1. Taylor & MacClaurin Series

2. Certain functions can be expressed as power series:
How can we find the values of a0, a1, a2, a3…?
Plug in 0 for x and watch all the other terms go away
Plug in 0 for x again

3. Plug in 0 for x
Can you find the pattern for finding an?

4. So the power series that converges to f (x) can be written as…
This is called the Taylor Series for f (x) centered at x = 0 because we used 0 to find
all the terms.
A series centered at x = 0 is also called a MacClaurin Series.
In class, we will discuss how to generate a series centered at a point other than x = 0

5. Find the Taylor Series centered at x = 0 for the function
We would start by finding the first few derivatives and
then looking for a pattern.
But in this case the derivatives are easy…

6. Find the Taylor Series centered at x = 0 for the function
We would start by finding the first few derivatives and
then looking for a pattern.
But in this case the derivatives are easy…
…and the answer is…

7. We can approximate the graph of ex by generating terms of the series…
f (x) = ex

8. We can approximate the graph of ex by generating terms of the series…
f (x) = ex

9. We can approximate the graph of ex by generating terms of the series…
f (x) = ex

10. We can approximate the graph of ex by generating terms of the series…
f (x) = ex
What happens every time we add a term?
The approximation gets better
Where does the overlap appear to be centered?
At x = 0

11. In class you will need to be prepared to generate other series
centered at any value of x
A number of MacClaurin Series for some familiar functions can be
found on page 477