Digital garden
Informal mathematical notes, historical curiosities, solutions to interesting problems, and occasional reflections on academic life.
We usually think of differentiation and inversion as completely different operations. But what happens if we force them to be the same? The solution to $f'(x) = f^{-1}(x)$ leads us unexpectedly to the Golden Ratio and generalized "Metallic Means".
Read articleA classic optimization problem proposed by Jose M.: What is the largest ellipse that fits inside a circle of radius $r$? By analyzing the tangency conditions of the curves, we derive the maximum area $A = \frac{2\sqrt{3}}{9}\pi r^2$.
Read articleComputing the sum $\sum_{k=1}^{n} \lfloor \sqrt{k} \rfloor$ term by term is inefficient for large $n$. In this post, we derive an elegant closed-form formula by regrouping the terms based on the intervals between consecutive perfect squares.
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