Overview
- Average rate of change on \([a,b]\): \( \displaystyle \frac{f(b)-f(a)}{\,b-a\,} \) (the secant line slope).
- Instantaneous rate at \(x=a\): the limit of secant line slopes as the interval \([a,b]\) becomes small and \(b\to a\) (the tangent line slope).
Average rate of change
Drag either blue point horizontally. The secant line updates and the readout shows \( \Delta x,\ \Delta y,\) and the average rate, which is computed using the formula \( \dfrac{f(b)-f(a)}{b-a} \).
Example
Suppose \(s(t)\) is position (meters) at time \(t\) (seconds) with \(s(2)=12.1\) and \(s(3)=17.0\).
The average velocity on \([2,3]\) is \( \dfrac{17.0-12.1}{3-2}=4.9\ \text{m/s}.\)
Instantaneous rate of change
Example: Estimating instantaneous rate from a model
A simple model for the height of an object is given by \(y(t)=v_0 t-\tfrac{1}{2}gt^2\), where \(g\) is some gravitational constant.
Consider the model \(y(t)=60t-1.9t^2\) (meters). What is the instantaneous velocity at \(t=1\)?
- The average velocity on \([1,2]\) is
\( \dfrac{y(2)-y(1)}{2-1} =\dfrac{(120-7.6)-(60-1.9)}{1}=54.3\ \text{m/s}.\) - The average velocity on \([1,1.5]\) is
\( \dfrac{(90-4.275)-(60-1.9)}{0.5}=55.65\ \text{m/s}.\) - The average velocity on \([1,1.01]\) is
\( \dfrac{y(1.01)-y(1)}{0.01}\approx 56.19\ \text{m/s}.\)
The instantaneous rate of change at \(t=1\) is the limit of these values, which is approximately 56.2 m/s.
Example: Estimating instantaneous rate from a data table
A device reports cumulative counts \(C(t)\) at discrete times. For example, \(C(t)\) could represent a person's total number of heartbeats after \(t\) minutes. Use nearby secants to estimate the instantaneous rate at \(t=42\) minutes.
| t (min) | 36 | 38 | 40 | 42 | 44 |
|---|---|---|---|---|---|
| \(C(t)\) | 2500 | 2642 | 2776 | 2910 | 3055 |
- Left secant: \(\dfrac{2910-2776}{42-40}=67.0\)
- Right secant: \(\dfrac{3055-2910}{44-42}=72.5\)