unj fmipa-fisika

1. Kalkulus II Teguh Budi P, M.Si Sesion#23-24 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam

2. Constantmultiple rule Higher Order Derivatives © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 2 Outline 1/9/2011

3. Multivariable Functions and Their Derivatives(part 1) © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 3 1/9/2011

4. The derivative of a constant is zero. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: 1/9/2011 4 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. (Pascal’s Triangle) If we find derivatives with the difference quotient: We observe a pattern: … 1/9/2011 5 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. We observe a pattern: … power rule examples: 1/9/2011 6 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

7. constant multiple rule: examples: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside. 1/9/2011 7 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. constant multiple rule: sum and difference rules: (Each term is treated separately) 1/9/2011 8 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. Horizontal tangents occur when slope = zero. Example: Find the horizontal tangents of: Plugging the x values into the original equation, we get: (The function is even, so we only get two horizontal tangents.) 1/9/2011 9 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

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15. First derivative (slope) is zero at: 1/9/2011 15 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

16. product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as: 1/9/2011 16 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

17. quotient rule: or 1/9/2011 17 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

18. is the first derivative of y with respect to x. is the second derivative. is the third derivative. is the fourth derivative. Higher Order Derivatives: (y double prime) We will learn later what these higher order derivatives are used for. 1/9/2011 18 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

19. Thank You 1/9/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 19