1. 9.7 Taylor Polynomials & Approximations
Objectives:
Find Taylor and Maclaurin polynomials
Use Taylor & Maclaurin Polynomials to approximate
Find the error of Taylor & Maclaurin approximations
©2004Roy L. Gover (

2. Definition A polynomial in x is any expression in the form
Where n is a nonnegative integer and coefficients
are complex numbers

3. Example The following are polynomials: 1st degree 2nd degree
3rd degree

4. Important Idea
Polynomial functions can be used to approximate other functions such as:
The approximation can be as accurate as we choose.

5. Example
Find a polynomial, P(x), that has the same value & slope as f(x) at x=0
Approximate
at values close to x=0
P(0)=f(0)=1
Slope of P(x)=f’(0)=1

6. Example P(0)=f(0)=1 Approximate at values close to x=0
Slope of P(x)=f’(0)=1
therefore
Remember

7. Example
Approximate
at values close to x=0 x
f(x)
P(x)
-.25
. 25

8. Definition x P(x) converges to f(x) at x=0. P(x) is centered at 0 1
is a 1st degree approximation

9. Important Idea
As x moves either side of center (x=0), the approximation of f(x) by P(x) decreases in accuracy x c

10. Important Idea
P(x) is an accurate approximationof f(x) at values of x close to c x c

11. Example Find such that:
Find a more accurate approximation of centered at x=0.

12. Example Find a more accurate approximation of centered at x=0. is a
2nd degree
approximation

13. Example Find a more accurate approximation of centered at x=0. Begins
to diverge
Notice how the polynomial more closely fits

14. Analysis x
f(x)
P2(x)
-.5
.6065
.625
-.25
.7788
.7813 1
.25
1.284
1.281 .5
1.649
1.625
A 2nd degree approx. is more accurate for more values of x

15. Example Find an even more accurate approx. for is a 3rd degree
approximation
Notice how the polynomial fits even more closely

16. Analysis
Do you see a pattern?

17. Important Idea

18. Try This
Graph a 5th degree polynomial approximation for using your graphing calculator. Use the table feature of your calculator to find the approximation for
and compare with the
calculator value for e.

19. Solution
looks good

20. Solution
per
approx.
per calculator

21. Important Idea
Can be generalized to center at values other than 0

22. Important Idea
is the nth polynomial approx. for centered at c.

23. Important Idea
Generalizing even more, the following polynomial can be used to approx any function at any center:

24. Important Idea
We just need a way to find the values of the coefficients for each we wish to approx.

25. Definition
The nth Taylor polynomial for f at c:

26. Definition
When referring to Taylor polynomials, we can talk about number of terms, order or degree.
For Example:
centered at 0.

27. Definition 1 2 3
This Taylor Polynomial has 3 terms

28. Definition
It is a 4th order Taylor polynomial, because it was found using the 4th derivative. (The 1st and 3rd derivative is 0)

29. Definition
It is a 4th degree Taylor polynomial, because the highest degree term is

30. Definition
The Maclaurin Polynomial is the Taylor Polynomial with c=0

31. Important Idea
The value of f and its first n derivatives must agree with the value of P and its first n derivatives at x=c. See p. 598

32. Example Find the Taylor Polynomials for and for centered at c=1 Steps:
1. Find the coefficients
2. Substitute into Taylor

33. Try This
Find the Taylor Polynomial
centered at c=1
for

34. Example Use the Taylor Polynomial centered at c=1 for
to approximate ln (1.1)
Ln(1.1) approx

35. Example
Compare the graphs of
with
&

36. Try This
Find the Maclaurin 6th degree approximation, , for Approximate
and compare with
the calculated value .

37. Important Idea x’s close to c Better approximations occur for:
Higher degree polynomials

38. Try This Which will give the better approximation of ln (1.1)?
The 4th Taylor Polynomial centered at c=1 and x=1.1 for ln(x)
The 4th Maclaurin Polynimial (centered at c=0) and x=.1 for ln(1+x)
See Example 7, page 653
Approximations are the same

39. Definition
The Remainder of a Taylor polynomial, is the difference between the exact value and the Taylor approximation.
Exact Value
Approx. Value

40. Important Idea
If you evaluate a Taylor polynomial to n terms as an approximation for f(x) centered at c, then the difference between the actual and approximate values, ,for f(x) is related to the n+1 term-as follows:

41. Definition The Error of a Taylor approximation is: where:
and z is in [c,x]
Lagrange Remainder (Error)

42. Definition This is the n+1 term of the Taylor polynomial
Choose that gives the max. value for
Maclaurin c=0.

43. Example Find the 3rd degree Maclaurin polynomial for .
Use to approximate sin(.01) and determine the accuracy of your approximation.
P 603,ex 8

44. Important Idea
When finding the Lagrange Form of the Remainder, you should not try to find the exact value of z. Rather, you should try to find a z that gives you an upper bound for

45. Definition Lagrange Remainder:
The error of an nth polynomial approximation is less than the absolute value of the maximum of the n+1 term of the polynomial approximation.

46. Example Estimate the error (Lagrange form of the remainder) when
is replaced by its
Maclaurin polynomial:
for x=0.1

47. Example
Use the 3rd degree Taylor polynomial centered at c=1 to approximate
Estimate the error.

48. Try This Find the 4rd degree Maclaurin polynomial for .
Use to approximate cos(.02) and determine the accuracy of your approximation.
where