the Taylor 

polynomials for f near 0 and let them keep going. 

Take g₅ from the previous example:

If we let the terms go forever, following this pattern, 

we get the Maclaurin series for f(x) = e^x:

Find the MacLaurin series for f(x) = sin x.

Start taking derivatives. If f(x) = sin x then

Evaluating these at 0, we get the Maclaurin series

We leave the factorials, instead of expanding, so we can see 

the pattern and how to write it in summation notation:

If we graph both f(x) and

on the same axes, it looks like we did things right:

Find the second-degree Taylor polynomial for  near x = 4.

We want to use a = 4 and we need to take two derivatives.

so

and

so

.

Putting this information into the formula we get

A graph confirms that this makes sense:

Find the Taylor series for the function sin x near .

We take f(x) = sin x, find some derivatives and evaluate 

them at , and finally plug things into the formula.

So

Let's graph f(x) and  

to make sure it looks right: