Instantaneous and Average
Your reasoning is very insightful and gets to the heart of the relationship between an average rate of change and the derivative. The core concept is exactly what you suggest:
the derivative is the limit of an average, not the average itself.
Here's a more detailed breakdown of the distinction:
| Concept | Average Rate of Change | Derivative |
|---|---|---|
| Calculation | The slope of the secant line between two points, 📷 Image (x1,f(x1)) (𝑥1,𝑓(𝑥1)) and 📷 Image (x2,f(x2)) (𝑥2,𝑓(𝑥2)) , on a curve. | The slope of the tangent line at a single point 📷 Image (x,f(x)) (𝑥,𝑓(𝑥)) on a curve. |
| Formula | 📷 Image mavg=f(x2)−f(x1)x2−x1 𝑚avg=𝑓(𝑥2)−𝑓(𝑥1)𝑥2−𝑥1 | 📷 Image f′(x)=limh→0f(x+h)−f(x)h 𝑓′(𝑥)=limℎ→0𝑓(𝑥+ℎ)−𝑓(𝑥)ℎ |
| Meaning | Measures the overall rate of change over a specified interval. | Measures the instantaneous rate of change at a single, precise point. |
| Process | It's a static calculation using two distinct points. | It is a dynamic process that uses a limit to define what happens as the interval shrinks to zero. |
How the average becomes the derivative
-
Start with an average: Pick two points on a curve,
📷 Image
(x,f(x))
(𝑥,𝑓(𝑥))
and
📷 Image
(x+h,f(x+h))
(𝑥+ℎ,𝑓(𝑥+ℎ))
. The slope of the line connecting them is the average rate of change, or the slope of the secant line:
📷 Image
f(x+h)−f(x)h
𝑓(𝑥+ℎ)−𝑓(𝑥)ℎ
.
-
Close the gap: You then use a limit to "close the gap" by moving the second point closer and closer to the first. This is done by letting the distance
📷 Image
h
ℎ
approach zero (
📷 Image
h→0
ℎ→0
).
-
Find the derivative: When
📷 Image
h
ℎ
becomes infinitesimally small, the secant line becomes the tangent line, which touches the curve at only one point. The slope of this tangent line is the derivative, which represents the instantaneous rate of change.
The derivative is the specific, final value that the average rate of change approaches as the interval closes. It is the result of that limiting process, not the process itself. Your understanding that "it is an average until then, no?" is a great way to describe this progression.