Example 5: Media Coverage

Click here to open an associated Mathcad worksheet:

Problem. The following graph displays the cumulative number of mentions in the New York Times of Barney, a lumpish purple dinosaur once popular with small children. The independent variable  t  gives the time, in months since January 1 of 1991, and the dependent variable  B = B(t)  gives the total number of mentions of Barney that had appeared in the paper up until time  t.

Where does  B(t)  have the largest value? Over what interval does  B(t)  show the greatest change in value? Over what interval does  B(t)  show the greatest average rate of change in value?

Solution. In the time period shown, B  takes on its largest value at the top of the graph, 40 months after Jan. 1, 1991, or around May 1, 1994. There, B  has a value of about 4300 mentions.

Since the function is never decreasing, the change in value is non-negative (positive or zero) over any interval, and so the greatest change in value is simply over the longest interval: from  t = 0  to  t = 40, during which Barney was mentioned an additional  DB = B(40) – B(0)    4300 – 100 = 4200  times.

Barney was being mentioned at the greatest rate, however, somewhere in the middle of the time period shown. It appears that between  t = 21  and  t = 24  the rate of change was approximately  DB/Dt = (2800 – 1900)/(24 – 21) = 300  mentions per month. Notice that this greatest rate occurs at the inflection point, where the concavity of the graph changes from up to down.

 
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