CRMS Calculus May 31, 2010

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Final Review

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CRMS Calculus May 31, 2010

  1. 1. The Year in Review What the heck is Calculus?
  2. 2. One of the supreme creations of human thought
  3. 3. One of the biggest controversies in history
  4. 4. The Culmination of K-12 mathematics Calculus Trigonometry Elementary Functions Geometry Algebra Arithmetic
  5. 5. Course Goals <ul><li>To introduce you to the four major concepts of calculus: </li></ul><ul><ul><li>Limits </li></ul></ul><ul><ul><li>Derivatives </li></ul></ul><ul><ul><li>Definite Integral </li></ul></ul><ul><ul><li>Indefinite Integral </li></ul></ul>
  6. 6. For each concept you will … <ul><li>know the precise definition; </li></ul><ul><li>have an intuitive understanding of what the concept means; </li></ul><ul><li>be able to “do” the concept; </li></ul><ul><li>be able to apply it in the real world or mathematical world. </li></ul>
  7. 7. You will develop this knowledge by using four multiple representations: <ul><li>Numerical (Table) </li></ul><ul><li>Visual (Graph) </li></ul><ul><li>Algebraic (Function) </li></ul><ul><li>Verbally (and written). </li></ul>
  8. 8. Do you Sir Know the Foundation For All of Calculus?
  9. 9. The foundation of Calculus is the concept of a limit . <ul><li>1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,... 99999/100000,... </li></ul>
  10. 10. Limits by definition
  11. 11. Limits by Delta - Epsilon definition
  12. 12. Limits by graph
  13. 13. Limits by numerical table
  14. 14. Limits by algebra
  15. 15. Limits involving
  16. 16. Intermediate Value Theorem
  17. 18. Differential Calculus The mathematics of change Physical Meaning Instantaneous rate of change
  18. 19. Slope of the tangent line at a point. Graphical Meaning
  19. 20. Do you Sir Know the definition of The Derivative ?
  20. 21. Derivative
  21. 23. If y = Displacement Then y’ = Velocity And y’’ = Acceleration
  22. 24. How’s My Deriving?
  23. 25. Power Rule for derivatives Function Notation: Operator Notation:
  24. 28. Related Rates
  25. 29. y = C + AsinB(x – D) y ‘ = ABcosB(x – D)
  26. 31. The Squeeze Theorem
  27. 32. If y = e x , then y’ = e x If y = b x , then y’ = b x · ln b
  28. 33. If y = ln x, then
  29. 34. I PITY DA FOO’ WHO CAN’T FIND DA DERIVATIVE !!!
  30. 35. Integral Calculus Areas and Volumes
  31. 36. Counting Squares
  32. 37. Archimedes Method of Exhaustion
  33. 38. Trapezoidal Rule
  34. 39. Do you sir, Know the difference between An indefinite integral And A definite integral?
  35. 40. Antiderivatives of y’ = x 2 Bring the Whole Indefinite Integral Family! Indefinite Integral A Family of Curves With a given Derivative
  36. 41. Definite Integral is a Number
  37. 43. Solving Differential Equations and Slope Fields
  38. 45. Sweetheart, Check out this definition ! Definite Integral;
  39. 46. Rolle’s Theorem The Mean Value Theorem Is just Rolle’s Theorem Rotated
  40. 49. What were those Properties Of Integrals?
  41. 50. Definite Integrals Allow us to find The area between curves
  42. 51. Naked Math Real World
  43. 55. Volumes of Revolution The Disk Method
  44. 57. An Illuminating Volume
  45. 58. This stuff isn’t so hard! In fact …

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