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Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All.

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Presentation on theme: "Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All."— Presentation transcript:

1 Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved.

2 Chapter 11  Differential Calculus

3 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.1  Limits

4 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

5 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

6 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

7 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. f(x) = x 2 + x + 1, find

8 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

9 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

10 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

11 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

12 Slide Find Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Note that So the limit of as x approaches 5 is the value of the function at 5: Quotient property Polynomial limit

13 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

14 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

15 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

16 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises

17 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Review

18 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.2  One-Sided Limits and Limits Involving Infinity

19 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

20 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

21 Slide Find each of the given limits. Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: (a) Since is not defined when the right- hand limit (which requires that does not exist. For the left-hand limit, write the square root in exponential form and apply the appropriate limit properties. Exponential form Power property Polynomial limit

22 Slide Find each of the given limits. Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: (b) Exponential form Power property Polynomial limits Sum property

23 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Examples

24 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Continued on next slide

25 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Continued from previous slide

26 Slide Infinite Limits Copyright ©2015 Pearson Education, Inc. All right reserved.

27 Slide Examples Copyright ©2015 Pearson Education, Inc. All right reserved.

28 Slide Review Copyright ©2015 Pearson Education, Inc. All right reserved. 1) 5) 4) 3) 2)

29 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.3  Rates of Change

30 Slide This is the situation described by the expression with The average rate of change is Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: If find the average rate of change of with respect to x as x changes from –2 to 3.

31 Slide Example Copyright ©2015 Pearson Education, Inc. All right reserved.

32 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

33 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

34 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example Exercise

35 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. The rate of change of the cost function is called the marginal cost. Similarly, the rate of change of the revenue and profit function are called the marginal revenue and marginal profit.

36 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises 1) 2)

37 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.4  Tangent Lines and Derivatives

38 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Secant lines

39 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

40 Slide According to the definition, the slope of the tangent line is Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Let a be any real number. Find the equation of the tangent line to the graph of at the point where

41 Slide Hence, the equation of the tangent line at the point is Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Let a be any real number. Find the equation of the tangent line to the graph of at the point where Thus, the tangent line is the graph of

42 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. 1) 3) 2)

43 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. If f’(x) is defined, then the function f is said to be differentiable at x. The process that produces f’ from f is called differentiation.

44 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

45 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Examples 1) 2) 3) 4) Label f and f’ correctly:

46 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Existence of the Derivative

47 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Select the graph of f’

48 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercise

49 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.5  Techniques for Finding Derivatives

50 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

51 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

52 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

53 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

54 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

55 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

56 Slide From the function p, the revenue function is given by Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The demand function for a certain product is given by The marginal revenue is Find the marginal revenue when units and p is in dollars.

57 Slide When the marginal revenue is Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The demand function for a certain product is given by or $1.20 per unit. Thus, the next unit sold (at sales of 10,000) will produce an additional revenue of about $1.20. Find the marginal revenue when units and p is in dollars.

58 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercise

59 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.6  Derivatives of Products and Quotients

60 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Derivative of product of 3 functions: [u(x)  v(x)  w(x)]’ = u’(x)  v(x)  w(x) + u(x)  v’(x)  w(x) + u(x)  v(x)  w’(x)

61 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Proof [f(x)g(x)]’

62 Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example:

63 Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Find the derivative of

64 Slide Example Copyright ©2015 Pearson Education, Inc. All right reserved. Given that: (sin(x))’ = cos(x) and (cos(x))’ = -sin(x) Find derivative of

65 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

66 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Proof

67 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example

68 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Complete the table without using the Quotient Rule

69 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The cost in dollars of manufacturing x hundred items is given by: C(x) = 4x 2 + 6x + 5 a)Find the average cost b)Find the marginal average cost c)Find the marginal cost d)Find the level of production at which the marginal average cost is zero.

70 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises y = 1 Use quotient rule to prove the Power Rule for negative integers n. i.e. (x n )’ = nx n-1 for negative integers n. Find derivative of g(x).

71 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Identify each graph. Sketch the graph of f’

72 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Review 1)Write equation of the tangent line to the graph of function f(x) = x 2 at (1, 1) 2)Write equation of the tangent line to the graph that passes through the point (-1, -1) f(x) = x 2

73 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.7  The Chain Rule

74 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example: find functions f and g so that f[g(x)] are the followings: f(x) = g(x) = Composite function Let f and g be functions. The composite function, or composition, of f and g is the function whose values are given by f[g(x)] for all x in the domain of g such that g(x) is in the domain of f.

75 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example: find functions f and g so that f[g(x)] are the followings:

76 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. y

77 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Proof c is in the domain of h

78 Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Use the chain rule to find

79 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

80 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Example:

81 Slide Summary of Differentiation Rules Copyright ©2015 Pearson Education, Inc. All right reserved.

82 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Exercises

83 Slide Exercises: find derivative of functions below. Copyright ©2015 Pearson Education, Inc. All right reserved. Find the equation of the tangent line to the graph of f at the given point.

84 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Label the graph as f or f’:

85 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.8  Derivatives of Exponential and Logarithmic Functions

86 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

87 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Proof: Example: y = e 2x – 1. Find y’.

88 Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Let Find Use the quotient rule: Exercise: for a > 0, y = a x. Find y’.

89 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

90 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Proof:

91 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Examples: find the derivatives

92 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. If y = ln|u|, then y’ = Example: y = log a (x)

93 Copyright ©2015 Pearson Education, Inc. All right reserved. Section 11.9  Continuity and Differentiability

94 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

95 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

96 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

97 Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Is the function in the following graph continuous on the given x-intervals? (a) The function g is discontinuous only at Hence, g is continuous at every point of the open interval which does not include 0 or 2. Solution: (b) The function g is not defined at so it is not continuous from the right there. Therefore, it is not continuous on the closed interval

98 Slide Solution: Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Is the function in the following graph continuous on the given x-intervals? (c) The interval contains a point of discontinuity at So g is not continuous on the open interval Solution: (d) The function g is continuous on the open interval continuous the right at and continuous from the left at Hence, g is continuous on the closed interval

99 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Proof:

100 Slide Copyright ©2015 Pearson Education, Inc. All right reserved.

101 Slide Copyright ©2015 Pearson Education, Inc. All right reserved. Match each function with each graph


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