Question

What does average rate of change mean?

Answer 1

Hint : We have to first describe the conditions for the average rate of change. We take an arbitrary function and find its two inputs and the outputs for those particular inputs. We find the difference or change in the values and then find the average rate of change as $\dfrac{\Delta y}{\Delta x}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\dfrac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}$ .

Complete step by step solution:
A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are output units per input units.
Let us take a function $y=f\left( x \right)$ . We have two values of $x$ as $x={{x}_{1}},{{x}_{2}}$ and for those values of $x$ we have two values of $y$ where ${{y}_{1}}=f\left( {{x}_{1}} \right)$ and ${{y}_{2}}=f\left( {{x}_{2}} \right)$ .
We find the change of values for $x$ and $y$ .
Therefore, $\Delta x={{x}_{2}}-{{x}_{1}}$ and $\Delta y={{y}_{2}}-{{y}_{1}}$ .
The average rate of change will be denoted by $\dfrac{\Delta y}{\Delta x}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\dfrac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}$ .
Obviously, the function is not a perfect straight line and it will change differently inside that interval but the average rate can only evaluate the change between the two given points not at each individual point.

Note : A decrease is expressed by a negative change or “negative increase.” A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.