Kalkulus 1 (01 -14)

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Kalkulus 1 (01 -14)

  1. 1. Kalkulus I<br />Drs. Tasman Abbas<br />Sesion#01-14<br />JurusanFisika<br />FakultasMatematikadanIlmuPengetahuanAlam<br />
  2. 2. Outline<br /><ul><li>Definition
  3. 3. Evaluation of Limits
  4. 4. Continuity
  5. 5. Limits Involving Infinity</li></ul>1/8/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />2<br />
  6. 6. Limits and Continuity <br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />3<br />1/8/2011<br />
  7. 7. Limit<br />L<br />a<br />1/8/2011<br />4<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  8. 8. Limits, Graphs, and Calculators<br />1/8/2011<br />5<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  9. 9. 1/8/2011<br />6<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  10. 10. c) Find<br />6<br />Note: f (-2) = 1 <br />is not involved <br /><ul><li>2</li></ul>1/8/2011<br />7<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  11. 11. 3) Use your calculator to evaluate the limits<br />Answer : 16<br />Answer : no limit<br />Answer : no limit<br />Answer : 1/2<br />1/8/2011<br />8<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  12. 12. The Definition of Limit<br />L<br />a<br />1/8/2011<br />9<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  13. 13. 1/8/2011<br />10<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  14. 14. Examples<br />What do we do with the x?<br />1/8/2011<br />11<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  15. 15. 1/2<br />1<br />3/2<br />1/8/2011<br />12<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  16. 16. One-Sided Limits<br />The right-hand limit of f (x), as x approaches a, equals L<br />written:<br />if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a.<br />L<br />a<br />1/8/2011<br />13<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  17. 17. The left-hand limit of f (x), as x approaches a, equals M<br />written:<br />if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a.<br />M<br />a<br />1/8/2011<br />14<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  18. 18. 1. Given<br />Find <br />Find <br />Examples of One-Sided Limit<br />1/8/2011<br />15<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  19. 19. More Examples<br />Find the limits:<br />1/8/2011<br />16<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  20. 20. A Theorem<br />This theorem is used to show a limit does not exist.<br />For the function<br />But<br />1/8/2011<br />17<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  21. 21. Limit Theorems<br />1/8/2011<br />18<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  22. 22. Examples Using Limit Rule<br />Ex.<br />Ex.<br />1/8/2011<br />19<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  23. 23. More Examples<br />1/8/2011<br />20<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  24. 24. Indeterminate Forms<br />Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. <br />Notice form<br />Ex.<br />Factor and cancel common factors<br />1/8/2011<br />21<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  25. 25. More Examples<br />1/8/2011<br />22<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  26. 26. The Squeezing Theorem<br />1/8/2011<br />23<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  27. 27. Continuity<br />A function f is continuous at the point x = a if the following are true:<br />f(a)<br />a<br />1/8/2011<br />24<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  28. 28. A function f is continuous at the point x = a if the following are true:<br />f(a)<br />a<br />1/8/2011<br />25<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  29. 29. Examples<br />At which value(s) of x is the given function discontinuous?<br />Continuous everywhere<br />Continuous everywhere except at <br />1/8/2011<br />26<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  30. 30. and<br />and<br />Thus F is not cont. at <br />Thus h is not cont. at x=1.<br />F is continuous everywhere else<br />h is continuous everywhere else<br />1/8/2011<br />27<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  31. 31. Continuous Functions<br />If f and g are continuous at x = a, then<br />A polynomial functiony = P(x) is continuous at every point x.<br />A rational function is continuous at every point x in its domain.<br />1/8/2011<br />28<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  32. 32. Intermediate Value Theorem<br />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. <br />f (b)<br />L<br />f (c) =<br />f (a)<br />a<br />b<br />c<br />1/8/2011<br />29<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  33. 33. Example<br />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.<br />1/8/2011<br />30<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  34. 34. Limits at Infinity<br />For all n > 0,<br />provided that is defined.<br />Divide by<br />Ex.<br />1/8/2011<br />31<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  35. 35. More Examples<br />1/8/2011<br />32<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  36. 36. 1/8/2011<br />33<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  37. 37. 1/8/2011<br />34<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  38. 38. Infinite Limits<br />For all n > 0,<br />1/8/2011<br />35<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  39. 39. Examples<br />Find the limits<br />1/8/2011<br />36<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  40. 40. Limit and Trig Functions<br />From the graph of trigs functions <br /> we conclude that they are continuous everywhere<br />1/8/2011<br />37<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  41. 41. Tangent and Secant <br />Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers <br />1/8/2011<br />38<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  42. 42. Examples<br />1/8/2011<br />39<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  43. 43. Limit and Exponential Functions<br />The above graph confirm that exponential functions are continuous everywhere.<br />1/8/2011<br />40<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  44. 44. Asymptotes<br />1/8/2011<br />41<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  45. 45. Examples<br />Find the asymptotes of the graphs of the functions<br />1/8/2011<br />42<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  46. 46. 1/8/2011<br />43<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />
  47. 47. Thank You<br />1/8/2011<br />© 2010 Universitas Negeri Jakarta | www.unj.ac.id |<br />44<br />

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