1. Calculus BC AP/Dual, Revised ©2016 5.8: Hyperbolic Functions
Section 5.8
Calculus BC AP/Dual, Revised ©2016
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5.8: Hyperbolic Functions

2. 5.8: Hyperbolic Functions
Hyperbolas
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5.8: Hyperbolic Functions

3. Definition of the Hyperbolic Functions
𝐬𝐢𝐧𝐡 𝒙 = 𝒆 𝒙 − 𝒆 −𝒙 𝟐
𝐜𝐬𝐜𝐡 𝒙 = 𝟏 𝐬𝐢𝐧𝐡 𝒙
𝐜𝐨𝐬𝐡 𝒙 = 𝒆 𝒙 + 𝒆 −𝒙 𝟐
𝐬𝐞𝐜𝐡 𝒙 = 𝟏 𝐜𝐨𝐬𝐡 𝒙
𝐭𝐚𝐧𝐡 𝒙 = 𝒆 𝒙 − 𝒆 −𝒙 𝒆 𝒙 + 𝒆 −𝒙
𝐜𝐨𝐭𝐡 𝒙 = 𝟏 𝐭𝐚𝐧𝐡 𝒙
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5.8: Hyperbolic Functions

4. 5.8: Hyperbolic Functions
Getting 𝒚=𝐬𝐢𝐧𝐡(𝒙)
Comparing 𝒚= 𝟏 𝟐 𝒆 𝒙 and 𝒚=− 𝟏 𝟐 𝒆 −𝒙 with 𝒇 𝒙 =𝐬𝐢𝐧𝐡 𝒙
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5.8: Hyperbolic Functions

5. 5.8: Hyperbolic Functions
Getting 𝒚=𝐜𝐨𝐬𝐡(𝒙)
Comparing 𝒚= 𝟏 𝟐 𝒆 𝒙 and 𝒚= 𝟏 𝟐 𝒆 −𝒙 with 𝒇 𝒙 =𝐜𝐨𝐬 𝐡 𝒙
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5.8: Hyperbolic Functions

6. 5.8: Hyperbolic Functions
Getting 𝒚=𝐭𝐚𝐧𝐡(𝒙)
Comparing 𝒚=𝟏 and 𝒚=−𝟏 with 𝒇 𝒙 = 𝐬𝐢𝐧𝐡(𝒙) 𝐜𝐨𝐬𝐡(𝒙)
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5.8: Hyperbolic Functions

7. Derivatives and Integrals
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5.8: Hyperbolic Functions

8. 5.8: Hyperbolic Functions
Example 1
Evaluate 𝒅𝒚 𝒅𝒙 =𝐬𝐢𝐧 𝐡(𝒙)
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5.8: Hyperbolic Functions

9. 5.8: Hyperbolic Functions
Example 2
Evaluate 𝒅 𝒅𝒙 𝐥𝐧 𝐜𝐨𝐬𝐡𝒙
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5.8: Hyperbolic Functions

10. 5.8: Hyperbolic Functions
Example 3
Evaluate 𝒅 𝒅𝒙 𝐬𝐢𝐧𝐡 𝒙 𝟐 −𝟑
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5.8: Hyperbolic Functions

11. 5.8: Hyperbolic Functions
Your Turn
Evaluate 𝒅 𝒅𝒙 𝒙 𝐬𝐢𝐧𝐡 𝒙−𝐜𝐨𝐬𝐡 𝒙
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5.8: Hyperbolic Functions

12. 5.8: Hyperbolic Functions
Example 4
Evaluate 𝐜𝐨𝐬 𝟐𝒙 𝐬𝐢𝐧 𝐡 𝟐 𝟐𝒙 𝒅𝒙
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5.8: Hyperbolic Functions

13. 5.8: Hyperbolic Functions
Your Turn
Evaluate 𝐬𝐢𝐧𝐡 𝟏−𝟐𝒙 𝒅𝒙
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5.8: Hyperbolic Functions

14. 5.8: Hyperbolic Functions
Assignment
Page odd, odd
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5.8: Hyperbolic Functions