1. Section 11.6 – Taylor’s Formula with Remainder

2. But there is some stuff we left out. The remainder.
We have created Taylor (Maclaurin) Polynomials to approximate other functions. For example,
But there is some stuff we left out. The remainder.
So how big is the remainder?
Which gives us the following result (after some work):

3. The Lagrange Remainder of a Taylor Polynomial
where z is some number between x and c
This is not the most useful thing in the world. However, it does lead to something good.
The Error of a Taylor Polynomial
where M is the maximum value of
on the interval [b, c] or [c, b]

4. Let f be a function that has derivatives of all orders on the
Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4,
f ’’’(0) = 3/8 and
for all x in the interval (0, 1).
a. Find the third-degree Taylor polynomial about x = 0 for f.
b. Use your answer to part a to estimate the value of f(0.5)

5. Let f be a function that has derivatives of all orders on the
Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4,
f ’’’(0) = 3/8 and
for all x in the interval (0, 1).
What is the maximum possible error for the approximation
made in part b?

6. Estimate the error that results when arctan x is replaced by

7. Estimate the error that results when ln(x + 1) is replaced by
F ‘’’ (x) has a maximum value at x = -0.1

8. Find an approximation of ln 1.1 that is accurate to three decimal
places.
We just determined that the error using the second degree
expansion is This is good enough.

9. Use this range for 2 decimal places.
Use a Taylor Polynomial to estimate cos(0.2) to 2 decimal places
If x = 0.2, Alternating Series Test works for convergence
Use this range for 2 decimal places.
Use P2

10. Use a Taylor Polynomial to estimate
with two decimal
place accuracy.
Satisfies Alternating Series Test

11. Suppose the function f is defined so that
a. Write a second degree Taylor polynomial for f about x = 1
b. Use the result from (a) to approximate f(1.5)

12. Suppose the function f is defined so that
for all x in [1, 1.5], find an upper bound for the
approximation error in part b if

13. The first four derivatives of
Find the third-degree Taylor
approximation to f at x = 0
Use your answer in (a) to find
an approximation of f(0.5)
Estimate the error involved in the
approximation in (b). Show your
reasoning.

14. The first four derivatives of
Find the third-degree Taylor
approximation to f at x = 0
Use your answer in (a) to find
an approximation of f(0.5)

15. The first four derivatives of
Estimate the error involved in the
approximation in (b). Show your
reasoning.

16. P5