Growth and Decay Problems

As we mentioned when we discussed the rate of change of the exponential function, $\exp:(-\infty,\infty)\rightarrow(-\infty,\infty)$, $\exp(x) = e^x$,there are many situations that are described in the following manner: ``The rate of change of AMOUNT (with respect to time) is proportional the quantity of AMOUNT present.'' AMOUNT can refer to number of bacteria, quantity of a radioactive substance, or, in the case of Newton's law of cooling, the temperature difference between an object and its surroundings. If we let A the function giving the quantity of interest, then in mathematical terms, we have

A'(t) = KA(t)

where t represents time and K is the proportionality constant. From our experience with the chain rule in problems such as ``Find y' if y =7e^2x'', where y' = 7e^2x(2) = 2y, we see that if A'(t) is to equal KA(t), one possibility is to have A(t) = A(0)e^Kt, just as you learned in College Algebra. A little later on, we will show that this is the only solution to this problem.