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Chapter 3 Exponential, Logistic, and Logarithmic Functions.

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Presentation on theme: "Chapter 3 Exponential, Logistic, and Logarithmic Functions."— Presentation transcript:

1 Chapter 3 Exponential, Logistic, and Logarithmic Functions

2 Slide 3- 2 Quick Review

3 Slide 3- 3 Quick Review Solutions

4 Slide 3- 4 Exponential Functions

5 Determine if they are exponential functions

6 Answers Yes No Yes no

7 Sketch an exponential function

8 Slide 3- 8 Example Finding an Exponential Function from its Table of Values

9 Slide 3- 9 Example Finding an Exponential Function from its Table of Values

10 Slide Exponential Growth and Decay

11 Sketch exponential graph and determine if they are growth or decay

12 Slide Example Transforming Exponential Functions

13 Slide Example Transforming Exponential Functions

14 Slide Example Transforming Exponential Functions

15 Group Activity

16 Slide The Natural Base e

17 Slide Exponential Functions and the Base e

18 Slide Exponential Functions and the Base e

19 Slide Example Transforming Exponential Functions

20 Slide Example Transforming Exponential Functions

21 Slide Logistic Growth Functions

22 Example: Graph and Determine the horizontal asymptotes

23 Answer Horizontal asymptotes at y=0 and y=7 Y-intercept at (0,7/4)

24 Group Work: Graph and determine the horizontal asymptotes

25 Answer Horizontal asymptotes y=0 and y=26 Y-intercept at (0,26/3)

26 Word Problems: Year ,248 people Year ,135 people Use this information to determine when the population will surpass 1 million people? (hint use exponential function)

27 Group Work Year ,530 people Year ,365 people Use this information and determine when the population will surpass 1.5 million people?

28 Word Problem

29 Answer A) 1,794,558 B) 19,161,673 C) 19,875,000

30 Group Work

31 Homework Practice P 286 #1-54 eoe

32 EXPONENTIAL AND LOGISTIC MODELING

33 Review We learned that how to write exponential functions when given just data. Now what if you are given other type of data? That would mean some manipulation

34 Slide Quick Review

35 Slide Quick Review Solutions

36 Slide Exponential Population Model

37 Example:

38 Slide Example Finding Growth and Decay Rates

39 Example

40 Slide Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year.

41 Group Work Suppose 50 bacteria is put into a petri dish and it doubles every hour. When will the bacteria be 350,000?

42 Answer

43 Slide Example Modeling Bacteria Growth

44 Group Work: half-life Suppose the half-life of a certain radioactive substance is 20 days and there are 10g initially. Find the time when there will be 1 g of the substance.

45 answer

46 Group Work

47 Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.

48 Slide Example Modeling a Rumor

49 Slide Example Modeling a Rumor: Answer

50 Key Word Maximum sustainable population What does this mean? What function deals with this?

51 Slide Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.

52 Homework Practice (Do in class also) P 296 #1-44 eoo

53 LOGARITHMIC FUNCTION, GRAPHS AND PROPERTIES

54 Slide Quick Review

55 Slide Quick Review Solutions

56 Slide Changing Between Logarithmic and Exponential Form

57 Group Work: transform logarithmic form into exponential form

58 Group Work: convert exponential form into logarithmic form

59 Slide Inverses of Exponential Functions

60 Slide Basic Properties of Logarithms

61 Slide An Exponential Function and Its Inverse

62 Slide Common Logarithm – Base 10 Logarithms with base 10 are called common logarithms. The common logarithm log 10 x = log x. The common logarithm is the inverse of the exponential function y = 10 x.

63 Slide Basic Properties of Common Logarithms

64 Slide Example Solving Simple Logarithmic Equations

65 Slide Example Solving Simple Logarithmic Equations

66 Slide Basic Properties of Natural Logarithms

67 Slide Graphs of the Common and Natural Logarithm

68 Slide Example Transforming Logarithmic Graphs

69 Slide Example Transforming Logarithmic Graphs

70 Slide Quick Review

71 Slide Quick Review Solutions

72 Slide What you’ll learn about Properties of Logarithms Change of Base Graphs of Logarithmic Functions with Base b Re-expressing Data … and why The applications of logarithms are based on their many special properties, so learn them well.

