1. The Reversal Problem — Why Antiderivatives Matter
The Central Question
Throughout this course you have learned to differentiate: given a function, find its rate of change. The antiderivative asks the reverse question:
Given a function $f(x)$, can we find a function $F(x)$ whose derivative is $f(x)$?
Motivation from Physics
Recall the chain of motion quantities:
$$ \text{Position } s(t) \;\xrightarrow{\text{differentiate}}\; \text{Velocity } v(t) \;\xrightarrow{\text{differentiate}}\; \text{Acceleration } a(t) $$Differentiation moves you left to right along this chain. But in many real problems you start on the right:
- A sensor measures acceleration $a(t)$. You need the velocity $v(t)$, so you must reverse differentiation.
- Given velocity $v(t)$, you want position $s(t)$ — again a reversal.
Each reversal is an antidifferentiation step:
$$ \text{Acceleration } a(t) \;\xrightarrow{\text{antidifferentiate}}\; \text{Velocity } v(t) \;\xrightarrow{\text{antidifferentiate}}\; \text{Position } s(t) $$Motivation from Geometry
Imagine you are told the slope of a curve at every point — a slope field. The antiderivative is the curve itself: the function whose tangent-line slopes match the given field everywhere. Finding the antiderivative means recovering the curve from its slopes.
2. Definition and Verification by Differentiation
Definition
A function $F$ is called an antiderivative of $f$ on an interval $I$ if
$$ F'(x) = f(x) \quad \text{for every } x \in I. $$