Foundations — Applications — increasing - decreasing, local extrema (first derivative test)

This page builds every symbol the parent note uses, starting from "what is a function's graph" and ending at "what does $f^{\prime }$ mean." If a word or squiggle appeared up there, it gets earned down here first.

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1. The graph — a road drawn from a function

Picture the horizontal axis as the ground you walk along (the input $x$) and the vertical axis as height (the output $f(x)$). As you step right, the dot moves along a curve — that curve is the road.

Why the topic needs it: the entire topic is about how this road rises and falls as you walk. Every later symbol describes a feature of this one picture.

• $x$ ::: the input; your left-right position on the ground.
• $f(x)$ ::: the output; your height above the ground at position $x$.

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2. Intervals and the symbols $(a,b)$, $[a,b]$, $\infty$

Think of a fence. $[a,b]$ has posts on the endpoints (they belong to the stretch). $(a,b)$ has the posts removed (only the strictly-inside ground counts).

• Meaning of $a<b$? ::: position $a$ is to the left of position $b$.
• $[a,b]$ vs $(a,b)$? ::: closed (includes ends) vs open (excludes ends).

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3. Slope — the number that measures tilt

Before any calculus, slope is a schoolyard idea: how much you rise for each step you take right.

What it looks like: draw the straight line joining the two points. Its steepness is this fraction. A positive value tilts uphill; negative tilts downhill; zero is flat.

• Slope sign for an uphill line? ::: positive.
• What does $\Delta$ mean? ::: "the change in."

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4. From average slope to instantaneous slope: the derivative $f^{\prime }(x)$

The slope in §3 is an average over a whole stretch $[a,b]$. But the road bends — its steepness differs at every point. We want the tilt at one exact spot.

What it looks like: start with a slanted line through two nearby points (a secant). Slide the second point toward the first. The line pivots until it just kisses the curve at one point — the tangent line. Its slope is $f^{\prime }(x)$.

Why the topic needs it: this is the engine. The parent note's every conclusion is "read the sign of $f^{\prime }$." That only makes sense once $f^{\prime }$ means "slope at a point," which we just built.

• $f^{\prime }(x)$ in plain words? ::: the slope of the graph at the single point $x$.
• Why a limit in its definition? ::: a lone point gives $0/0$; the limit finds the slope two nearby points approach.
• $f^{\prime }(x)=0$ means the tangent is...? ::: flat (horizontal).

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5. "Undefined" derivative — corners and cusps

$f^{\prime }(x)=0$ is one way the slope test triggers. The other is the derivative simply not existing.

The graph of $f(x)=|x|$ (absolute value — distance from zero, always $\ge 0$) is the classic V. Its lowest point is a genuine valley, yet $f^{\prime }(0)$ never equals zero; it fails to exist. That is why the parent defines a critical point as "$f^{\prime }(c)=0$ or $f^{\prime }(c)$ undefined" — both cases must be hunted.

• Why is $f^{\prime }(0)$ undefined for $f(x)=|x|$? ::: the left slope is $-1$, the right slope is $+1$; no single tangent exists.
• What does $|x|$ mean? ::: the distance of $x$ from zero (never negative).

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6. Symbols glossary — quick reference

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How these foundations feed the topic

The whole chain terminates in the parent topic Mean Value Theorem powers the middle link (average slope → instantaneous behaviour), and Fermat's Theorem on Stationary Points explains why extrema hide only at critical points.

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Equipment checklist

Cover the right side and test yourself — if any line stumps you, reread its section above.

• Read a graph as a road: horizontal = input $x$, vertical = height $f(x)$. ::: §1 — the graph.
• Compute slope between two points as $\frac{f(b)-f(a)}{b-a}$. ::: §3 — rise over run.
• Explain why we divide by $b-a$. ::: §3 — to measure steepness fairly per step.
• Tell open $(a,b)$ from closed $[a,b]$ intervals. ::: §2 — brackets exclude vs include ends.
• Say what ${\lim }_{h\to 0}$ means in words. ::: §4 — the value a quantity approaches as $h$ shrinks to $0$.
• State what $f^{\prime }(x)$ is in one sentence. ::: §4 — the slope of the graph at the single point $x$.
• Match each sign of $f^{\prime }$ to rising/falling/flat. ::: §4 — $+$ up, $-$ down, $0$ flat.
• Give a point where $f^{\prime }$ is undefined and why. ::: §5 — the corner of $|x|$; left and right slopes differ.
• Read $\pm$, $\Rightarrow$, $\Delta$, $\infty$. ::: §6 — glossary.