Learning objectives
The average rate of change of a function is like finding the slope between two points on the graph. It helps us understand how the output values change when the input values change over a certain interval.
The average rate of change for a function over the interval [a,b] is represented algebraically as:
The rate of change at a single point, or instantaneous rate of change, can be approximated by finding the slope of the line that just touches the graph at that point. This is called a tangent line. We can estimate this rate of change by looking at the average rates of change over very small intervals around that point.
The water level of a water tank as it drains is given by the function W\left(t\right), measured in meters. Time, t, is measured in minutes. This is represented by the following table.
t \text{ minutes} | W(t) \text{ meters} |
---|---|
0 | 12 |
1 | 11.5 |
2 | 10.8 |
3 | 9.9 |
4 | 8.8 |
5 | 7.5 |
6 | 6 |
How fast is the water tank draining, on average, over the six-minute interval?
Estimate the rate at which the water level is changing at t=3.
Consider the function: g\left(x\right) = \sqrt{5x + 2}
Find the average rate of change of g\left(x\right) on the interval [0,\, 3].
Which has the smaller instantaneous rate of change: x=7 or x=9?
Average rate of change over an interval [a,\,b] is:
\dfrac{\text{change in }y}{\text{change in }x}=\dfrac{\Delta y}{\Delta x}=\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
Instantaneous rate of change at a point can be approximated by finding the average rates of change of small intervals near the point.