a population p(t) is changing at a rate of P(t)=10,000e^(-t/4) with t in years. If the population is initially 80,000 at t=0, how large is the population as t--> + infinity
The rate of change of a function is the first derivative of the function.
If p(t) is the population function, then `p'(t)=P(t)=10000e^(- t /4)`
To find the population function we integrate P(t) with respect to t:
`=10000int e^(-t/4)dt` Let `u=-t/4,du=-1/4dt` then
Now p(0)=80000 so `80000=-40000e^0+C==>C=120000`
We are asked to find the limit of p(t) as t goes to infinity:
The population approaches 120000 as t goes to infinity.