Question

Often sales of a new product grow rapidly at first and then level off with time. This is the case with the sales represented by the function S(t)=500−400t−1​, where t represents time in years. Find the rate of change of sales for the following number of years.

a.1
b.10

Answer 1

Here we have the function:

S(t) = 500 - 400*t^(-1)

Then the rate of change at the value t, will be:

S'(t) = dS(t)/dt

This differentiation will be:

S'(t) = -400/t^2

Then:

a) the rate of change at t = 1 is:

S'(1) = -400/1^2 = -400

The rate of change after one year is -400

b) t = 10

S'(10) = -400/10^2 = -400/100 = -4

The rate of change after 10 years is -4, it reduced as the years passed, as expected.