1. 5.8 Exponential Growth and Decay Mon Dec 7 Do Now In the laboratory, the number of Escherichia coli bacteria grows exponentially with growth constant k = 0.41. Assume that 1000 bacteria are present at time = 0 1) Find the formula for the # of bacteria P(t) at time t 2) How large is the population after 5 hours? 3) When will the population reach 10,000?

2. HW Review p.339

3. Exponential growth and decay When P0 is the initial size at t = 0 If k > 0, then P(t) grows exponentially If k < 0 then P(t) decreases exponentially K is either known as the growth or decay constant. We’ve done these things in Pre-calc

4. Y’ = ky If y(t) is a differentiable function satisfying the differential equation then, where P0 is the initial value P0 = y(0)

5. Notes This theorem tells us that a process obeys an exponential law precisely when its rate of change is proportional to the amount present. A population grows exponentially because each present organism contributes to growth

6. Ex Find all solutions of y’ = 3y. Which solution satisfies y(0) = 9?

7. Ex Pharmacologists have shown that penicillin leaves a person’s bloodstream at a rate proportional to the amount present. A) Express this as a differential equation B) Find the decay constant if 50 mg of penicillin remains in the bloodstream 7 hours after an injection of 450 mg C) Under the hypothesis of (B), at what time was 200 mg present?

8. Doubling Time / Half-life If, then is the doubling time if k >0 is the half-life if k < 0

9. Compound Interest If P0 dollars are deposited into an account earning interest at an annual rate r, compounded M times yearly, then the value of the account after t years is If compounded continuously, is

10. Present Value The concept of present value (PV) is used to compare payments made at different times The PV of P dollars received at time t is Having money now means you can get interest from it right away. Getting money later means you will lose out on any interest you could have gotten

11. Ex Is it better to receive $2000 today or $2200 in 2 years? Consider a 3% and 7% compounded interest rate

12. Ex You must decide whether to upgrade your company’s computer system. The upgrade costs $400,000 and will save $150,000 per year for the next 3 years. Is this a good investment if r = 7%?

13. Income stream An income stream is a sequence of periodic payments that continue over an interval of T years This is like winning the lottery and taking several payments over the years vs taking a lump sum

14. PV of an income stream If the interest rate is r, the present value of an income stream paying out R(t) dollars per year continuously for T years is

15. Ex An investment pays out 800,000 pesos per year, continuously for 5 years. Find the PV of the investment for r = 0.04

16. Closure For the investment that paid out 800,000 pesos per year for 5 years, find the PV of the investment if r = 0.06 HW: p.350 #3 7 13 14 15 39 44 45