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Chapter 3 Exponential, Logistic, and Logarithmic Functions

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Answers

- Yes
- No
- Yes
- Yes
- no

Group Activity

- Use this formula
- Group 1 calculate when x=1
- Group 2 calculate when x=2
- Group 3 calculate when x=4
- Group 4 calculate when x=12
- Group 5 calculate when x=365
- Group 6 calculate when x=8760
- Group 7 calculate when x=525600
- Group 8 calculate when x=31536000
- What do you guys notice?

Answer

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

Answer

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

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)

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?

Word Problem

- The population of New York State can be modeled by
- A) What’s the population in 1850?
- B) What’s the population in 2010?
- C) What’s the maximum sustainable population?

Answer

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

Group Work

In chemistry, you are given half-life formulas

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?

Homework Practice

- P 286 #1-54 eoe

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

Example:

- You are given
- Is this a growth or decay? What is the rate?

Example

- You are given
- Is this a growth or decay? What is the rate?

Example Finding an Exponential Function

Determine the exponential function with initial value=10,

increasing at a rate of 5% per year.

Group Work

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

Answer

- t=12.77 hours

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.

answer

- Just the setting up

Group Work

- You are given
- When will this become 150000?

Example Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

Key Word

- Maximum sustainable population
- What does this mean? What function deals with this?

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.

Homework Practice (Do in class also)

- P 296 #1-44 eoo

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.

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.

Group Work

- Expand

Group Work

- Express as a single logarithm

Group Work

- Express as a single logarithm

Homework Practice

- Pg 317 #1-50 eoe

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.

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.

Example:

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

Example:

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

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

Answer

- 2 orders of magnitude
- 4 orders of magnitude
- 6 orders of magnitude

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

Answer

- 6 orders of magnitude
- 2 orders of magnitude

Example:

- How many times more severe was the 2001 earthquake in Gujarat, India (=7.9) than the 1999 earthquake in Athens, Greece (=5.9)

Group Work: Show work

- How many times more severs was the earthquake in SF ( than the earthquake in PS ()?

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.

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?

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?

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?

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?

Group Work

- A substance is at temperature is placed in . Four minutes later the temperature of the egg is Use Newton’s Law of Cooling to determine when the egg will be

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

Homework Practice

- Pg 331 #1-51 eoe

Example Compounding Monthly

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

Example Compounding Monthly

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

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.

Example Compounding Continuously

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

Example Compounding Continuously

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

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.

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.

Example Computing Annual Percentage Yield

Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

Example Computing Annual Percentage Yield

Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

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

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?

Example

- Mr. Liu bought a new car for $20000. What are the monthly payment for a 5 year loan with 0 down payment if the annual interest rate (APR) is 2.9%?

Homework Practice

- Pg341 #2-56 eoe

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