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change is akin to your average speed on the trip. Your speedometer measures your speed at any
one instant along the trip, your instantaneous rates of change, and this is one of the central
themes of Calculus.7
When interpreting rates of change, we interpret them the same way we did slopes. In the context
of functions, it may be helpful to think of the average rate of change as:
change in outputs
change in inputs
Example 2.1.7. The revenue of selling x units at a price p per unit is given by the formula R = xp.
Suppose we are in the scenario of Examples 2.1.5 and 2.1.6.
1. Find and simplify an expression for the weekly revenue R as a function of weekly sales, x.
2. Find and interpret the average rate of change of R over the interval [0, 50].
3. Find and interpret the average rate of change of R as x changes from 50 to 100 and compare
that to your result in part 2.
4. Find and interpret the average rate of change of weekly revenue as weekly sales increase from
100 PortaBoys to 150 PortaBoys.
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