73 Slide Properties of Logarithms

74 Slide Example Proving the Product Rule for Logarithms

75 Slide Example Proving the Product Rule for Logarithms

76 Slide Example Expanding the Logarithm of a Product

77 Slide Example Expanding the Logarithm of a Product

78 Slide Example Condensing a Logarithmic Expression

79 Slide Example Condensing a Logarithmic Expression

80 Group Work

81

82

83

84 Slide Change-of-Base Formula for Logarithms

85 Slide Example Evaluating Logarithms by Changing the Base

86 Slide Example Evaluating Logarithms by Changing the Base

87 Solving

88

89

90

91

92

93 Homework Practice Pg 317 #1-50 eoe

94 EQUATION SOLVING AND MODELING

95 Slide Quick Review

96 Slide Quick Review Solutions

97 Slide One-to-One Properties

98 Slide Example Solving an Exponential Equation Algebraically

99 Slide Example Solving an Exponential Equation Algebraically

100 Slide Example Solving a Logarithmic Equation

101 Slide Example Solving a Logarithmic Equation

102 Group Work

103 Group Work: Solve for x

104 Group Work: Solve

105

106 Slide Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: A kilometer is 3 orders of magnitude longer than a meter. A dollar is 2 orders of magnitude greater than a penny. New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.

107 Note: In regular cases, how you determine the magnitude is by how many decimal places they differ In term of Richter scale and pH level, since the number is the power or the exponent, you just take the difference of them.

108 Example: What’s the difference of the magnitude between kilometer and meter? It is 3 orders of magnitude longer than a meter

109 Example: The order of magnitude between an earthquake rated 7 and Richter scale rated 5.5. The difference of magnitude is 1.5

110 Group Work Find the order of magnitude: Between A dollar and a penny A horse weighing 500 kg and a horse weighing 50g 8 million people vs population of 8

111 Answer 2 orders of magnitude 4 orders of magnitude 6 orders of magnitude

112 Group Work Find the difference of the magnitude: Sour vinegar a pH of 2.4 and baking soda pH of 8.4 Earthquake in India 7.9 and Athens 5.9

113 Answer 6 orders of magnitude 2 orders of magnitude

114 Slide Richter Scale

115 Example:

116 Group Work: Show work

117 Slide pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H + ]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH=-log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

118 Example: Sour vinegar has pH of 2.4 and a box of Leg and Sickle baking soda has a pH of 8.4. A) what are their hydrogen-ion concentration? B) How many more times greater is the hydrogen-ion concentration of the vinegar than of the baking soda?

119 Group Work A substance with pH of 3.4 and another with pH of 8.1 A) what are their hydrogen-ion concentration? B) How many more times greater is the hydrogen-ion concentration?

120 Slide Newton’s Law of Cooling

121 Slide Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

122 Slide Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

123 Group Work

124 Slide Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + blnx Exponential regression:y = a·b x Power regression:y = a·x b

125 Slide Three Types of Logarithmic Re- Expression

126 Slide Three Types of Logarithmic Re- Expression (cont’d)

127 Slide Three Types of Logarithmic Re- Expression (cont’d)

128 Homework Practice Pg 331 #1-51 eoe

129 MATHEMATICS OF FINANCE

130 Slide Interest Compounded Annually

131 Slide Interest Compounded k Times per Year

132 Slide Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

133 Slide Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

134 Group Work Suppose you have $10000, you invest in a place where they give you 12% interest compounded quarterly. Find the value of your investment after 40 years.

135 Slide Compound Interest – Value of an Investment

136 Slide Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

137 Slide Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

138 Group Work Suppose you have $10000, you invest in a company where they give you 12% interest compounded continuously. Find the value of your investment after 40 years.

139 Slide Annual Percentage Yield A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.

140 Slide Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

141 Slide Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

142 Slide Future Value of an Annuity

143 At the end of each quarter year, Emily makes a $500 payment into the Lanaghan Mutual Fund. If her investments earn 7.88% annual interest compounded quarterly, what will be the value of Emily’s annuity in 20 years? Remember i=r/k

144 Group Work You are currently 18 and you want to retire at age 65. You decide to invest in your future. You are putting in $35 month. If your investment earn 12% annual interest compounded monthly, what will the value of your annuity when you retire?

145 Slide Present Value of an Annuity

146 Example Mr. Liu bought a new car for $ What are the monthly payment for a 5 year loan with 0 down payment if the annual interest rate (APR) is 2.9%?

147 Homework Practice Pg 341 #2-56 eoe


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