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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

AFind the avg. value of S(t) during the first year.
BDuring what month does S'(t) equal the avg value during the first year? ...?

Verified Answer :

AThe "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

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