Derivatives of Parametric-Functions.pptx
- 3. Parametric Functions Some relationships between two quantities or variables are so complicated that we sometimes introduce a third quantity or variable in order to make things easier to handle. In mathematics this third quantity is called a parameter
- 4. Parametric Functions Instead of one equation relating two variables, here, we have two equations relating each one of them with the parameter. For curves which are defined in this way, their rates of change can be found using parametric differentiation.
- 5. Parametric Functions If , and , differentiate both and with respect to Then, 𝒅𝒚 𝒅𝒙 = 𝒅𝒚 𝒅𝒕 𝒅𝒙 𝒅𝒕 from g(y) from f(x)
- 6. Parametric Functions For Higher Derivatives: 𝒅 𝒏 𝒚 𝒅 𝒙𝒏 = 𝒅 𝒅𝒕 (𝒅 𝒏−𝟏 𝒚 𝒅 𝒙 𝒏− 𝟏 ) 𝒅𝒙 𝒅𝒕 Note that 𝒅 𝟐 𝒚 𝒅 𝒙 𝟐 = 𝒅 𝒅𝒕 ( 𝒅𝒚 𝒅𝒙 ) 𝒅𝒙 𝒅𝒕 𝒅 𝟑 𝒚 𝒅 𝒙𝟑 = 𝒅 𝒅𝒕 (𝒅 𝟐 𝒚 𝒅 𝒙 𝟐 ) 𝒅 𝒙 𝒅𝒕
- 7. Ex. 1 Find 𝒅𝒚 𝒅𝒙 , 𝒅𝟐 𝒚 𝒅𝒙 𝟐 𝒙=𝒕𝟑 +𝟐𝒕 −𝟒 and 𝒚=𝒕𝟑 −𝒕+𝟐. 𝒅𝒙 𝒅𝒕 =𝟑 𝒕 𝟐 +𝟐 𝒅𝒚 𝒅𝒕 =𝟑 𝒕 𝟐 −𝟏 𝒅𝒚 𝒅𝒙 = 𝒅𝒚 𝒅𝒕 𝒅𝒙 𝒅𝒕 ¿ 𝟑 𝒕𝟐 −𝟏 𝟑 𝒕 𝟐 +𝟐
- 8. Ex. 1 Find 𝒅𝒚 𝒅𝒙 , 𝒅𝟐 𝒚 𝒅𝒙 𝟐 𝒙=𝒕𝟑 +𝟐𝒕 − 𝟒 and 𝒚=𝒕𝟑 − 𝒕+𝟐 . 𝒅𝒙 𝒅𝒕 =𝟑 𝒕 𝟐 +𝟐 𝒅𝒚 𝒅𝒕 =𝟑 𝒕 𝟐 −𝟏
- 9. Ex. 2 Find 𝒅𝒚 𝒅𝒙 , 𝒅𝟐 𝒚 𝒅𝒙 𝟐 . 𝒅𝒙 𝒅𝒕 =− 𝟑𝐬𝐢𝐧 𝒕 𝒅𝒚 𝒅𝒕 =𝟑𝐜𝐨𝐬𝒕 𝒅𝒚 𝒅𝒙 = 𝒅𝒚 𝒅𝒕 𝒅𝒙 𝒅𝒕 ¿ 𝟑 𝐜𝐨𝐬 𝒕 −𝟑 𝐬𝐢𝐧 𝒕 ¿ − 𝐜𝐨𝐭 𝒕
- 10. Ex. 2 Find 𝒅𝒚 𝒅𝒙 , 𝒅𝟐 𝒚 𝒅𝒙 𝟐 . 𝒙 𝒅𝒕 =− 𝟑𝐬𝐢𝐧 𝒕 𝒅𝒚 𝒅𝒕 =𝟑 𝐜𝐨𝐬 𝒕 𝒅𝒚 𝒅𝒙 ¿ − 𝐜𝐨𝐭 𝒕
- 12. Seatwork 1. Find , from and 1.