Foundations — Antiderivative — definition, family of solutions (+C)

This page assumes you have seen nothing. Before you can read the parent note Antiderivative — family of solutions, every squiggle it uses has to be earned. Below is each symbol, in build-order: plain words → the picture → why the topic needs it.

0. What is a function $f$? — the machine

Picture. Imagine a plot of land. At each horizontal position $x$ (how far east you walk), the machine tells you the height $f(x)$ of the ground there. Drawing every $(x,f(x))$ pair gives a curve — the shape of the hill.

Why the topic needs it. Antiderivatives are entirely about turning one curve into another curve, so we must first agree that a curve is a function: one height per position.

1. The coordinate plane and $(x,y)$ — where pictures live

Picture. Look at figure s01 again: the flat arrow pointing right is the $x$-axis, the arrow pointing up is the $y$-axis. The curve is the collection of all points $(x,f(x))$.

Why the topic needs it. The parent's phrase "the point $(0,5)$ selects one curve" is meaningless unless you can read $(0,5)$ as "input $0$, height $5$."

2. Slope — how steep is the hill?

Picture. Pick two nearby points on the curve and draw the straight line through them. Its steepness is the slope. Slide the two points until they almost touch — the line becomes the tangent line, the direction the curve is heading right at that spot.

Why the topic needs it. The whole antiderivative question is "I know the slope everywhere — rebuild the curve." Slope is the raw material.

3. The derivative $F^{\prime }$ and $\frac{d}{dx}$ — the slope-machine

Two notations, one meaning.

• $F^{\prime }(x)$ — the prime mark ${}^{\prime }$ just means "the slope-version of $F$."
• $\frac{d}{dx}(\cdots )$ — read as "the slope, with respect to $x$, of whatever is inside." The $d$'s are a shorthand for tiny change: a tiny change in the inside per tiny change in $x$.

Picture. Above the hill $F$, draw a second curve whose height at each $x$ equals the steepness of the hill there. Where $F$ climbs steeply, $F^{\prime }$ is high; where $F$ is flat (a peak or valley), $F^{\prime }$ touches zero.

Why the topic needs it. The parent's core equation $F^{\prime }(x)=f(x)$ says exactly: "the slope-machine applied to my mystery curve $F$ gives the known curve $f$." Antidifferentiation is running this backwards.

4. The constant $C$ — the hidden starting height

Picture. A flat line never rises or falls, so its slope is $0$ at every point — this is the picture behind $\frac{d}{dx}(C)=0$.

Why the topic needs it. Since a flat line has zero slope, gluing it on top of any curve shifts the curve up by $C$ without touching any slope. That is precisely the freedom the antiderivative can never see — the famous $+C$.

5. The integral sign $\int f(x)dx$ — "run the machine backwards"

Picture. Figure s03 read right-to-left: you are handed the lower slope-curve $f$ and must reconstruct an upper curve whose steepness matches it everywhere — knowing you can slide the answer up or down freely.

Why the topic needs it. This is the entire notation of the chapter. Its answer is always written $F(x)+C$, tying together every symbol above.

6. Interval $I$ and $x\in I$ — where the rule is allowed to hold

Picture. A single solid segment with no gaps. Contrast with $\{x\ne 0\}$, which is two segments $(-\infty ,0)$ and $(0,\infty )$ separated by a hole at $0$.

Why the topic needs it. The proof "zero slope everywhere ⇒ constant" only works on one connected piece. Across a gap, the two pieces can sit at different heights — this is exactly the parent's sneaky Example 3 with $\ln |x|$, where each side of $0$ gets its own constant $C_1,C_2$. That subtlety is guaranteed by Mean Value Theorem.

7. The last few symbols, quickly

Prerequisite map

Equipment checklist

Test yourself — reveal only after answering aloud.