1. Taylor Polynomials – Day 2
Section 9.7
Calculus BC AP/Dual, Revised ©2018
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§9.7: Taylor Polynomials Day 2

2. §9.7: Taylor Polynomials Day 2
Example 1
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers.
Solve for 𝒇 𝟒 and 𝒇′′ 𝟒
Write the second-degree Taylor polynomial for 𝒇′ about 4 and use it to approximate 𝒇 ′ 𝟒.𝟑
Write the fourth-degree Taylor polynomial for 𝒈 𝒙 = 𝟒 𝒙 𝒇 𝒕 𝒅𝒕 at 𝒙=𝟒.
Can the exact value of 𝒇 𝟑 be determined from the information given? Justify your answer.
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§9.7: Taylor Polynomials Day 2

3. §9.7: Taylor Polynomials Day 2
Example 1a
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers.
Solve for 𝒇 𝟒 and 𝒇′′ 𝟒
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§9.7: Taylor Polynomials Day 2

4. §9.7: Taylor Polynomials Day 2
Example 1b
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (b) Write the second-degree Taylor polynomial for 𝒇′ about 4 and use it to approximate 𝒇 ′ 𝟒.𝟑
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§9.7: Taylor Polynomials Day 2

5. §9.7: Taylor Polynomials Day 2
Example 1b
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (b) Write the second-degree Taylor polynomial for 𝒇′ about 4 and use it to approximate 𝒇 ′ 𝟒.𝟑
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§9.7: Taylor Polynomials Day 2

6. §9.7: Taylor Polynomials Day 2
Example 1c
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (c) Write the fourth-degree Taylor polynomial for 𝒈 𝒙 = 𝟒 𝒙 𝒇 𝒕 𝒅𝒕 at 𝒙=𝟒.
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§9.7: Taylor Polynomials Day 2

7. §9.7: Taylor Polynomials Day 2
Example 1c
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (c) Write the fourth-degree Taylor polynomial for 𝒈 𝒙 = 𝟒 𝒙 𝒇 𝒕 𝒅𝒕 at 𝒙=𝟒.
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§9.7: Taylor Polynomials Day 2

8. §9.7: Taylor Polynomials Day 2
Example 1d
Suppose 𝑷 𝟒 𝒙 =𝟕−𝟑 𝒙−𝟒 +𝟓 𝒙−𝟒 𝟐 −𝟐 𝒙−𝟒 𝟑 + 𝟔 𝒙−𝟒 𝟒 be a Taylor Polynomial for the order of four for the function 𝒇 at 𝒙=𝟒. Assume 𝒇 has derivatives of all orders for all real numbers. (d) Can the exact value of 𝒇 𝟑 be determined from the information given? Justify your answer.
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§9.7: Taylor Polynomials Day 2

9. §9.7: Taylor Polynomials Day 2
Example 2
Suppose that the function 𝒇 𝒙 is approximated near 𝒙=𝟎 by a third-degree Taylor Polynomial 𝑷 𝟑 𝒙 =𝟐−𝟓 𝒙 𝟐 +𝟖 𝒙 𝟑 .
Find the value of 𝒇 𝟎 , 𝒇 ′ 𝟎 , 𝒇 ′′ 𝟎 , and 𝒇 ′′′ 𝟎
Does 𝒇 have a local maximum, a local minimum, or neither at 𝒙=𝟎? Justify response.
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§9.7: Taylor Polynomials Day 2

10. §9.7: Taylor Polynomials Day 2
Example 2
Suppose that the function 𝒇 𝒙 is approximated near 𝒙=𝟎 by a third-degree Taylor Polynomial 𝑷 𝟑 𝒙 =𝟐−𝟓 𝒙 𝟐 +𝟖 𝒙 𝟑 .
Find the value of 𝒇 𝟎 , 𝒇 ′ 𝟎 , 𝒇 ′′ 𝟎 , and 𝒇 ′′′ 𝟎
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§9.7: Taylor Polynomials Day 2

11. §9.7: Taylor Polynomials Day 2
Example 2b
Suppose that the function 𝒇 𝒙 is approximated near 𝒙=𝟎 by a third-degree Taylor Polynomial 𝑷 𝟑 𝒙 =𝟐−𝟓 𝒙 𝟐 +𝟖 𝒙 𝟑 . (b) Does 𝒇 have a local maximum, a local minimum, or neither at 𝒙=𝟎? Justify response.
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§9.7: Taylor Polynomials Day 2

12. §9.7: Taylor Polynomials Day 2
Example 3
Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 :
Find the third-degree polynomial
Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙
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§9.7: Taylor Polynomials Day 2

13. §9.7: Taylor Polynomials Day 2
Example 3a
Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 ,
Find the third degree polynomial
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§9.7: Taylor Polynomials Day 2

14. §9.7: Taylor Polynomials Day 2
Example 3b
Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙
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§9.7: Taylor Polynomials Day 2

15. §9.7: Taylor Polynomials Day 2
Example 3b
Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙
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§9.7: Taylor Polynomials Day 2

16. §9.7: Taylor Polynomials Day 2
Example 3b
Using the MacLaurin polynomial for 𝒇 𝒙 = 𝒆 𝒙 , B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝒇 𝒙 −𝟏 𝟐𝒙
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§9.7: Taylor Polynomials Day 2

17. §9.7: Taylor Polynomials Day 2
Your Turn
Using the MacLaurin polynomial for 𝒇 𝒙 =𝐬𝐢𝐧 𝒙:
Find the third-degree polynomial
Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙
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§9.7: Taylor Polynomials Day 2

18. §9.7: Taylor Polynomials Day 2
Your Turn
Using the MacLaurin polynomial for 𝒇 𝒙 =𝐬𝐢𝐧 𝒙:
Find the third degree polynomial
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§9.7: Taylor Polynomials Day 2

19. §9.7: Taylor Polynomials Day 2
Your Turn
Using the MacLaurin polynomial for 𝒇 𝒙 =𝐬𝐢𝐧 𝒙, B) Use answer to find 𝐥𝐢𝐦 𝒙→𝟎 𝐬𝐢𝐧 𝒙 𝒙
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§9.7: Taylor Polynomials Day 2

20. AP Multiple Choice Practice Question 1 (non-calculator)
The coefficient of 𝒙 𝟑 in the Taylor series for 𝒆 𝟑𝒙 about 𝒙=𝟎 is (A) 𝟏 𝟑 (B) 𝟏 𝟐 (C) 𝟑 𝟐 (D) 𝟗 𝟐
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§9.7: Taylor Polynomials Day 2

21. AP Multiple Choice Practice Question 1 (non-calculator)
The coefficient of 𝒙 𝟑 in the Taylor series for 𝒆 𝟑𝒙 about 𝒙=𝟎 is
Vocabulary
Connections and Process
Answer
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§9.7: Taylor Polynomials Day 2

22. §9.7: Taylor Polynomials Day 2
Assignment
Worksheet
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§9.7: Taylor Polynomials Day 2