1. What Rate of Change Means
The rate of change tells how one quantity varies with respect to another. In simple notes-style: it shows how fast something is increasing or decreasing when the input changes slightly. This idea appears naturally in motion, growth, and many real-life situations.
2. Average Rate of Change
The average rate of change measures the overall change in the function over a given interval.
2.1. Formula
\text{Average rate of change over } [a, b] = \frac{f(b) - f(a)}{b - a}
2.2. Interpretation
This is the slope of the secant line joining the points (a, f(a)) and (b, f(b)) on the graph.
3. Instantaneous Rate of Change
The instantaneous rate of change tells how fast the function is changing at a specific point. This is exactly what the derivative measures.
3.1. Definition
\text{Instantaneous rate at } x = a = f'(a)
It is the limit of the average rate of change as the interval shrinks.
3.2. Graphical View
Instantaneous rate of change equals the slope of the tangent line at that point on the curve.
4. Using Derivatives for Rate of Change
The derivative gives a quick way to find how fast a quantity is changing. For a function y = f(x), the rate of change is:
\frac{dy}{dx} = f'(x)
This tells how much y changes for a very small change in x.
5. Examples of Rate of Change
These short examples show how derivatives measure change in different contexts.
5.1. Example 1 — Distance and Speed
5.2. Example 2 — Growth of an Area
If the area of a circle depends on radius:
A(r) = \pi r^2
Then the rate of change of area with respect to radius is:
A'(r) = 2 \pi r
5.3. Example 3 — Population Trend
If population is modeled as:
P(t) = 500 e^{0.02t}
The rate of change at time t is:
P'(t) = 10 e^{0.02t}
6. Instantaneous vs Average Rate — Quick Comparison
| Average Rate | Instantaneous Rate |
|---|---|
| Over an interval | At one point |
| Slope of secant line | Slope of tangent line |
| Uses finite change | Uses derivative |
7. Why Rate of Change Is Important
Understanding rate of change helps in analyzing motion, growth, decay, and almost every situation where quantities shift continuously. The derivative gives a precise way to track these changes at any moment.