Suppose that the size S of a tumor in cubic millimeters is g

Solution

(a) The problem is basically asking us to find S|_t=4 = f(4). We can easily substitute t = 4 into f(t) = 3^t to get S|_t=4 = f(4) = 3⁴ = 81. The units of S are given in cubic millimeters, so our final answer is that during the first four months after discovery, the tumor will increase in size by 81 cubic millimeters.

(b) Finding the instantaneous rate of change at a given time is a matter of being able to calculate a derivative (and being able to interpret units). If S = f(t), then S\' = f\'(t). Thus, when we are asked to find S|_t=3, we need to calculate f\'(3). Recall that the derivative of f(t) = 3^t is f\'(t) = 3^tln3, so our answer is S\'|_t=3 = f\'(3) = 3³ln3 29.7; i.e. after 3 months, the tumor\'s instantaneous rate of change will be about 29.7 cubic millimeters per month.