# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### Do Dogs Know Calculus of Variation

This article generalizes a problem of dogs fetching a ball in the water and presents it as a problem of calculus of variation.

##### On Rational Function Approximations to Square Roots

Using a result on periodic continued fractions, the author presents a rational function method of approximating square roots that is faster than Newton's method.

##### Averaging Sums of Powers of Integers

When is the average of sums of powers of integers itself a sum of the ﬁrst $$n$$ integers raised to a power? We provide all solutions when averaging two sums, and provide some conditions regarding when larger averages may have solutions.

##### Amalgamation of Formulae for Sequences

Many calculations in number theory have different rules for different residue class mod an integer $$r$$. The author shows how to combine these cases using a single amalgamated formula.

##### Universality in Mathematical Modeling: A Comment on "Surprising Dynamics From a Simple Model"

A few simple models in the fields of neuroscience and high voltage engineering are presented and discussed.

##### Introducing the Sums of Powers

This is an attempt to motivate as well as prove formulae for sums of powers of integers.