Calculate average rate of change

Average rate of change describes the net change in a function’s output per unit change in its input over a finite interval, calculated as $\frac{f(b)-f(a)}{b-a}$ for $a\ne b$ . Learners interpret this as the slope of the secant line between two points on a graph, including its sign and units, and distinguish it from the instantaneous rate at a single point; advanced limit techniques and formal derivative rules are not included.

Detailed Explanation: Calculate average rate of change

Average rate of change measures how much the output changes, on average, for each one-unit change in the input over an interval.

For a function $f(x)$ from $x=a$ to $x=b$ ,

\text{Average rate of change}=\frac{f(b)-f(a)}{b-a}, \qquad a\ne b.

This is the slope of the secant line joining the two points $(a,f(a))$ and $(b,f(b))$ on the graph.

Example

Suppose the position of an object is modeled by

s(t)=t^2+2t,

where $s$ is measured in metres and $t$ is measured in seconds. Find the average rate of change of position from $t=1$ to $t=4$ .

Identify the endpoints.

Here, $a=1$ and $b=4$ .
Find the output values at the endpoints.

s(1)=1^2+2(1)=3

s(4)=4^2+2(4)=24

Substitute into the formula.

\frac{s(4)-s(1)}{4-1} =\frac{24-3}{3} =\frac{21}{3} =7

State the answer with units.

The average rate of change is

\boxed{7\text{ metres per second}}.

This means that from $t=1$ to $t=4$ , the object’s position increased by an average of $7$ metres for every second. Because the answer is positive, the overall change in position is increasing. This is the slope of the secant line over the interval, not the instantaneous rate of change at just one time.

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Average Rate of Change - Function and Point plus Delta to Slope

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