Calculus BC AP/Dual, Revised ©2016 5.8: Hyperbolic Functions Section 5.8 Calculus BC AP/Dual, Revised ©2016 viet.dang@humble.k12.tx.us 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Hyperbolas 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

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 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

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 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

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 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Getting 𝒚=𝐭𝐚𝐧𝐡(𝒙) Comparing 𝒚=𝟏 and 𝒚=−𝟏 with 𝒇 𝒙 = 𝐬𝐢𝐧𝐡(𝒙) 𝐜𝐨𝐬𝐡(𝒙) 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

Derivatives and Integrals 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Example 1 Evaluate 𝒅𝒚 𝒅𝒙 =𝐬𝐢𝐧 𝐡(𝒙) 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Example 2 Evaluate 𝒅 𝒅𝒙 𝐥𝐧 𝐜𝐨𝐬𝐡𝒙 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Example 3 Evaluate 𝒅 𝒅𝒙 𝐬𝐢𝐧𝐡 𝒙 𝟐 −𝟑 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Your Turn Evaluate 𝒅 𝒅𝒙 𝒙 𝐬𝐢𝐧𝐡 𝒙−𝐜𝐨𝐬𝐡 𝒙 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Example 4 Evaluate 𝐜𝐨𝐬 𝟐𝒙 𝐬𝐢𝐧 𝐡 𝟐 𝟐𝒙 𝒅𝒙 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Your Turn Evaluate 𝐬𝐢𝐧𝐡 𝟏−𝟐𝒙 𝒅𝒙 5/4/2019 5:52 AM 5.8: Hyperbolic Functions

5.8: Hyperbolic Functions Assignment Page 390 23-29 odd, 43-47 odd 5/4/2019 5:52 AM 5.8: Hyperbolic Functions