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Calculus 1 · Axiom Academy
Where comes from — the Pythagorean theorem applied to a curve, one infinitesimal piece at a time. 1. The Problem: A Curve Beats Its Shortcut Pick two points on a curve y=f(x) . The straight chord between them is the shortest route — so the arc that actually follows the curve is always longer . Watch the traveler crawl along the bend: it covers far more ground than the dashed shortcut beneath it. 2. Approximate It With Straight Pieces Split [a,b] into n equal slices of width and connect the dots on the curve with straight segments. The total length of that polygonal path approximates the arc. Watch the count climb : the corners hug the curve tighter and the running length creeps up toward the true value. 3. One Segment Is a Right Triangle Zoom in on a single piece. It rises by as it runs across by , so the segment is the hypotenuse of a right triangle. The Pythagorean theorem gives its length directly: Now factor out from under the root — that single algebraic move is what makes a derivative appear: As the segment shrinks, and its slope becomes the exact derivative f'(x) — the slope of the tangent line. The little segment becomes the infinitesimal arc element ds . Watch the secant chord swing into the tangent as the second point slides home: Slope , the average rate over the whole piece. Slope , the exact rate at the point. This is the length of an infinitesimally short piece of the curve — a tiny hypotenuse with horizontal leg dx and slope f'(x) .
This is the written version of the interactive lesson above. See the full Calculus 1 course.