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**Chapter 3 Exponential, Logistic, and Logarithmic Functions**

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Quick Review

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**Quick Review Solutions**

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**Exponential Functions**

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**Determine if they are exponential functions**

𝑓 𝑥 = 4 𝑥 𝑔 𝑥 =6 𝑥 −9 𝑡 𝑥 =−2∗ 1.5 𝑥 ℎ 𝑥 =7∗ 3 −𝑥 𝑞 𝑥 =5∗ 6 3

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Answers Yes No no

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**Sketch an exponential function**

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**Example Finding an Exponential Function from its Table of Values**

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**Example Finding an Exponential Function from its Table of Values**

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**Exponential Growth and Decay**

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

𝑓 𝑥 = 2 𝑥 𝑔 𝑥 = 𝑥 ℎ 𝑥 = 4 −𝑥

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**Example Transforming Exponential Functions**

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**Example Transforming Exponential Functions**

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**Example Transforming Exponential Functions**

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**Group Activity Use this formula 1+ 1 𝑥 𝑥 Group 1 calculate when x=1**

What do you guys notice?

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The Natural Base e

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**Exponential Functions and the Base e**

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**Exponential Functions and the Base e**

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**Example Transforming Exponential Functions**

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**Example Transforming Exponential Functions**

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**Logistic Growth Functions**

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**Example: Graph and Determine the horizontal asymptotes**

𝑓 𝑥 = 7 1+3∗ .6 𝑥

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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**

𝐺(𝑥)= 𝑒 −4𝑥

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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 2000 782,248 people Year 2010 923,135 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 1990 156,530 people Year 2000 531,365 people**

Use this information and determine when the population will surpass 1.5 million people?

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**Word Problem The population of New York State can be modeled by**

𝑓 𝑡 = 𝑒 − 𝑡 𝑓 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑚𝑖𝑙𝑙𝑖𝑜𝑛𝑠 𝑎𝑛𝑑 𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1800 A) What’s the population in 1850? B) What’s the population in 2010? C) What’s the maximum sustainable population?

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

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Group Work In chemistry, you are given half-life formulas 𝑃 𝑡 = 𝑃 0 𝑏 𝑡 𝑟 𝑟=ℎ𝑎𝑙𝑓 𝑙𝑖𝑓𝑒, 𝑡=𝑡𝑖𝑚𝑒, 𝑃 0 =𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 If you are given a certain chemical have a half-life of 56.3 minutes. If you are given 80 g first, when will it become 16 g?

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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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Quick Review

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**Quick Review Solutions**

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**Exponential Population Model**

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**Example: You are given 𝑃 𝑡 =300 1.05 𝑡**

Is this a growth or decay? What is the rate?

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**Example Finding Growth and Decay Rates**

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**Example You are given 𝑃 𝑡 =800 .15 𝑡**

Is this a growth or decay? What is the rate?

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**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 𝑃 𝑡 = 𝑡 350000= 𝑡 t=12.77 hours

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**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 Just the setting up 𝑃 𝑡 =𝑃 𝑏 𝑡 ℎ𝑎𝑙𝑓𝑙𝑖𝑓𝑒 1= 𝑡 20

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**Group Work You are given 𝑃 𝑡 =150 1.025 𝑡**

When will this become ?

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**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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**Example Modeling a Rumor**

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**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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**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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Quick Review

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**Quick Review Solutions**

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**Changing Between Logarithmic and Exponential Form**

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**Group Work: transform logarithmic form into exponential form**

B) 𝑙𝑜𝑔 = 1 2 C) 𝑙𝑜𝑔 2 60=𝑥 D) 𝑙𝑜𝑔𝑥=8

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**Group Work: convert exponential form into logarithmic form**

5 𝑥 =34 5 −2 = 1 25 4 0 =1 16 1 =𝑝

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**Inverses of Exponential Functions**

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**Basic Properties of Logarithms**

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**An Exponential Function and Its Inverse**

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**Common Logarithm – Base 10**

Logarithms with base 10 are called common logarithms. The common logarithm log10x = log x. The common logarithm is the inverse of the exponential function y = 10x.

