Calculus
Second Derivatives & Curvature
Take two points either side of where you are and join them with a straight line. The curve does not pass through the middle of that line: it sags below it, or bulges above it, and that gap is the second derivative. Shrink the gap between the two points and the gap to the line falls four times as fast.
the chord, and how far the curve misses its middlethe circle that fits herethe tangent
Drag along the curve to move the point · run to walk it across
Second derivative—
—
Curve
almost touchingfar apart
Values
f′′ from the formula—
f′′ from the three heights—
the chord clears it by—
curvature κ—
radius of the fitting circle—
What to observe
- Halve how far the neighbours sit and the gap under the chord falls to a quarter, not a half. A second derivative is a difference of differences, so it divides by h twice.
- On the cubic at x = 0 the bending stops: f′′ is zero, the chord runs through the curve and the fitting circle becomes a straight line of infinite radius. Either side of it the gap changes sign.
- Where the curve is steep, κ falls well below f′′: f′′ measures sag against a horizontal ruler, κ how sharply the curve actually turns. On the cubic at x = 2, f′′ is 3 but κ is about 0.24.
- Tick the parabola that matches to second order: told only the height, slope and bending, it hugs the curve near the point. That is the second-order Taylor series.
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