1. Taylor Series – Day 2 Section 9.6 Calculus BC AP/Dual, Revised ©2014
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

2. Definition of Nth degree Taylor Polynomial
In a Maclaurin polynomial, it starts at zero.
In a Taylor polynomial, it starts at one.
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

3. 9.7 - Taylor Polynomials and Approximations
What Do You Notice?
y1 = sin x
y2 = 𝒙− 𝒙 𝟑 𝟑! + 𝒙 𝟓 𝟓!
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

4. 9.7 - Taylor Polynomials and Approximations
Link
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

5. 9.7 - Taylor Polynomials and Approximations
Example 1
Find the Maclaurin polynomial of degree n = 6 for f(x) = cos x. Then, use P6(x) to approximate the value of cos (0.1).
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

6. 9.7 - Taylor Polynomials and Approximations
Example 1
Find the Maclaurin polynomial of degree n = 6 for f(x) = cos x. Then, use P6(x) to approximate the value of cos (0.1).
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

7. 9.7 - Taylor Polynomials and Approximations
Example 2
Find the Taylor polynomial of degree n = 6 for f(x) = ex at c = 0
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

8. 9.7 - Taylor Polynomials and Approximations
Example 3
Suppose that g is a function which as continuous derivatives and that g(2) = 3, g’(2) = –4, g’’(2) = 7, and g’’’(2) = –5 with a degree of 3 and for g to be centered at 2.
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

9. 9.7 - Taylor Polynomials and Approximations
Your Turn
Find the Taylor polynomial of degree n = 6 for f(x) = ln x at c = 1
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

10. 9.7 - Taylor Polynomials and Approximations
Example 4a
Using the MacLaurin polynomial for f(x) = ex, A) Find the third degree and B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

11. 9.7 - Taylor Polynomials and Approximations
Example 4b
Using the MacLaurin polynomial for f(x) = ex, A) Find the third degree and B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

12. 9.7 - Taylor Polynomials and Approximations
Example 5a
Suppose that the function f(x) is approximated near x = 0 by a third-degree Taylor Polynomial P3(x) = 2 – 5x2 + 8x3. A) Find the value of f(0), f ’(0), f ’’(0), and f ’’’(0) and B) does f have a local maximum, a local minimum, or neither at x = 0? Justify response.
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

13. 9.7 - Taylor Polynomials and Approximations
Example 5b
Suppose that the function f(x) is approximated near x = 0 by a third-degree Taylor Polynomial P3(x) = 2 – 5x2 + 8x3. A) Find the value of f(0), f ’(0), f ’’(0), and f ’’’(0) and B) does f have a local maximum, a local minimum, or neither at x = 0? Justify response.
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations

14. 9.7 - Taylor Polynomials and Approximations
Assignment
Worksheet
12/4/ :47 AM
9.7 - Taylor Polynomials and Approximations