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Slide 3- 1

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Chapter 3 Exponential, Logistic, and Logarithmic Functions

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3.1 Exponential and Logistic Functions

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Slide 3- 4 Quick Review

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Slide 3- 5 Quick Review Solutions

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Slide 3- 6 What you’ll learn about Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models … and why Exponential and logistic functions model many growth patterns, including the growth of human and animal populations.

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Slide 3- 7 Exponential Functions

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Slide 3- 8 Exponential Functions Rules For Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

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Slide 3- 9 Use the rules for exponents to solve for x 4 x = 128 (2) 2x = 2 7 2x = 7 x = 7/2 2 x = 1/32 2 x = 2 -5 x = -5 Exponential Functions

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Slide 3- 10 (x 3 y 2/3 ) 1/2 x 3/2 y 1/3 27 x = 9 -x+1 (3 3 ) x = (3 2 ) -x+1 3 3x = 3 -2x+2 3x = -2x+ 2 5x = 2 x = 2/5 Exponential Functions

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Slide 3- 11 Example Finding an Exponential Function from its Table of Values

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Slide 3- 12 Example Finding an Exponential Function from its Table of Values

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Slide 3- 13 5432154321 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x y = 2 x If b > 1, then the graph of b x will: Rise from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

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Slide 3- 14 5432154321 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x y = ( 1 / 2 ) x If 0 < b < 1, then the graph of b x will: Fall from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

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Slide 3- 15 Example Transforming Exponential Functions Describe how to transform the graph of f(x) = 2 x into the graph g(x) = 2 x-2 The graph of g(x) = 2 x-2 is obtained by translat ing the graph of f(x) = 2 x by 2 units to the right.

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Slide 3- 16 Example Transforming Exponential Functions

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Slide 3- 17 Example Transforming Exponential Functions

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Slide 3- 18 The Natural Base e

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Slide 3- 19 Exponential Functions and the Base e

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Slide 3- 20 Exponential Functions and the Base e

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Slide 3- 21 Example Transforming Exponential Functions

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Slide 3- 22 Example Transforming Exponential Functions

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Slide 3- 23 Logistic Growth Functions

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Slide 3- 24 Exponential Growth and Decay

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Slide 3- 25 Exponential Functions Definitions Exponential Growth and Decay The function y = k a x, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. y new amount y O original amount b base t time h half life

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Slide 3- 26 Exponential Functions An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. (a)Find the amount remaining after t hours. (b)Find the amount remaining after 60 hours. a. y = y o b t/h y = 2 (1/2) (t/15) b. y = y o b t/h y = 2 (1/2) (60/15) y = 2(1/2) 4 y =.125 g

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Slide 3- 27 Exponential Functions A bacteria double every three days. There are 50 bacteria initially present (a)Find the amount after 2 weeks. (b)When will there be 3000 bacteria? a. y = y o b t/h y = 50 (2) (14/3) y = 1269 bacteria

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Slide 3- 28 Exponential Functions A bacteria double every three days. There are 50 bacteria initially present When will there be 3000 bacteria? b. y = y o b t/h 3000 = 50 (2) (t/3) 60 = 2 t/3

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3.2 Exponential and Logistic Modeling

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Slide 3- 30 Quick Review

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Slide 3- 31 Quick Review Solutions

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Slide 3- 32 What you’ll learn about Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

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Slide 3- 33 Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.

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Slide 3- 34 Exponential Population Model

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Slide 3- 35 Example Finding Growth and Decay Rates

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Slide 3- 36 Example Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.

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Slide 3- 37 Example Modeling Bacteria Growth

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Slide 3- 38 Example Modeling Bacteria Growth

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Slide 3- 39 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

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Slide 3- 40 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

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Slide 3- 41 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.

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Slide 3- 42 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor which spreads logistically so that s(t) = 1500/(1 + 29 e.-.09t ) models the number of students who have heard the rumor at the end of t days, where t = 0 is the day the rumor begins to spread (a)How many students have heard the rumor by the end of Day 0? (b)How long does it take for 1000 students to hear the rumor?

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Slide 3- 43 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor which spreads logistically so that s(t) = 1500/(1 + 29 e.-.09t ) models the number of students who have heard the rumor at the end of t days, where t = 0 is the day the rumor begins to spread (a)How many students have heard the rumor by the end of Day 0? (b)How long does it take for 1000 students to hear the rumor?

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3.3 Logarithmic Functions and Their Graphs

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Slide 3- 45 Quick Review

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Slide 3- 46 What you’ll learn about Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

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Slide 3- 47 Logarithmic Functions The inverse of an exponential function is called a logarithmic function. Definition: x = a y if and only if y = log a x

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Slide 3- 48 Changing Between Logarithmic and Exponential Form

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Slide 3- 49 Logarithmic Functions log 4 16 = 2 ↔ 4 2 = 16 log 3 81 = 4 ↔ 3 4 = 81 log 10 100 = 2 ↔ 10 2 = 100

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Slide 3- 50 Inverses of Exponential Functions

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Slide 3- 51 Logarithmic Functions The function f (x) = log a x is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote: x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)

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Slide 3- 52 Basic Properties of Logarithms

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Slide 3- 53 An Exponential Function and Its Inverse

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Slide 3- 54 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.

