1. Table of Contents 5. Section 5.8 Exponential Growth and Decay

2. 5.8 Exponential Growth or Decay Essential Question – What are some applications to exponential functions?

3. Law of Exponential Growth If y changes at a rate proportional to the amount present, and if y = y 0 when t = 0, then k is growth constant if k>0 and decay constant if k<0

4. Bacteria example E coli bacteria increase exponentially with a growth constant of k=0.41. Assume there are 1000 bacteria present at t=0. How large is population after 5 hrs? When will population reach 10000?

5. If you differentiate exponential growth law…. What this tells us is that a process obeys an exponential law when its rate of change is proportional to the amount present at time t. So a population grows exponentially because its growth rate is proportional to the size of the population (each organism contributes to the growth through reproduction)

6. Example Find all solutions to y’ = 3y. Find particular solution for y(0)=9.

7. Penicillin example Find decay constant if 50 mg of penicillin remain in the bloodstream 7 hrs after an initial injection of 450 mg. What time was 200 mg of penicillin left?

8. Doubling Time (time it takes for population to double)

9. Doubling Time example From 1955 to 1970, physics degrees grew exponentially with k = 0.1. Find doubling time then find how long it would take to increase 8 fold. Double in 7, quadruple in 14, 8 fold in 28 years

10. Half life example Half life follows same model as double time. An isotope of Radon-222 has half-life of 3.825 days. Find decay constant and determine how long it will take for 80% of isotope to decay.

11. Carbon-14 dating C 14 decays exponentially in organisms after death, but C 12 does not. So by measuring ratio, we can determine when death occurred. The decay constant is k=-0.000121

12. Carbon dating example A charcoal sample from paintings in a prehistoric cave had C 14 to C 12 ratio of 15%. What is the age of paintings in cave?

13. Money

14. Example Suppose you deposit $800 in account that pays 6.3% annual interest. How much will you have 8 years later if it is compounded a) continuously b) quarterly?

15. Present Value of money

16. Present Value Example Is it better to receive $2000 today or $2200 in 2 years if r=7%? What if r=3%?

17. Income Stream An income stream is money that is paid out equally and continuously over a number of years. The present value of an income stream is given below where payout is R(t) and T is number of years paid out.

18. Income Stream You can only use this formula if money is paid continuously. If money is paid once per year, you need to use PV formula for each year and add them up

19. Income Stream Example An investment pays out $800/yr for 5 years, assume income is continuous. Find PV of investment for r=6%.

20. Assignment Pg. 366: #1-13 odd, 17-23 odd, 31-47 odd