1. Taylor Series
Section 9.2b

2. Do Now Find the fourth order Taylor polynomial that approximates near
Before finding a bunch of derivatives, remember that we
can use a known power series to generate another…
From the Table of Maclaurin Series on p.477:
Therefore,

3. Do Now Find the fourth order Taylor polynomial that approximates near
The Taylor polynomial:
Support by graphing both functions in [–3, 3] by [–2, 2]

4. Practice Problems: #16 on p.478
Let f be a function that has derivatives of all orders for all
real numbers. Assume
Write the third order Taylor polynomial for f at x = 0 and
use it to approximate f(0.2).

5. Practice Problems: #16 on p.478
Let f be a function that has derivatives of all orders for all
real numbers. Assume
(b) Write the second order Taylor polynomial for , the
derivative of f, at x = 0 and use it to approximate
Second order Taylor polynomial for :

6. Practice Problems: #18 on p.478
The Maclaurin series for f(x) is
(a) Find and
In this case,
Similarly,

7. Practice Problems: #18 on p.478
The Maclaurin series for f(x) is
(b) Let g(x) = x f(x). Write the Maclaurin series for g(x),
showing the first three nonzero terms and the general term.
Multiply each term of f(x) by x:
(c) Write g(x) in terms of a familiar function without using
series.

8. Practice Problems: #20 on p.478
Let
and
Find the first four terms and the general term for the
Maclaurin series generated by f.
From the Table of Maclaurin Series on p.477:
So factor out 2 and substitute for

9. Practice Problems: #20 on p.478
Let
and
Find the first four terms and the general term for the
Maclaurin series generated by f.

10. Practice Problems: #20 on p.478
Let
and
(b) Find the first four nonzero terms and the Maclaurin series
for G.
The first term:
To find the other terms, integrate the terms of the series for f:

11. Practice Problems: #20 on p.478
Let
and
(b) Find the first four nonzero terms and the Maclaurin series
for G.