# functions

## Devil’s Staircase

Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically...

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## Multidimensional Newton’s Method

You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did...

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## Sum of Cubes and Beyond

We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the...

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## Continuous but Nowhere Differentiable

You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have...

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## Euler’s Product Formula

Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it...

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## Connected Sums

A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,...

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## Fourier Ears Only

Did you know that every sufficiently smooth function on an interval can be expressed as an infinite sum of sines...

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## Quick Square Roots

Here’s a very quick way to generate the square root of N. Let A0=N. Then generate a sequence of numbers A1, A2,...

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## Sine of (1/55555….)

Take your calculator, and enter a number consisting of several 5’s, such as 5555555. Now take the reciprocal of this number. Now take the...

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## Area of a Hyperbolic Wedge

The unit circle can be parametrized by (cos w, sin w). Given a point on it, the region cut out...

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