Exponential Growth and Decay

1. Exponential Growth and Decay Calculus Lesson 7-2 Mr. Hall

2. Direct Proportion Property of Exponentials If f is an exponential function, where a and b are positive constants, then f’(x) is directly proportional to f (x). So, if f’(x) is directly proportional to f (x) then f is an exponential function.

3. Application When money is left in a savings account, it earns interest equal to a certain percent of what is there. The more money you have there, the faster it grows. If the interest is compounded continuously, the interest is added to the account the instant it is earned. So, the instantaneous rate of change dm/dt where (m) represents money and (t) time, is directly proportional to the amount of money invested.

4. Compounding Interest where k is a constant Differential Equation - an equation that contains the derivative of a function.

5. Remember, we are taking a derivative that is directly proportional to a function to see if we can get an exponential function. An Exponential!

6. Converse of the Direct Proportion Property of Exponentials So, if f’(x) is directly proportional to f (x) then f is an exponential function. What we did in the previous slide is solve a differential equation by separating the variables.

7. General Solution We can clean up this solution a bit. since a positive # like e, raised to any number is positive, ec is positive. If we replace ec with a new constant m0 and allow it to be positive or negative. This gives us a general solution representing an entire family of functions.

8. Compounding Interest Suppose we start saving with $1000. What would m0 be? The initial condition. But what is “k” in the differential ?

9. Law of Exponential Change • If y changes at a rate proportional to the amount present (dy/dt = kt) and y = y0 when t = 0, then y = y0ekt, where k > 0 represents growth and k < 0 represents decay. The number k is the rate of constant of the equation.

10. Compounding Interest If we put money (m) in an account and the amount changes with time (dm/dt) the “k” might represent an interest rate. Suppose the interest rate is 5% per yr. and we begin with $1000. How much money will we have after 5 years? After 50 yrs?

11. Compounding Interest How long would it take for our initial $1000 to double?

12. Example: • You pour yourself a cup of coffee. When it is poured it is at D=130°F above room temperature. Three minutes later it has cooled to D=117°F above room temperature. As the coffee cools, the instantaneous rate of change D with respect to time t (minutes) is directly proportional to D. • Write the general solution for D. Then use the given data to find the constants in the equation.

13. Example: At

14. Example continues At Finally,

15. Example continues… The coffee is “drinkable” if it is at least 50°F above room temperature. Will it still be drinkable if you let it sit for 15 minutes after you pour it? At So, when will it not be drinkable?

16. Will we ever get to drink it? The coffee is “drinkable” if it is at least 50°F above room temperature. Will it still be drinkable if you let it sit for 15 minutes after you pour it? Ω or after about 27 minutes.