What's the intuition for understanding that the second derivative does correspond to the curvature of the function?
For first derivatives one can think of the tangent lines, but what to think for second derivatives?
2026-03-30 15:53:14.1774885994
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Intution: second derivative corresponds to curvature?
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How about the definition of curvature? Let $y=f(x)$ be a function. Then we have:
$sgn(\kappa) = sgn(y'')$
The second derivative is the instantaneous rate of change of the first derivative.
So if the second derivative is large and positive, then the slope of the tangent line is increasing quickly, which means the graph is curving sharply.