Table of Contents 1. Section 5.8 Exponential Growth and Decay

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

Applications of integrals so far Area between curves Volumes (cross sections and rotations) Average value of functions Differential equations (separable and slope fields and Euler) Still to go Exponential growth Logistic growth Length of arcs

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

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?

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)

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

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?

Doubling Time (time it takes for population to double)

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

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

Assignment Pg. 366: #1-13 odd, 16, 17, 21, 23