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Slide 3- 55 Basic Properties of Common Logarithms

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Slide 3- 56 Example Solving Simple Logarithmic Equations

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Slide 3- 57 Basic Properties of Natural Logarithms

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Slide 3- 58 Graphs of the Common and Natural Logarithm

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Slide 3- 59 Example Transforming Logarithmic Graphs

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Slide 3- 60 Example Transforming Logarithmic Graphs

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Slide 3- 61 Decibels

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3.4 Properties of Logarithmic Functions

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Slide 3- 63 Quick Review

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Slide 3- 64 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.

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Slide 3- 65 1. log a (a x ) = x for all x 2. a log a x = x for all x > 0 3. log a (xy) = log a x + log a y 4. log a (x/y) = log a x – log a y 5. log a x n = n log a x Common Logarithm: log 10 x = log x Natural Logarithm: log e x = ln x All the above properties hold. Logarithmic Functions

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Slide 3- 66 Properties of Logarithms

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Slide 3- 67 Example Proving the Product Rule for Logarithms

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Slide 3- 68 Example Proving the Product Rule for Logarithms

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Slide 3- 69 Example Expanding the Logarithm of a Product

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Slide 3- 70 Example Expanding the Logarithm of a Product

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Slide 3- 71 Example Condensing a Logarithmic Expression

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Slide 3- 72 Example Condensing a Logarithmic Expression

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Slide 3- 73 Logarithmic Functions Product Rule

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Slide 3- 74 Quotient Rule Logarithmic Functions

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Slide 3- 75 Power Rule Logarithmic Functions

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Slide 3- 76 Expand Logarithmic Functions

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Slide 3- 77 Change-of-Base Formula for Logarithms

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Slide 3- 78 Example Evaluating Logarithms by Changing the Base

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Slide 3- 79 Example Evaluating Logarithms by Changing the Base

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3.5 Equation Solving and Modeling

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Slide 3- 81 Quick Review

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Slide 3- 82 Quick Review Solutions

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Slide 3- 83 What you’ll learn about Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression … and why The Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.

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Slide 3- 84 One-to-One Properties

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Slide 3- 85 Example Solving an Exponential Equation Algebraically

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Slide 3- 86 Example Solving an Exponential Equation Algebraically

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Slide 3- 87 Example Solving a Logarithmic Equation

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Slide 3- 88 Example Solving a Logarithmic Equation

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Slide 3- 89 Solving Exponential Equations To solve exponential equations, pick a convenient base (often base 10 or base e) and take the log of both sides. Solve:

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Slide 3- 90 Take the log of both sides: Power rule: Solving Exponential Equations

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Slide 3- 91 Solve for x: Divide: Solving Exponential Equations

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Slide 3- 92 To solve logarithmic equations, write both sides of the equation as a single log with the same base, then equate the arguments of the log expressions. Solve: Solving Exponential Equations

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Slide 3- 93 Write the left side as a single logarithm: Solving Exponential Equations

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Slide 3- 94 Equate the arguments: Solving Exponential Equations

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Slide 3- 95 Solve for x: Solving Exponential Equations

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Slide 3- 96 Solving Exponential Equations

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Slide 3- 97 Check for extraneous solutions. x = -3, since the argument of a log cannot be negative Solving Exponential Equations

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Slide 3- 98 To solve logarithmic equations with one side of the equation equal to a constant, change the equation to an exponential equation Solve: Solving Exponential Equations

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Slide 3- 99 Write the left side as a single logarithm: Solving Exponential Equations

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Slide 3- 100 Write as an exponential equation: Solving Exponential Equations

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Slide 3- 101 Solve for x: Solving Exponential Equations

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Slide 3- 102 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.

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Slide 3- 103 Richter Scale

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Slide 3- 104 5.5 Graphs of Logarithmic Functions What is the magnitude on the Richter scale of an earthquake if a = 300, T = 30 and B = 1.2?

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Slide 3- 105 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.

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Slide 3- 106 Newton’s Law of Cooling

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Slide 3- 107 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?

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Slide 3- 108 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?

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Slide 3- 109 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

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Slide 3- 110 Three Types of Logarithmic Re-Expression

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Slide 3- 111 Three Types of Logarithmic Re-Expression (cont’d)

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Slide 3- 112 Three Types of Logarithmic Re-Expression (cont’d)

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3.6 Mathematics of Finance

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Slide 3- 114 Quick Review

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Slide 3- 115 Quick Review Solutions

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Slide 3- 116 What you’ll learn about Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value … and why The mathematics of finance is the science of letting your money work for you – valuable information indeed!

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Slide 3- 117 Interest Compounded Annually

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Slide 3- 118 Interest Compounded k Times per Year

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Slide 3- 119 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

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Slide 3- 120 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

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Slide 3- 121 Compound Interest – Value of an Investment

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Slide 3- 122 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

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Slide 3- 123 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

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Slide 3- 124 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.

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Slide 3- 125 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

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Slide 3- 126 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

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Slide 3- 127 Future Value of an Annuity

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Slide 3- 128 Present Value of an Annuity

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Slide 3- 129 Chapter Test

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Slide 3- 130 Chapter Test

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Slide 3- 131 Chapter Test

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Slide 3- 132 Chapter Test Solutions

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Slide 3- 133 Chapter Test Solutions

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Slide 3- 134 Chapter Test Solutions

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