Calculus / Unit 3: The Derivative
3.1 Average and Instantaneous Rate of Change
Notes
The average rate of change of $f$ from $x = a$ to $x = b$ is the slope of the secant line: $\displaystyle \frac{f(b) - f(a)}{b - a}$.
- The instantaneous rate of change at $x = a$ is the limit of average rates as the interval shrinks to zero length.
- Example: for $f(x) = x^2$ from 1 to 3, the average rate is $\displaystyle \frac{9 - 1}{2} = 4$.
- Units are output units per input unit, such as miles per hour.
Practice
Find the average rate of change of $f(x) = x^2 + x$ from $x = 2$ to $x = 5$.
$f(2) = 6$ and $f(5) = 30$, so $\displaystyle \frac{30 - 6}{3} = 8$.
A car travels 120 miles in 2 hours. What does 60 mph represent?
Total change divided by elapsed time is the average rate of change.
How is the instantaneous rate of change at $x = a$ found?
Shrinking the interval turns the secant slope into the slope at a single point.
Ready for more? Take the Unit 3 test.