# Browse Classroom Capsules and Notes

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The authors exhibit two differentiable functions for which the integration by parts formula does not apply.
The author analyzes the motion of a hollow ball with a viscous fluid center as it slowly rolls down a slightly inclined plane.

Three new proofs of the fact that the point $$A$$ on the circular arc $$BC$$ for which $$AB + AC$$ is maximum is the midpoint of the arc are presented.

This note uses the Gauss sums to evaluate certain sums of trigonometric functions. It includes a short history of Gauss sums.

The result in the title is demonstrated graphically.

By exploiting the geometry of the cobweb plot, the author provides a simple and elementary derivation of the parameter for the period-three cycle of the logistic map.

The author gives a mapping of the ordered pairs of positive integers onto the positive integers which is immediately recognizable as a bijection.

The author uses Newton`s impact method to compute the centripetal force exerted on a particle moving uniformly on a circumference of a noneuclidean circle.

A combinatorial proof of a formula for the sum in the title is provided.
How do sequences of the form $$(1+x/n)^{n + \alpha}$$ with $$x > 0$$ approach their limits?