Question

The marketing division of a large firm has found that it can model the sales generated by an advertising campaign as S(u) = 0.25 u + 1.4 millions of dollars when the firm invests u thousand dollars in advertising. The firm plans to invest u(x) = −2.5x2.56 + 56x + 200 thousand dollars each month where x is the number of months after the beginning of the advertising campaign. How quickly will the sales for this firm be changing when x = 12?

Answer 1

To determine how quickly sales are changing when x = 12, one must compute the derivative of the sales function, using the chain rule to combine the derivatives of S(u) and u(x), and then evaluate this at x = 12.

Step-by-step explanation:

The question is asking how quickly sales will be changing based on an investment model when x equals 12 months after the beginning of an advertising campaign. To find this rate of change, we can compute the derivative of the sales function with respect to time, S(u(x)), and then evaluate it at x = 12.

First, take the derivative of the function u(x) = -2.5x2.56 + 56x + 200 with respect to x to find u'(x). Then, take the derivative of S(u) = 0.25u + 1.4 million with respect to u to find S'(u). The chain rule tells us that the rate of change of sales with respect to time, dS/dx, is given by S'(u(x)) × u'(x).

After calculating the derivatives, we plug in x = 12 into u'(x) to get the investment rate of change, and then multiply this by S'(u) to obtain the rate of change of sales. This resulting value will tell us how quickly sales are changing twelve months after the start of the advertising campaign. Calculus is the mathematical tool used for these computations.