1. Announcements Topics: -finish section 4.2; work on sections 4.3, 4.4, and 4.5 * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

2. Evaluating Limits We can evaluate the limit of a function in 3 ways: 1.Graphically 2.Numerically 3.Algebraically

3. Evaluating Limits Algebraically BASIC LIMITS Limit of a Constant Function Example: Limit of the Identity Function Example:

4. LIMIT LAWS [used to evaluate limits algebraically] Suppose that c is a constant and the limits exist. Then 1. 2. 3.

5. LIMIT LAWS [used to evaluate limits algebraically] Continued… 4. 5.

6. Evaluating Limits Algebraically Example: Evaluate the limit and justify each step by indicating the appropriate Limit Laws.

7. Evaluating Limits Algebraically Example: Evaluate the limit and justify each step by indicating the appropriate Limit Laws.

8. Evaluating Limits Algebraically Example: Evaluate the limit and justify each step by indicating the appropriate Limit Laws.

9. Evaluating Limits Algebraically Example: Evaluate the limit and justify each step by indicating the appropriate Limit Laws.

10. Direct Substitution Property From the previous slide, we have Notice that we could have simply found the value of the limit by plugging in x=1 into the function.

11. Direct Substitution Property Direct Substitution Property: If f(x) is an algebraic, exponential, logarithmic, trigonometric, or inverse trigonometric function, and is in the domain of f(x), then

12. Equal Limits Property Consider the functions: * Note: f(x)=g(x) everywhere except at x=2

13. Equal Limits Property Example: Calculate FACT: If when, then provided the limits exist. Note: direct substitution does not work

14. Strategy for Evaluating Limits

15. Evaluating Limits Algebraically Evaluate each limit or state that it does not exist. (a)(b) (c) (d)

16. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.1 0.01 0.001 0undefined -0.001 -0.01 -0.1

17. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01 0.001 0undefined -0.001 -0.01 -0.1

18. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01100 0.001 0undefined -0.001 -0.01 -0.1

19. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01100 0.0011000 0undefined -0.001 -0.01 -0.1

20. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01100 0.0011000 0undefined -0.001 -0.01 -0.1

21. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01100 0.0011000 0undefined -0.001-1000 -0.01-100 -0.1-10

22. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01100 0.0011000 0undefined -0.001-1000 -0.01-100 -0.1-10

23. Infinite Limits Example: Use a table of values to estimate the value of xf(x) 0.110 0.01100 0.0011000 0undefined -0.001-1000 -0.01-100 -0.1-10

24. Infinite Limits Definition: “the limit of f(x), as x approaches a, is infinity” means that the values of f(x) (y-values) increase without bound as x becomes closer and closer to a (from either side of a), but x a. Definition: “the limit of f(x), as x approaches a, is negative infinity” means that the values of f(x) (y-values) decrease without bound as x becomes closer and closer to a (from either side of a), but x a.

25. Infinite Limits Definition: “the limit of f(x), as x approaches a, is infinity” means that the values of f(x) (y-values) increase without bound as x becomes closer and closer to a (from either side of a), but x a. Definition: “the limit of f(x), as x approaches a, is negative infinity” means that the values of f(x) (y-values) decrease without bound as x becomes closer and closer to a (from either side of a), but x a.

26. Infinite Limits Example: Determine the infinite limit. (a) (b) Note: Since the values of these functions do not approach a real number L, these limits do not exist.

27. Vertical Asymptotes Definition: The line x=a is called a vertical asymptote of the curve y=f(x) if either Example: Basic functions we know that have VAs:

28. Limits at Infinity The behaviour of functions “at” infinity is also known as the end behaviour or long-term behaviour of the function. What happens to the y-values of a function f(x) as the x-values increase or decrease without bounds?

