Question

when a new product is advertised, more and more people try it. however, the rate at which new people try it slows as time goes on. consider the following six graphs. which of these graphs could be a reasonable sketch of the graph of the total number of people who have tried such a product against time? what can you say about the concavity of the graph? it is

Explanation:

First Sub - Question: Identifying the Graph

Step 1: Analyze the trend of total people over time

The total number of people who have tried the product should be an increasing function of time (since more and more people try it as time goes on). Also, the rate of increase (the slope of the tangent to the graph) should be decreasing (because the rate at which new people try it slows as time goes on).

• Graph 1: Decreasing, so not valid.
• Graph 2: Decreasing (linear), not valid.
• Graph 3: Decreasing, not valid.
• Graph 4: Increasing, but the slope is increasing (concave up), which would mean the rate of new people trying is increasing, not slowing. So not valid.
• Graph 5: Increasing at a constant rate (linear), so the rate of new people trying is constant, not slowing. Not valid.
• Graph 6: Increasing, and the slope is decreasing (the curve is getting less steep as time goes on), which matches the description that the rate of new people trying slows as time goes on.

Step 2: Confirm the selection

Based on the analysis of the trend (increasing with a decreasing rate of increase), Graph 6 is the reasonable sketch.

Second Sub - Question: Concavity of the Graph

Step 1: Recall the definition of concavity

A function is concave down if the slope of its tangent line is decreasing as \(x\) (time, in this case) increases. A function is concave up if the slope of its tangent line is increasing as \(x\) increases.

Step 2: Relate to the problem

For the total number of people who have tried the product, the rate of increase (slope) is decreasing as time goes on. When the slope of the tangent is decreasing, the function is concave down.

Answer:

(First Sub - Question): 6. The graph with the increasing curve that becomes less steep (Graph 6)