1. Calculus BC AP/Dual, Revised ©2015
Taylor Series – Day 1
Section 9.9
Calculus BC AP/Dual, Revised ©2015
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9.9: Taylor Series Day 1

2. Definition of Taylor Series
Taylor Series Equation: 𝑷 𝒏 𝒙 =𝒇 𝒄 + 𝒇 ′ 𝒄 𝒙−𝒄 + 𝒇 ′′ 𝒄 𝟐! 𝒙−𝒄 𝟐 𝒙−𝒄 𝟐 + 𝒇 ′′′ 𝒄 𝟑! 𝒙−𝒄 𝟑 +…+ 𝒇 𝒏 𝒄 𝒏! 𝒙−𝒄 𝒏
Taylor Series General Term: 𝒏=𝟎 ∞ 𝒇 𝒏 𝒄 𝒏! 𝒙−𝒄 𝒏
MacLaurin Series Equation when 𝒄=𝟎: 𝑷 𝒏 𝒙 =𝒇 𝟎 + 𝒇 ′ 𝟎 𝒙+ 𝒇 ′′ 𝟎 𝟐! 𝒙 𝟐 + 𝒇 ′′′ 𝟎 𝟑! 𝒙 𝟑 +…+ 𝒇 𝒏 𝟎 𝒏! 𝒙 𝒏
MacLaurin Series General Term: 𝒏=𝟎 ∞ 𝒙 𝒏 𝒏!
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9.9: Taylor Series Day 1

3. Example 1
Find the MacLaurin Series of 𝒇 𝒙 = 𝒆 𝟓𝒙 centered at 𝒄=𝟎. Give the first four nonzero terms and the general term.
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9.9: Taylor Series Day 1

4. Example 2
Find the Taylor Series of 𝒇 𝒙 = 𝒆 𝟓𝒙 centered at 𝒄=𝟐. Give the first four nonzero terms and the general term.
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9.9: Taylor Series Day 1

5. Your Turn
Find the MacLaurin Series of 𝒇 𝒙 = 𝒆 𝟑𝒙 centered at 𝒄=𝟎. Give the first four nonzero terms and the general term.
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9.9: Taylor Series Day 1

6. Special Case of Maclaurin Series: 𝒆𝒙
To derive a series for 𝒆 𝒙 :
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9.9: Taylor Series Day 1

7. Special Case of Maclaurin Series: 𝐬𝐢𝐧 𝒙
B. To derive a series for 𝐬𝐢𝐧⁡𝒙 :
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9.9: Taylor Series Day 1

8. Special Case of Maclaurin Series: 𝐜𝐨𝐬 𝒙
C. To derive a series for 𝐜𝐨𝐬⁡𝒙 :
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9.9: Taylor Series Day 1

9. Special Maclaurin Series which must be Memorized:
𝒆 𝒙 =𝟏+𝒙+ 𝒙 𝟐 𝟐! + 𝒙 𝟑 𝟑! 𝒙 𝒏 𝒏! ; 𝒏=𝟎 ∞ 𝒙 𝒏 𝒏!
𝐬𝐢𝐧 𝒙=𝒙− 𝟏 𝟑! 𝒙 𝟑 + 𝟏 𝟓! 𝒙 𝟓 − 𝟏 𝟕! 𝒙 𝟕 −𝟏 𝒏 𝒙 𝟐𝒏+𝟏 𝟐𝒏+𝟏 ! ; 𝒏=𝟎 ∞ −𝟏 𝒏 𝒙 𝟐𝒏+𝟏 𝟐𝒏+𝟏 !
𝐜𝐨𝐬 𝒙=𝟏− 𝟏 𝟐! 𝒙 𝟐 + 𝟏 𝟒! 𝒙 𝟒 − 𝟏 𝟔! 𝒙 𝟔 −𝟏 𝒏 𝒙 𝟐𝒏 𝟐𝒏 ! ; 𝒏=𝟎 ∞ −𝟏 𝒏 𝒙 𝟐𝒏 𝟐𝒏 !
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9.9: Taylor Series Day 1

10. Manipulation Techniques
One can manipulate these three special series (or any series we are given) to find other series by using the techniques, called manipulation techniques. These include:
Substituting into the series
Multiplying or dividing the series by a constant and/or a variable
Adding or subtracting two series
Differentiating or integrating a series
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9.9: Taylor Series Day 1

11. Example 3
Find the MacLaurin Series of 𝒇 𝒙 =𝐬𝐢𝐧 𝒙 𝟐 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

12. Example 3
Find the MacLaurin Series of 𝒇 𝒙 =𝐬𝐢𝐧 𝒙 𝟐 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

13. Example 3
Find the MacLaurin Series of 𝒇 𝒙 =𝐬𝐢𝐧 𝒙 𝟐 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

14. Example 4
Find the MacLaurin Series of 𝒇 𝒙 =𝐜𝐨𝐬 𝒙 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

15. Example 4
Find the MacLaurin Series of 𝒇 𝒙 =𝐜𝐨𝐬 𝒙 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

16. Example 5
Find the MacLaurin Series of 𝒇 𝒙 =𝒙 𝐜𝐨𝐬 𝒙. Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

17. Example 6
Find the MacLaurin Series of 𝒇 𝒙 = 𝒆 𝒙 + 𝒆 −𝒙 𝟐 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

18. Example 6
Find the MacLaurin Series of 𝒇 𝒙 = 𝒆 𝒙 + 𝒆 −𝒙 𝟐 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

19. Your Turn
Find the MacLaurin Series of 𝒇 𝒙 =𝒆 𝟐𝒙 . Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

20. Example 7
Find the Taylor Series for each of the following functions by finding the derivative and using the Taylor formula of 𝒇 𝒙 =𝒆 𝒙−𝟑 where 𝒙=𝟑. Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

21. Example 8
Find the Taylor Series for each of the following functions by finding the derivative and using the Taylor formula of 𝒇 𝒙 =𝐬𝐢𝐧 𝒙+ 𝝅 𝟔 where 𝒙=𝟎. Give the first four nonzero terms.
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9.9: Taylor Series Day 1

22. Your Turn
Find the Taylor Series for each of the following functions by finding the derivative and using the Taylor formula of 𝒇 𝒙 =𝒆 𝒙−𝟑 where 𝒙=𝟎. Give the first four nonzero terms and general term.
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9.9: Taylor Series Day 1

23. Assignment
Worksheet
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9.9: Taylor Series Day 1