Suppose that a certain television program has ana average point rating of 5.9 (rating measure the number of viewers), and that the program's executives estimate that a 0.2 drop in ratings will correspond to a $4.7 million drop in revenue. if f(p) is the revenue earned in millions of dollars in terms of p, the rating points, estimate f'(p) and explain the meaning of the number you found (including units).

Hi!

We know that the program has an average of 5.9 points in rating.

The executives estimate that a 0.2 drop means that they lose $4.7 million.

this means that, if we define f(p) as the revenue:

f(5.9) = x  

f(5.7) = x - $4.7 millions

If we assume that this has a linear relationship, we can assume that:

0.2*k = $4.7 mill

k = $4.7/0.2 mill

k = $23.5 mill.

this means that for every point of rating, they lose $23.5 million.

Now we can describe our function as:

f( 5.9 - j) = x - ($23.5 mill)*j

where j is the change with respect to the mean of 5.9 points of rating, if j is positive means that the rating is decreasing, then the profit is decreasing.

Now if we derivate this we can obtain.

f'(5.9 - j) = -$23.5 mill

this means that the rate of change of the profit with respect to the change in the rating is of -$23.5 million, meaning that if the tv program lost one point of rating, they would lose 23.5 million dollars.