2. Exponential Growth & Decay Section 4.5 JMerrill, 2005 Revised 2008

3. Review Is this okay? Arguments must be positive

4. Review 500e 0.3x = 600 e 0.3x = 1.2 ln 1.2 = 0.3x x = 0.608

5. Exponents and Logarithms How are exponents and logarithms related? They are inverses of each other Why is this important? Using inverses allow us to solve problems (we use subtraction to solve addition problems & division to solve multiplication) Many real-life scenarios are exponential in nature and logarithms allow us to solve for the unknown.

6. Examples Using Logarithmic Scales The Richter scale is used to determine the intensity of an earthquake. Measuring acidity using the pH scale, or concentration of ions. Carbon dating. Modeling population growth/decay--just to name a few…

7. Exponential Decay Model A(t) = A0ekt A0 is the initial amount K is the growing/decay entity. If k>0, the entity is growing (an increasing function). If k<0, the entity is decaying (a decreasing function). Looks like A(t) = Pert? It works the same way.

8. Population Model In 1970, the US population was 203.3 million. In 2003, the population was 294 million. 1.F ind the exponential growth model 2.B y which year will the US population reach 315 million?

9. Population Model t is the number of years after 1970. t=0 represents 1970. t = 33 represents 2003 When t = 33, A = 294 A(t) = A0ekt 294 = 203.3ek(33)

10. Population Cont 294 = 203.3e k(33) 294 = 203.3e k(33) What do you do when the exponent is a variable? What does lne = ? So, k 0.011, which is exponential growth The growth model is A(t) = 203.3e0.011t

11. Population Cont When will the population reach 315 million? A(t) = 203.3e0.011t 315 = 203.3e0.011t You finish… Did you get approximately 40? That means that in the year 2010 the population will be approx. 315 million!

12. Carbon Dating The natural base, e, is used to estimate the ages of artifacts. Plants and animals absorb Carbon-14 from the atmosphere. When a plant or animal dies, the amount of carbon-14 it contains decays in such a way that exactly half of the initial amount is present after 5,715 years. The natural base, e, is used to estimate the ages of artifacts. Plants and animals absorb Carbon-14 from the atmosphere. When a plant or animal dies, the amount of carbon-14 it contains decays in such a way that exactly half of the initial amount is present after 5,715 years.

13. Carbon Dating The function that models the decay of carbon-14, where A 0 is the initial amount of carbon-14, and A(t) is the amount present t years after the plant or animal dies, is The function that models the decay of carbon-14, where A 0 is the initial amount of carbon-14, and A(t) is the amount present t years after the plant or animal dies, is

14. Carbon Dating Example Archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14. Could the scrolls be 2000 years old? Using the same process as the last example, we find k to be -0.00012. Finding k is written out in the book on P449.

15. 78% of the original amount

16. You Do A wooden chest is found and said to be from the 2nd century BCE. Tests on a sample of wood from the chest reveal that it contains 92% of its original carbon-14. Could the chest be from the 2nd century BCE? Use the same k as the last example.

17. You do

18. Logistic Growth Model The spread of disease is exponential in nature. However, there arent an infinite number of people. Eventually, the disease has to level off. Growth is always limited. A logistic growth model is used in this type of situation: Y = c is the horizontal asymptote. Thus c is the limiting value of the function.

19. Modeling the Spread of the Flu The function below describes the number of people, f(t), who have become ill with influenza t weeks after its initial outbreak in a town with a population of 30,000 people.

20. 1.How many people became ill with the flu when the epidemic began? 2.How many people were ill by the end of the fourth week? 3.What is the limiting size of f(t), the population that become ill?

21. Modeling the Spread of the Flu 1.H ow many people became ill with the flu when the epidemic began? In the beginning, t = 0:

22. Modeling the Spread of the Flu 2. How many people were ill by the end of the fourth week?

23. 3. What is the limiting size of f(t), the population that become ill? C represents the limiting size that f(t) can obtain. There are only 30,000 people in the town, therefore, the limiting size must be 30,000!