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Chapter 3 Exponential, Logistic, and Logarithmic Functions

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Slide 3- 2 Quick Review

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Slide 3- 3 Quick Review Solutions

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Slide 3- 4 Exponential Functions

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Determine if they are exponential functions

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Answers Yes No Yes no

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Sketch an exponential function

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

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

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Slide Exponential Growth and Decay

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Sketch exponential graph and determine if they are growth or decay

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Slide Example Transforming Exponential Functions

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Slide Example Transforming Exponential Functions

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Slide Example Transforming Exponential Functions

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Group Activity

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Slide The Natural Base e

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Slide Exponential Functions and the Base e

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Slide Exponential Functions and the Base e

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Slide Example Transforming Exponential Functions

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Slide Example Transforming Exponential Functions

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Slide Logistic Growth Functions

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Example: Graph and Determine the horizontal asymptotes

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

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Group Work: Graph and determine the horizontal asymptotes

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

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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)

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

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Word Problem

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

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Group Work

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Homework Practice P 286 #1-54 eoe

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EXPONENTIAL AND LOGISTIC MODELING

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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

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Slide Quick Review

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Slide Quick Review Solutions

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Slide Exponential Population Model

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Example:

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Slide Example Finding Growth and Decay Rates

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Example

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

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Answer

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Slide Example Modeling Bacteria Growth

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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.

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answer

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Group Work

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

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Slide Example Modeling a Rumor

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Slide Example Modeling a Rumor: Answer

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

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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.

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

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LOGARITHMIC FUNCTION, GRAPHS AND PROPERTIES

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Slide Quick Review

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Slide Quick Review Solutions

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Slide Changing Between Logarithmic and Exponential Form

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Group Work: transform logarithmic form into exponential form

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Group Work: convert exponential form into logarithmic form

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Slide Inverses of Exponential Functions

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Slide Basic Properties of Logarithms

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Slide An Exponential Function and Its Inverse

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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.

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Slide Basic Properties of Common Logarithms

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Slide Example Solving Simple Logarithmic Equations

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Slide Example Solving Simple Logarithmic Equations

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Slide Basic Properties of Natural Logarithms

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Slide Graphs of the Common and Natural Logarithm

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Slide Example Transforming Logarithmic Graphs

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Slide Example Transforming Logarithmic Graphs

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Slide Quick Review

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Slide Quick Review Solutions

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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.

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Slide Properties of Logarithms

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Slide Example Proving the Product Rule for Logarithms

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Slide Example Proving the Product Rule for Logarithms

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Slide Example Expanding the Logarithm of a Product

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Slide Example Expanding the Logarithm of a Product

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Slide Example Condensing a Logarithmic Expression

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Slide Example Condensing a Logarithmic Expression

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Group Work

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Slide Change-of-Base Formula for Logarithms

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Slide Example Evaluating Logarithms by Changing the Base

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Slide Example Evaluating Logarithms by Changing the Base

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Solving

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Homework Practice Pg 317 #1-50 eoe

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EQUATION SOLVING AND MODELING

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Slide Quick Review

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Slide Quick Review Solutions

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Slide One-to-One Properties

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Slide Example Solving an Exponential Equation Algebraically

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Slide Example Solving an Exponential Equation Algebraically

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Slide Example Solving a Logarithmic Equation

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Slide Example Solving a Logarithmic Equation

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Group Work

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Group Work: Solve for x

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Group Work: Solve

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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.

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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.

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

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

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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

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

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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

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Answer 6 orders of magnitude 2 orders of magnitude

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Slide Richter Scale

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Example:

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Group Work: Show work

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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.

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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?

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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?

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Slide Newton’s Law of Cooling

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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?

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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?

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Group Work

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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

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Slide Three Types of Logarithmic Re- Expression

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

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

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Homework Practice Pg 331 #1-51 eoe

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MATHEMATICS OF FINANCE

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Slide Interest Compounded Annually

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Slide Interest Compounded k Times per Year

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Slide 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 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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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.

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Slide Compound Interest – Value of an Investment

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Slide 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 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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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.

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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.

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Slide 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 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 Future Value of an Annuity

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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

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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?

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Slide Present Value of an Annuity

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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%?

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Homework Practice Pg 341 #2-56 eoe

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