Rates of change on curves
For a straight line, the rate of change (slope) is the same everywhere. For a curve, it changes from point to point.
Average rate of change
The average rate of change from x = a to x = b is:
[f(b) − f(a)] / (b − a)
It’s the slope of the secant line through those two points.
Example: f(x) = x², from x = 1 to x = 3
[9 − 1]/(3 − 1) = 8/2 = 4
Context: A car’s position (km) is d(t). From t = 1 h to t = 3 h it goes from 60 km to 200 km.
Average speed = 140/2 = 70 km/h
Instantaneous rate of change
The instantaneous rate of change is the rate at a single moment, like a speedometer reading. Graphically, it’s the slope of the tangent line.
You can’t use two points that are the same, so we estimate with points that are very close.
Estimating with shrinking intervals
f(x) = x² at x = 3. Use the difference quotient:
[f(3 + h) − f(3)] / h
| h | Slope |
|---|---|
| 1 | (16 − 9)/1 = 7 |
| 0.1 | (9.61 − 9)/0.1 = 6.1 |
| 0.01 | (9.0601 − 9)/0.01 = 6.01 |
| 0.001 | 6.001 |
The slopes approach 6. That’s the instantaneous rate of change at x = 3.
Algebraically: [(3 + h)² − 9]/h = (6h + h²)/h = 6 + h → 6 as h → 0.
From estimates to calculus
The derivative of f, written f′(x), gives the instantaneous rate of change at every point. For f(x) = x², f′(x) = 2x, so at x = 3 the rate is 6 ✓.
Reading rates from graphs
- Steep secant or tangent → fast change.
- Positive slope → increasing; negative → decreasing.
- Zero slope → a peak or valley (turning point).
Applications
| Situation | Rate of change means |
|---|---|
| position vs time | velocity |
| velocity vs time | acceleration |
| cost vs units | marginal cost |
| population vs time | growth rate |
| temperature vs time | heating or cooling rate |
Worked examples
Example 1: f(x) = 2x³. Average rate from x = 0 to x = 2? (16 − 0)/2 = 8.
Example 2: A ball’s height h(t) = −5t² + 20t. Average velocity from t = 1 to t = 3? h(1) = 15, h(3) = 15 → 0 m/s.
Example 3: Estimate the instantaneous rate of f(x) = x² at x = 5 using h = 0.01. (25.1001 − 25)/0.01 = 10.01 (exact: 10).
Common mistakes
- Subtracting in different orders in the numerator and denominator.
- Confusing average and instantaneous rates.
- Using h = 0 directly (division by zero).
Try it yourself
- Average rate of change of f(x) = x² + 1 from x = 2 to x = 5.
- A stock price goes from $40 to $52 in 4 days. Average rate?
- Estimate the instantaneous rate of f(x) = x² at x = 1 using h = 0.001.
- What does a horizontal tangent line tell you?
Answers: 1) 7 2) $3 per day 3) ≈ 2.001 4) The instantaneous rate is 0 (a possible max or min).
Quick recap
- Average rate of change = slope of a secant line.
- Instantaneous rate of change = slope of a tangent line, estimated with the difference quotient.
- The derivative gives instantaneous rates everywhere.