3.2 Exponential and Logistic Modeling

1. 3.2 Exponential and Logistic Modeling

2. List similarities and differences between the following graphs:

3. Definition: Exponential Growth and Decay For any exponential function where x is any real number… • If a > 0 and b > 1, the function f is increasing and is an exponential growth function. The base b is its growth factor. • If a> 0 and b <1, f is decreasing and is an exponential decay function. The base b is its decay factor.

4. Example 1: Finding Growth and Decay Rates Tell whether the population model is an exponential growth function or exponential decay function. Find the rate of each. • (a) San Jose: P(t) = 782,284*(1.10136)t • (b) Detroit: P(t) = 1,203,368*(0.9858)t

5. General Exponential Population Model • If a population P is changing at a constant percentage rate r each year, then where Pois the initial population, r is the percent expressed as a decimal, and t is time in years.

6. Example 2 Finding an Exponential Function • Determine the exponential function with initial value = 12, increases at a rate of 8% per year.

7. Example 3 Modeling Bacteria Growth • Suppose a culture of 100 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. • Write function for P(t) • Find t when P(t)=350,000

8. Example 4 Modeling Radio Active Decay • Suppose the half-life of a certain radioactive substance is 20 days and there are 5 g present initially. Find the time when there will be 1 g of the substance remaining. • Dealing with half-lives: the exponent will be the number of half-lives!

9. Exit Pass • Name at least two ways you can determine if a table of data, a graph, or a function has exponential decay.

10. Homework • 3.2 #1-5 odd, 29-33 odd, 39, 45