A company introduces a new product for which the number of units sold S is

S (t) = 200 (5 - 9 / (2+t))

where t is the time in months

(a) Find the avg. value of S (t) during the first year.

(b) During what month does S' (t) equal the avg value during the first year? ... ?

(a) The "average value" of a function over an interval [a, b] is defined to be

(1 / (b-a)) times the integral of f from the limits x = a to x = b.

Now S = 200 (5 - 9 / (2+t))

The average value of S during the first year (from t = 0 months to t = 12 months) is then:

(1/12) times the integral of 200 (5 - 9 / (2+t)) from t = 0 to t = 12

or 200/12 times the integral of (5 - 9 / (2+t)) from t = 0 to t = 12

This equals 200/12 * (5t - 9ln (2+t))

Evaluating this with the limits t = 0 to t = 12 gives:

708.113 units., which is the average value of S (t) during the first year.

(b). We need to find S' (t), and then equate this with the average value.

Now S' (t) = 1800 / (t+2) ^2

So you're left with solving 1800 / (t+2) ^2 = 708.113

I'll leave that to you