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 rate of change of $S$ during the first year. (b) During what month of the first year does $S^{\prime}(t)$ equal the average rate of change?

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 rate of change of $S$ during the first year.
(b) During what month of the first year does $S^{\prime}(t)$ equal the average rate of change?

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Key Concepts

Average Rate of Change

The average rate of change of a function over a particular interval measures how much the function’s output changes per unit change in the input over that interval. It is calculated as the difference in the function’s values at the endpoints divided by the length of the interval and provides the slope of the secant line connecting those points.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a specific point, given by its derivative, indicates how fast the function is changing at that exact moment. It corresponds to the slope of the tangent line to the curve at the point of interest.

Mean Value Theorem

The Mean Value Theorem is a fundamental result in calculus which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval at which the derivative (instantaneous rate of change) equals the average rate of change over the entire interval. This theorem bridges the gap between the instantaneous and average rates of change.

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