Question

at the end of the holiday season in january, the sales at a department store are expected to fall. it is estimated that for the x day of january the sales will be ( s(x)=2+\frac{9}{(x + 1)^2} ) thousand dollars.
a) find the total sales for january 9 and determine the rate at which sales are falling on that day.
b) compare the rate of change of sales on january 2 to the rate on january 9. what can you infer about the rate of change of sales?
a) the total sales for january 9 were 2.090 thousand dollars.
(round to three decimal places as needed.)
the sales are falling at a rate of (square) thousand dollars per day on january 9.
(round to three decimal places as needed.)

Explanation:

Step1: Differentiate the function \(S(x)\)

We have \(S(x)=2 + 9(x + 1)^{-2}\).
Using the power rule \((u^n)^\prime=nu^{n - 1}u^\prime\), where \(u=x + 1\), \(n=-2\) and \(u^\prime = 1\).
The derivative \(S^\prime(x)=9\times(-2)(x + 1)^{-3}\times1=-\frac{18}{(x + 1)^{3}}\)

Step2: Evaluate \(S^\prime(x)\) at \(x = 9\)

Substitute \(x = 9\) into \(S^\prime(x)\).
\(S^\prime(9)=-\frac{18}{(9+ 1)^{3}}=-\frac{18}{1000}=-0.018\)

Answer:

The sales are falling at a rate of \(-0.018\) thousand dollars per day on January 9.