1. 4.5Modeling with Exponential and Logarithmic Functions Discuss cell division.

2. The mathematical model for exponential growth or decay is given by f (t) = A 0 e kt or A = A 0 e kt. If k > 0, the function models the amount or size of a growing entity. A 0 is the original amount or size of the growing entity at time t = 0. A is the amount at time t, and k is a constant representing the growth rate. If k < 0, the function models the amount or size of a decaying entity. A 0 is the original amount or size of the decaying entity at time t = 0. A is the amount at time t, and k is a constant representing the decay rate. The mathematical model for exponential growth or decay is given by f (t) = A 0 e kt or A = A 0 e kt. If k > 0, the function models the amount or size of a growing entity. A 0 is the original amount or size of the growing entity at time t = 0. A is the amount at time t, and k is a constant representing the growth rate. If k < 0, the function models the amount or size of a decaying entity. A 0 is the original amount or size of the decaying entity at time t = 0. A is the amount at time t, and k is a constant representing the decay rate. decreasing A0A0 x y increasing y = A 0 e kt k > 0 x y y = A 0 e kt k < 0 A0A0 Exponential Growth and Decay Models

3. Ex: Given the exponential model to describe the population of the indicated country, A, in millions, t years after 2003: India: Russia: What is the initial population for each country? Which country is growing at a faster rate?

4. Ex cont: Given the exponential model to describe the population of the indicated country, A, in millions, t years after 2003: India: Russia: What will Russia’s population be in 2006? When will India’s population reach 1,600,000,000?

5. Example The value of houses in your neighborhood follows a pattern of exponential growth. In the year 2006, you purchased a house in this neighborhood. The value of your house, in thousands of dollars, t years after 2006 is given by the exponential growth model V = 250e.07t How much did we pay for the house? Out of curiosity, how could we find the value in 2016? When will your house be worth $500,000? (Which variable will we substitute? For which will we solve?)

6. Cont… Solution: V = 250e.07t In the year ____________our house should be worth $500,000.

7. p456 #26, 8 26)A logistic growth model for world population, f(x), in billions, x years after 1949 is: How many people populated the Earth initially (in 1949)? How many people are expected in 2049? What is the limiting size of the population that Earth will eventually sustain (see # 26)? (If time 22 & 24)

8. 8. a) In 2000, the population of the Palestinians in the West Bank, Gaza Strip, and East Jerusalem was approximately 3.2 million and by 2050 it is projected to grow to 12 million. Use the exponential growth model in which t is the number of years after 2000, to find the exponential growth function that models the data. b) In which year will the Palestinian population be 9 million?

9. Use the fact that after 5715 years a given amount of carbon- 14 will have decayed to half the original amount to find the exponential decay model for carbon-14. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls. Solution We begin with the exponential decay model A = A 0 e kt. (ans. on next slide – can you do it first?) Text Example

10. Substituting for k in the decay model, the model for carbon-14 is A = A 0 e –0.000121t. k = ln(1/2)/5715=-0.000121 Solve for k. Divide both sides of the equation by A 0. Solution A 0 /2= A 0 e k5715 After 5715 years, A = A 0 /2 Take the natural logarithm on both sides. ln e x = x. Text Example cont.

11. Solution The Dead Sea Scrolls are approximately ___________ years old plus the number of years between 1947 and the current year. A = A 0 e -0.000121t This is the decay model for carbon-14. ______ = A 0 e -0.000121t A = ___________ since 76% of the initial amount remains. ______ = e -0.000121t Divide both sides of the equation by A 0. ln 0.76 = ln e -0.000121t Take the natural logarithm on both sides. ln e x = x. Now for part 2: In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls. t=ln(0.76)/(-0.000121) Solve for t.

12. *Ex: Use the exponential growth model to find the time it takes a population of 12 cells to quadruple in number at a growth rate of 8% per second. Round your answer to the nearest tenth.

13. Ex: (see notes on test review) Ex: Find how long it would take for a $2000 investment to double if it is invested at 14% compounded continuously.

14. (see previous slide) Repeat the above to find how long it would take that $2000 investment to double if it is invested at 14% compounded bi-annually.

15. The graph below shows the growth of the Mexico City metropolitan area from 1970 through 2000. In 1970, the population of Mexico City was 9.4 million. By 1990, it had grown to 20.2 million. Find the exponential growth function that models the data. By what year will the population reach 40 million? Use: A = A 0 e kt What is the original amount? What is the initial time? The final time given? What is the final amount given? 20 15 10 5 25 30 1970198019902000 Population (millions) Year (Opt)Example Which variable in our equation does each of these answers fill? What information is missing?

16. Solution a. We use the exponential growth model A = A 0 e kt in which t is the number of years since 1970. This means that 1970 corresponds to t = 0. At that time there were 9.4 million inhabitants, so we substitute 9.4 for _________ in the growth model: We are given that there were 20.2 million inhabitants in 1990. Because 1990 is 20 years after 1970, when ________ = 20 the value of _________ is 20.2. Substituting these numbers into the growth model will enable us to find k, the growth rate. We know that k > 0 because the problem involves growth. We can now write: Example cont.

17. Solution We substitute 0.038 for k in the growth model to obtain the exponential growth function for Mexico City. It is A = 9.4 e 0.038t where t is measured in years since 1970. What is our next step? (Remember our goal is to find when the population will reach 40 million.) Example cont. 0.038 = k 20.2 = 9.4e k20

18. Solution A = 9.4 e 0.038t b. To find the year in which the population will grow to 40 million, we substitute _______ in for _______ in the model from part (a) and solve for ________. Because _________ is the number of years after 1970, the model indicates that the population of Mexico City will reach 40 million by ____________. Example cont. t = 38.10973065…