Learning Objectives
- Explain how change can be measured at a single instant.
- Differentiate between average and instantaneous rates of change.
- Describe how a limit produces the slope of a tangent line.
1. Motivation: The Idea of Instantaneous Change
In algebra and physics, you’ve used average rates of change. If a car travels 120 miles in 2 hours, then \( \text{Average rate}=\frac{\Delta \text{distance}}{\Delta \text{time}}=\frac{120}{2}=60\ \text{mph} \). But to know speed at a single instant (say at \(t=1\) hr), we study what happens as the time interval shrinks to \(0\).
2. Secant Lines vs. Tangent Lines
For position \( s(t) \):
- Average velocity on \([a,b]\): \( \displaystyle \frac{s(b)-s(a)}{\,b-a\,} \) (slope of the secant).
- Instantaneous velocity at \( t=a \): \( \displaystyle \lim_{h\to 0}\frac{s(a+h)-s(a)}{h} \) (slope of the tangent).
The fundamental shift—from average change to instantaneous change—is the heart of calculus.
3. Numerical Example: Instantaneous Velocity
A ball is dropped: \( s(t)=100-16t^2 \) feet.
Average velocity from \(t=2\) to \(t=2.1\) is \( \displaystyle \frac{s(2.1)-s(2)}{0.1}=\frac{100-16(2.1)^2 - [100-16(2)^2]}{0.1}=\frac{-6.56}{0.1}=-65.6\ \text{ft/s}\).
As the interval shrinks (\(2\to 2.01\to 2.001\)), the average velocities approach \(-64\). Hence \( \displaystyle v(2)=\lim_{h\to 0}\frac{s(2+h)-s(2)}{h}=-64\ \text{ft/s}\).
| h | \( \displaystyle \frac{s(2+h)-s(2)}{h} \) |
|---|
4. Graphical Interpretation
A secant line passes through two points on a curve. As the second point approaches the first, the secant “rotates” into the tangent. The slope of the tangent is the instantaneous rate of change.
Use the interactive graphs below to watch secant slopes approach tangent slopes.
Interactive A: Secant → Tangent on \( f(x)=x^2 \)
Here the tangent slope is \(f'(a)=2a\). Watch the secant slope approach \(2a\) as \(h\to 0\).
Interactive B: Instantaneous Velocity for \( s(t)=100-16t^2 \)
Here the tangent slope is \(s'(t)=-32t\). At \(t=2\) the instantaneous velocity is \(-64\ \text{ft/s}\).
5. Historical Note
Newton and Leibniz independently recognized that instantaneous change can be captured as the limit of average change—leading to the invention of calculus.
Worked Example
Problem. If \( s(t)=t^2 \) meters, find the instantaneous velocity at \(t=3\).
Solution. \( \displaystyle v(3)=\lim_{h\to 0}\frac{(3+h)^2-3^2}{h} =\lim_{h\to 0}\frac{6h+h^2}{h} =\lim_{h\to 0}(6+h)=6\ \text{m/s}. \)
Quick Reference
- Secant slope (average): \( \displaystyle \frac{f(b)-f(a)}{\,b-a\,} \)
- Tangent slope (instantaneous): \( \displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h} \)
- For \(f(x)=x^2\): \(f'(x)=2x\). For \(s(t)=100-16t^2\): \(s'(t)=-32t\).