29. Limits at Infinity Possibility: y-values also approach infinity or - infinity Examples:.

30. Limits at Infinity Possibility: y-values also approach infinity or - infinity Examples:.

31. Limits at Infinity Possibility: y-values approach a unique real number L Examples:.

32. Limits at Infinity Possibility: y-values approach a unique real number L Examples:.

33. Limits at Infinity Possibility: y-values oscillate and do not approach a single value Example:.

34. Limits at Infinity Possibility: y-values oscillate and do not approach a single value Example:.

35. Limits at Infinity Definition: “the limit of f(x), as x approaches, equals L” means that the values of f(x) (y-values) can be made as close as we’d like to L by taking x sufficiently large. Definition: “the limit of f(x), as x approaches, equals L” means that the values of f(x) (y-values) can be made as close as we’d like to L by taking x sufficiently small.

36. Limits at Infinity Definition: “the limit of f(x), as x approaches, equals L” means that the values of f(x) (y-values) can be made as close as we’d like to L by taking x sufficiently large. Definition: “the limit of f(x), as x approaches, equals L” means that the values of f(x) (y-values) can be made as close as we’d like to L by taking x sufficiently small.

37. Calculating Limits at Infinity *The Limit Laws listed previously are still valid if “ ” is replaced by “ ” Limit Laws for Infinite Limits (abbreviated): where c is any non-zero constant

38. Calculating Limits at Infinity Theorem: If r>0 is a rational number, then If r>0 is a rational number such that is defined for all x, then

39. Calculating Limits at Infinity Examples: Find the limit or show that it does not exist. (a) (b)

40. Horizontal Asymptotes Definition: The line y=L is called a horizontal asymptote of the curve y=f(x) if either Example: Basic functions we know that have HAs:.

41. Limits at Infinity What about the limits at infinity of these functions? (a)(b) Which part (top or bottom) goes to infinity faster?

42. Limits at Infinity

43. Comparing Functions That Approach at Suppose and 1.f(x) approaches infinity faster than g(x) if 2.f(x) approaches infinity slower than g(x) if 3.f(x) and g(x) approach infinity at the same rate if where L is any finite number other than 0.

44. Comparing Functions That Approach at The Basic Functions in Increasing Order of Speed Note: The constant can be any positive number and does not change the order of the functions. FunctionComments Goes to infinity slowly withApproaches infinity faster for larger withApproaches infinity faster for larger

45. Comparing Functions That Approach at

46. Limits at Infinity What about the limits at infinity of these functions? (a)(b) Which part (top or bottom) goes to 0 faster?

47. Limits at Infinity Semilog Graphs

48. Comparing Functions That Approach at Suppose and 1.f(x) approaches 0 faster than g(x) if 2.f(x) approaches 0 slower than g(x) if 3.f(x) and g(x) approach 0 at the same rate if where L is any finite number other than 0.

49. Comparing Functions That Approach at The Basic Functions in Increasing Order of Speed Note: Again, a can be any positive constant and this will not affect the ordering. FunctionComments withApproaches 0 faster for larger withApproaches 0 faster for larger withApproaches 0 really fast

50. Comparing Functions That Approach at

51. Semilog Graphs

52. Continuity Intuitive idea: A process is continuous if it takes place without interruptions or an abrupt change. Geometrically, a function is continuous if it’s graph has no break in it.

53. Continuity Definition: A function f is continuous at the point x=a if f(x) approaches f(a) as x approaches a, i.e.

54. Continuity Implicitly requires 3 things: 1. exists 2. is defined 3. If f is not continuous at a (i.e. f fails to meet at least one of the three conditions above), then we say that f is discontinuous at x=a.

55. Continuity Example: Find the discontinuities of the function and explain why it is discontinuous there.. Start by looking at x-values where f(x) is not defined and then check the 3 conditions of continuity.

56. Continuity Example: Find the discontinuities of the function and explain why it is discontinuous there.. Start by looking at x-values where f(x) is changes from one ‘piece’ to another and then check the 3 conditions of continuity.

