Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Let \(S\) represent monthly sales of a new digital audio player. Write a statement describing \(S^{\prime}\) and \(S^{\prime \prime}\) for each of the following. (a) The rate of change of sales is increasing. (b) Sales are increasing, but at a greater rate. (c) The rate of change of sales is steady. (d) Sales are steady. (e) Sales are declining, but at a lower rate. (f) Sales have bottomed out and have begun to rise.

Step-by-Step Solution

Answer

In summary: (a) \(S'\) positive, \(S''\) positive; (b) \(S'\) positive, \(S''\) positive; (c) \(S'\) constant, \(S''\) zero; (d) \(S'\) zero, \(S''\) zero; (e) \(S'\) negative, \(S''\) positive; (f) \(S'\) positive, \(S''\) positive.

Step 1: Interpret (a)

(a) The rate of change of sales is increasing. In this case, \(S'\) is positive (sales are changing) and \(S''\) is positive (that change is increasing).

Step 2: Interpret (b)

(b) Sales are increasing, but at a greater rate. Here, both \(S'\) and \(S''\) are positive. Sales (\(S'\)) are going up and the increase (\(S''\)) is getting faster.

Step 3: Interpret (c)

(c) The rate of change of sales is steady. This indicates that \(S'\) is not changing, implying \(S''\) is zero.

Step 4: Interpret (d)

(d) Sales are steady. In this case, \(S'\) is zero because sales are not changing. Similarly, \(S''\) is zero because the rate of change is also not changing.

Step 5: Interpret (e)

(e) Sales are declining, but at a lower rate. For this scenario, \(S'\) is negative (sales are decreasing) and \(S''\) is positive (the rate of decrease is slowing down).

Step 6: Interpret (f)

(f) Sales have bottomed out and have begun to rise. This means that \(S'\) is positive (sales have started to increase) and \(S''\) is positive (the rate of increase is accelerating).