Quiz 7-1,2: 1. Where does the graph cross the y-axis? 2. f(1) = ? 3. Horizontal asymptote = ? 4. How was the function : transformed to get f(x) above? to get f(x) above? 5. Domain = ? 6. range = ?

7-3 Functions involving “e”

What you’ll learn  The Natural Base e  Population Models … and why Exponential and logistic functions model many growth patterns: Exponential and logistic functions model many growth patterns: human and animal population human and animal population money money …and patterns of decline …and patterns of decline radioactive decay dilution of chemical solutions dilution of chemical solutions

Exponential Function Review: ‘a’ is the initial value  f(0) = ‘a’ ‘b’ is called the growth factor Initial value: 10 f(0) = = 12 Growth factor: 4 ‘d’ shifts graph up/down and is the horizontal asymptote Horizontal asymptote: 2

Identifying the Parts of the function: ‘a’ is the initial value  f(0) = ‘a’ ‘ ’ is the growth factor but since all are base ‘e’ we say that “k” is the growth factor. “k” is the growth factor. Initial value: 10 f(0) = = 12 Growth factor: 3 ‘d’ shifts graph up/down and is the horizontal asymptote Horizontal asymptote: 2

Exponential Functions and the number ‘e’ Any exponential function of the form: Can be written in the form: Exponential Growth: k > 0 Exponential Decay: k < 0

Where does the number ‘e’ come from? Named after the Swiss mathematician Leonard Euler (1707 – 1783). Leonard Euler (1707 – 1783). The number ‘e’ can be found on your calculator by “2nd” + “ln” + “1” = … by “2nd” + “ln” + “1” = … ‘e’ is an irrational number like pi or the square root of a prime number. The numbers after the root of a prime number. The numbers after the decimal point go on forever without any repetition decimal point go on forever without any repetition of number patterns. of number patterns.

The slope of the tangent line at any point on the line at any point on the curve is curve is The number ‘e’ (Named after Leonard Euler, a Swiss mathematician) ‘e’ is a very unique number e = ….

Exponential Functions and the number ‘e’ Any exponential function of the form: Can be written in the form: Exponential Growth: k > 0 Exponential Decay: k < 0

Exponential Functions and ‘e’

The “natural” number ‘e’ works perfectly with natural processes Exponential growth of populations Exponential decay of radioactive material

What processes does ‘e’ have to do with? Think of a bacteria cell. Over a certain period of time it splits a certain period of time it splits from one cell into two cells. The from one cell into two cells. The number of bacteria doubles. number of bacteria doubles. This type of growth occurs in “spurts”. At one instant of time it is a single bacteria. The next instant it is two bacteria. The number of bacteria doubles. A(t) = 2 A(t) = 2 In “t” time periods it doubles ‘t’ times. t = 1 t = 1

What processes does ‘e’ have to do with? The number e (2.718…) represents the maximum compound rate of growth from a process that grows at 100% for one time period. Sure, you start out expecting to grow from 1 to 2. But with each tiny step forward you create a little “dividend” that starts growing on its own. When all is said and done, you end up with e (2.718…) at the end of 1 time period, not 2. Instead of in “spurts,” this type of growth occurs continuously  growth of money in an account that pays interest continuously  growth of money in an account that pays interest continuously  the decay of radioactive material which occurs continuously  the decay of radioactive material which occurs continuously t = 1 t = 1 e= 2.718… e= 2.718…

Your turn: 2. simplify 2. simplify 1. simplify 1. simplify 3. simplify

Putting it all together: If negative: Reflect across x-axis Initial value: Crosses y-axis here Growth factor: If negative: decay Horizontal asymptote vertical shift

Exponential Function (base ‘e’) What is the “initial value” ? Describe the transformation: What is the horizontal asymptote ? Is it growth or decay? Where does g(x) cross the y-axis ?

Your turn: For each of the following what is the: a. “initial value”? a. “initial value”? b. Growth or decay? b. Growth or decay? c. “horizontal asymptote” c. “horizontal asymptote” d. Any reflections (across x-axis) d. Any reflections (across x-axis)

Population Growth We can rewrite the change in the population as some percentage of the original population. as some percentage of the original population. The population at the end of one period of time equals the initial population plus the growth/decay of the population. initial population plus the growth/decay of the population. If “r” is the % change in population (decimal equivalent), then we can rewrite that as: then we can rewrite that as: Factoring out the common factor results in:

Population Growth over “t” time periods: Population (as a function of time) function of time) Initial population population Growth rate rate time It’s just a formula!!! The initial population of a colony of bacteria is The population increases by 50% is The population increases by 50% every hour. What is the population after 5 hours? every hour. What is the population after 5 hours? Percent rate of change (in decimal form) (in decimal form)

Your turn: 7. The population of a small town was 1500 in the year The population increases by 3% every year. What is the population of the town in 2009? 8. The population of bacteria in a petri dish is 175 at the start of an experiment. The population doubles every 2 hours. What is the population after 6 hours?

The population of a small town is modeled by: The population of a small town is modeled by: What is the % change in population for every time period ‘t’ ?

Your turn: 9. The population of a small town can be modeled by: What is the % change in population for every time period ‘t’ ? 10. The population of a large city can be modeled by:

Simple Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”)Interest rate rate time A bank account pays 3.5% interest per year. If you initially invest $200, how much money If you initially invest $200, how much money will you have after 5 years? will you have after 5 years?

Your turn: A bank account pays 14% interest per year. If you initially invest $2500, how much money If you initially invest $2500, how much money will you have after 7 years? will you have after 7 years? 11.

Value of a depreciating asset. According to tax law, the value of a piece of equipment can be “depreciated” and the depreciation can be used as a “business “depreciated” and the depreciation can be used as a “business expense” to reduce the amount of taxes that you pay. expense” to reduce the amount of taxes that you pay. A company buys a car for $20,000. It can depreciate the value of the car by 20% per year. What is the value of the car after 2 years?

Your turn: A company is in debt for $300,000. It is reducing its debt at the annual rate of 15%. What is the company’s debt at the annual rate of 15%. What is the company’s debt after 10 years? debt after 10 years?12. $59,062.32

Vocabulary Compounding: interest is paid at the end of a specified time period (instead of a year). This results is MORE MONEY at the end of the year.

Compound Interest (savings account) Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”) Interest rate time A bank account pays 3.5% interest per year. Compounded quarterly. If you initially invest $200, how much money will you have after 5 years? will you have after 5 years? # of periods per year

Continuously Compounded Interest Amount (as a function of time) function of time) Initial amount (“principle”) (“principle”) Interest rate time A bank account pays 3.5% interest per year. Compounded continuously. If you initially invest $200, how much money will you have after 5 years?

Your turn: A bank account pays 14% interest per year compounded monthly. If you initially invest $2500, how much money will you have after 7 years? will you have after 7 years? 13. A bank account pays 14% interest per year compounded continuously. If you initially invest $2500, how much money will you have after 7 years? 14.

HOMEWORK Section 7-1 (page 482) 28, 30 28, 30 Section 7-3 (page 495) , 24, , , 24, , 56 (17 total problems)