unj fmipa-fisika

1. Kalkulus I Drs. Tasman Abbas Sesion#01-14 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam

3. Evaluation of Limits

4. Continuity

5. Limits Involving Infinity1/8/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 2

6. Limits and Continuity © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 3 1/8/2011

7. Limit L a 1/8/2011 4 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. Limits, Graphs, and Calculators 1/8/2011 5 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. 1/8/2011 6 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. 3) Use your calculator to evaluate the limits Answer : 16 Answer : no limit Answer : no limit Answer : 1/2 1/8/2011 8 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

12. The Definition of Limit L a 1/8/2011 9 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

13. 1/8/2011 10 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

14. Examples What do we do with the x? 1/8/2011 11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

15. 1/2 1 3/2 1/8/2011 12 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

16. One-Sided Limits The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a 1/8/2011 13 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

17. The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. M a 1/8/2011 14 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

18. 1. Given Find Find Examples of One-Sided Limit 1/8/2011 15 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

19. More Examples Find the limits: 1/8/2011 16 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

20. A Theorem This theorem is used to show a limit does not exist. For the function But 1/8/2011 17 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

21. Limit Theorems 1/8/2011 18 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

22. Examples Using Limit Rule Ex. Ex. 1/8/2011 19 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

23. More Examples 1/8/2011 20 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

24. Indeterminate Forms Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Notice form Ex. Factor and cancel common factors 1/8/2011 21 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

25. More Examples 1/8/2011 22 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

26. The Squeezing Theorem 1/8/2011 23 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

27. Continuity A function f is continuous at the point x = a if the following are true: f(a) a 1/8/2011 24 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

28. A function f is continuous at the point x = a if the following are true: f(a) a 1/8/2011 25 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

29. Examples At which value(s) of x is the given function discontinuous? Continuous everywhere Continuous everywhere except at 1/8/2011 26 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

30. and and Thus F is not cont. at Thus h is not cont. at x=1. F is continuous everywhere else h is continuous everywhere else 1/8/2011 27 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

31. Continuous Functions If f and g are continuous at x = a, then A polynomial functiony = P(x) is continuous at every point x. A rational function is continuous at every point x in its domain. 1/8/2011 28 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

32. Intermediate Value Theorem If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. f (b) L f (c) = f (a) a b c 1/8/2011 29 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

33. Example f (x) is continuous (polynomial) and since f (1) < 0 and f (2) > 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0. 1/8/2011 30 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

34. Limits at Infinity For all n > 0, provided that is defined. Divide by Ex. 1/8/2011 31 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

35. More Examples 1/8/2011 32 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

36. 1/8/2011 33 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

37. 1/8/2011 34 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

38. Infinite Limits For all n > 0, 1/8/2011 35 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

39. Examples Find the limits 1/8/2011 36 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

40. Limit and Trig Functions From the graph of trigs functions we conclude that they are continuous everywhere 1/8/2011 37 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

41. Tangent and Secant Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers 1/8/2011 38 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

42. Examples 1/8/2011 39 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

43. Limit and Exponential Functions The above graph confirm that exponential functions are continuous everywhere. 1/8/2011 40 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

44. Asymptotes 1/8/2011 41 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

45. Examples Find the asymptotes of the graphs of the functions 1/8/2011 42 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

46. 1/8/2011 43 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

47. Thank You 1/8/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 44