57. Which Functions Are Continuous? Definition: A function is said to be continuous if it is continuous at every point in its domain. Basic Continuous Functions: polynomials ex: rational functions ex: root functions ex:

58. Which Functions Are Continuous? Basic Continuous Functions: algebraic functions ex: absolute value function exponential and logarithmic functions ex: trigonometric and inverse trigonometric functions ex:

59. Which Functions Are Continuous? Combining Continuous Functions: The sum, difference, product, quotient, and composition of continuous functions is continuous where defined. Example: Determine where is continuous..

60. Limits of Continuous Functions Example: Evaluate Note: By the definition of continuity, if a function is continuous at x=a, then we can evaluate the limit simply by direct substitution.

61. Rates of Change The rate of change of a function tells us how the dependent variable changes when there is a change in the independent variable. Geometrically, the rate of change of a function corresponds to the slope of it’s graph.

62. Average Rate of Change = Slope of Secant Line The average rate of change of f(t) from t=t 1 to t=t 2 corresponds to the slope of the secant line PQ.

63. Average Rate of Change = Slope of Secant Line Alternative Notation: The average rate of change of f(t) from the base point t=t 0 to t=t 0 +Δt is

64. Average Rate of Change = Slope of Secant Line Example: Find the average rate of change of over the interval [-1, 1]..

65. Instantaneous Rate of Change = Slope of Tangent Line The instantaneous rate of change of f(t) at t=t 0 corresponds to the slope of the tangent line at t=t 0. Note: The slope of the curve y=f(t) at P is the slope of its tangent line at P.

66. Instantaneous Rate of Change = Slope of Tangent Line This special limit is called the derivative of f at t 0 and is denoted by f’(t 0 ) (read “f prime of t 0 ”). Alternative notation:

67. Instantaneous Rate of Change = Slope of Tangent Line Example: (a) Determine the instantaneous rate of change of h(t) when t=1. (b) Use this to find the equation of the tangent line to h(t) at t=1..

68. The Derivative Function Definition: Given a function f(x), the derivative of f with respect to x is the function f’(x) defined by The domain of this function is the set of all x- values for which the limit exists.

69. The Derivative Function Definition: Given a function f(x), the derivative of f with respect to x is the function f’(x) defined by The domain of this function is the set of all x- values for which the limit exists.

70. The Derivative Function Interpretations of f’: 1.The function f’(x) tells us the instantaneous rate of change of f(x) with respect to x for all x-values in the domain of f’(x). 2. The function f’(x) tells us the slope of the tangent to the graph of f(x) at every point (x, f(x)), provided x is in the domain of f’(x).

71. The Derivative Function Example: Find the derivative of and use it to calculate the instantaneous rate of change of f(x) at x=1. Sketch the curve f(x) and the tangent to the curve at (1,2).

72. The Derivative Function Example: Find the derivative of Sketch the graph of f(x) and the graph of f’(x).

73. Relationship between f’ and f If f is increasing on an interval (c,d): The derivative f’ is positive on (c,d). The rate of change of f is positive for all x in (c,d). The slope of the tangent is positive for all x in (c,d). If f is decreasing on an interval (c,d): The derivative f’ is negative on (c,d). The rate of change of f is negative for all x in (c,d). The slope of the tangent is negative for all x in (c,d).

74. Critical Numbers Definition: c is a critical number of f if c is in the domain of f and either f’(c)=0 or f’(c) D.N.E.

75. Differentiable Functions A function f(x) is said to be differentiable at x=a if we are able to calculate the derivative of the function at that point, i.e., f(x) is differentiable at x=a if exists.

76. Differentiable Functions Geometrically, a function is differentiable at a point if its graph has a unique tangent line with a well- defined slope at that point. 3 Ways a Function Can Fail to be Differentiable:

77. Graphs Example: (a)Sketch the graph of (b) By looking at the graph of f, sketch the graph of f’(x).

78. Relationship Between Differentiability and Continuity If f is differentiable at a, then f is continuous at a.