Sales A company introduces a new product for which the...

Sales A company introduces a new product for which the number of units sold $S$ is $S(t)=200\left(5-\frac{9}{2+t}\right)$ where $t$ is the time in months. (a) Find the average value of $S(t)$ during the first year. (b) During what month does $S^{\prime}(t)$ equal the average value during the first year?

Sales A company introduces a new product for which the number of units sold $S$ is
$S(t)=200\left(5-\frac{9}{2+t}\right)$
where $t$ is the time in months.
(a) Find the average value of $S(t)$ during the first year.
(b) During what month does $S^{\prime}(t)$ equal the average value during the first year?

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Key Concepts

Average Value of a Function

The average value of a function over an interval is computed by integrating the function over that interval and dividing the result by the interval's length. This process yields a single value that represents the overall behavior or output of the function throughout that period.

Differentiation and Rate of Change

Differentiation involves finding the derivative of a function, which represents its instantaneous rate of change with respect to its variable. This concept is key in understanding how quickly a quantity, like sales, is increasing or decreasing at any particular moment.

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point at which the derivative equals the function's average rate of change over the interval. This theorem underpins problems that relate a derivative to an average value computed over an interval.

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