1. Calculus Mrs. Dougherty’s Class

2. drivers Start your engines

3. 3 Big Calculus Topics Limits Derivatives Integrals

4. Chapter 2

5. 2.1 Limits and continuity

6. Limits can be found Graphically

7. Limits can be found Graphically Numerically

8. Limits can be found Graphically Numerically By direct substitution

9. Limits can be found Graphically Numerically By direct substitution By the informal definition

10. Limits can be found Graphically Numerically By direct substitution By the informal definition By the formal definition

11. Limits Informal Def.

12. Limits Informal Def. Given real numbers c and L, if the values f(x) of a function approach or equal L

13. Limits Informal Def. Given real numbers c and L, if the values f(x) of a function approach or equal L as the values of x approach ( but do not equal c),

14. Limits Informal Def. Given real numbers c and L, if the values f(x) of a function approach or equal L as the values of x approach ( but do not equal c), then f has a limit L as x approaches c.

15. Limits notation

16. LIFE IS GOOD

17. Theorem 1 Constant Function f(x)=k Identity Function f(x)=x

18. Theorem 2 Limits of polynomial functions can be found by direct substitution.

19. Properties of Limits

20. If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c

21. Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Sum Rule: lim [f(x) + g(x)]= lim f(x) +lim g(x)=L1 + L2

22. Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Difference Rule: lim [f(x) - g(x)]= L1 - L2

23. Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Product Rule: lim [f(x) * g(x)]= L1 * L2

24. Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Constant multiple Rule: lim c f(x) = c L1

25. Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Quotient Rule: lim [f(x) / g(x)]= L1 / L2, L1=0 NOT

26. Theorem 3 Many ( not all ) limits of rational functions can be found by direct substitution.

27. Right-hand and Left-hand Limits

28. Theorem 4 A function, f(x), has a limit as x approaches c

29. Theorem 4 A function, f(x), has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist

30. Theorem 4 A function, f(x), has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal.

31. Calculus 2.2 Continuity

32. Definition f(x) is continuous at an interior point of the domain if

33. Definition f(x) is continuous at an interior point of the domain if lim f(x) = f(c ) x->c

34. Definition f(x) is continuous at an endpoint of the domain if

35. A “continuous” function is continuous at each point of its domain.

36. Definition Discontinuity If a function is not continuous at a point c, then c is called a point of discontinuity.

37. Types of Discontinuities Removable

38. Types of Discontinuities Removable Non-removable A) jump

39. Types of Discontinuities Removable Non-removable A) jump B) oscillating

40. Types of Discontinuities Removable Non-removable A) jump B) oscillating C) infinite

41. Test for Continuity

42. y=f(x) is continuous at x=c iff 1.

43. Test for Continuity y=f(x) is continuous at x=c iff 1. f(c) exists

44. Test for Continuity y=f(x) is continuous at x=c iff 1. f(c) exists 2. lim f(x) exists x-> c

45. Test for Continuity y=f(x) is continuous at x=c iff 1. f(c) exists 2. lim f(x) exists x -> c 3. f(c ) = lim f(x) x -> c

46. Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x)

47. Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x)

48. Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x) 3. f (x) g(x)

49. Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x) 3. f (x) g(x) 4. k g(x)

50. Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x) 3. f (x) g(x) 4. k g(x) 5. f(x)/g(x), g(x)/=0 are continuous

51. Theorem 6 If f and g are continuous at c, Then g f and f g are continuous at c

52. Theorem 7 If f(x) is continuous on [a,b], then f(x) has an absolute maximum,M, and an absolute minimum,m, on [a,b].

53. Intermediate Value Theorem for continuous functions A function that is continuous on [a,b] takes on every value between f(a) and f(b).

54. Calculus 2.3 The Sandwich Theorem

55. If g(x) < f(x) < h(x) for all x /=c and lim g(x) = lim h(x) = L, then lim f(x) = L.

56. Use sandwich theorem to find lim sin x x->0 x

57. Sandwich theorem examples So you can see the light.

58. Calculus 2.4 Limits Involving Infinity

59. Limits at + infinity are also called “end behavior” models for the function.

60. Definition y=b is a horizontal asymptote of f(x) if

61. Horizontal Tangents Case 1 degree of numerator < degree of denominator

62. Case 2 degree of numerator = degree of denominator

63. Case 3 degree of numerator > degree of denominator

64. Theorem Polynomial End Behavior Model

65. Calculus 2.6 The Formal Definition of a Limit

66. Now this is mathematics!!!