Question 3.8.4: Estimating a Rate of Change in Economics A small company est...
Estimating a Rate of Change in Economics
A small company estimates that when it spends x thousand dollars for advertising in a year, its annual sales will be described by s=60-40 e^{-0.05 x} thousand dollars. The four most recent annual advertising totals are given in the following table.
Estimate the current (year 4) value of x'(t) and the current rate of change of sales.
| Year | 1 | 2 | 3 | 4 |
| Advertising Dollars | 14,500 | 16,000 | 18,000 | 20,000 |
From the table, we see that the recent trend is for advertising to increase by $2000 per year. A good estimate is then x'(4) ≈ 2. Starting with the sales equation
s(t)=60-40 e^{-0.05 x(t)},
we use the chain rule to obtain
s^{\prime}(t)=-40 e^{-0.05 x(t)}\left[-0.05 x^{\prime}(t)\right]=2 x^{\prime}(t) e^{-0.05 x(t)}.
Using our estimate that x'(4) ≈ 2 and since x(4) = 20, we get s^{\prime}(4) \approx 2(2) e^{-1} \approx 1.472.
Thus, sales are increasing at the rate of approximately $1472 per year.