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**Basic Properties of Common Logarithms**

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**Example Solving Simple Logarithmic Equations**

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**Example Solving Simple Logarithmic Equations**

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**Basic Properties of Natural Logarithms**

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**Graphs of the Common and Natural Logarithm**

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**Example Transforming Logarithmic Graphs**

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**Example Transforming Logarithmic Graphs**

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Quick Review

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**Quick Review Solutions**

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**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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**Properties of Logarithms**

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**Example Proving the Product Rule for Logarithms**

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**Example Proving the Product Rule for Logarithms**

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**Example Expanding the Logarithm of a Product**

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**Example Expanding the Logarithm of a Product**

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**Example Condensing a Logarithmic Expression**

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**Example Condensing a Logarithmic Expression**

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Group Work 𝐸𝑥𝑝𝑎𝑛𝑑 log(7 𝑥 2 𝑦 𝑧 5 )

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Group Work Expand 𝑙𝑜𝑔 𝑦 2 𝑥

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Group Work Express as a single logarithm 𝑙𝑜𝑔 𝑧 𝑡− 𝑙𝑜𝑔 𝑧 𝑥 +5 𝑙𝑜𝑔 𝑧 𝑚

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**Group Work Express as a single logarithm**

4 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 2 𝑦−3 𝑙𝑜𝑔 2 𝑧

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**Change-of-Base Formula for Logarithms**

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**Example Evaluating Logarithms by Changing the Base**

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**Example Evaluating Logarithms by Changing the Base**

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Solving 4 𝑥 =51

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Solving ln 𝑒

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Solving log 1

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Solving log 5𝑥= log 4+ log (𝑥−3)

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Solving 𝑙𝑜𝑔 =𝑥

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Solving 2 5+3𝑥 =16

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Homework Practice Pg 317 #1-50 eoe

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**Equation Solving and Modeling**

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Quick Review

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**Quick Review Solutions**

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**One-to-One Properties**

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**Example Solving an Exponential Equation Algebraically**

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**Example Solving an Exponential Equation Algebraically**

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**Example Solving a Logarithmic Equation**

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**Example Solving a Logarithmic Equation**

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Group Work ln 3𝑥−2 + ln 𝑥−1 =2𝑙𝑛𝑥

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Group Work: Solve for x 𝑥 3 =5

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Group Work: Solve 𝑒 𝑥 − 𝑒 −𝑥 2 =5

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Group Work: Solve 1.05 𝑥 =8

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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**

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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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Richter Scale

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Example: How many times more severe was the 2001 earthquake in Gujarat, India ( 𝑅 1 =7.9) than the 1999 earthquake in Athens, Greece ( 𝑅 2 =5.9)

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Group Work: Show work How many times more severs was the earthquake in SF ( 𝑅 1 =6.5) than the earthquake in PS ( 𝑅 2 =3.6)?

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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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**Newton’s Law of Cooling**

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**Example Newton’s Law of Cooling**

A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC?

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**Example Newton’s Law of Cooling**

A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC?

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Group Work A substance is at temperature 96℃ is placed in 16℃. Four minutes later the temperature of the egg is 45℃. Use Newton’s Law of Cooling to determine when the egg will be 20℃

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**Regression Models Related by Logarithmic Re-Expression**

Linear regression: y = ax + b Natural logarithmic regression: y = a + blnx Exponential regression: y = a·bx Power regression: y = a·xb

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**Three Types of Logarithmic Re-Expression**

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

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**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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**Interest Compounded Annually**

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**Interest Compounded k Times per Year**

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**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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**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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**Compound Interest – Value of an Investment**

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**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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**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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**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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**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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**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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**Future Value of an Annuity**

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**Future Value of an Annuity**

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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**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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