# 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

##### The Cobb-Douglas Function and Hölder's Inequality

Hölder’s inequality is here applied to the Cobb-Douglas production function to provide simple estimates to total production.

##### Fetching Water with Least Residues

In a classic pouring problem, given two unmarked jugs with capacities $$m$$ and $$n$$ pints, where $$m$$ and $$n$$ are relatively prime integers, and an unlimited supply of water, the goal is to obtain exactly $$p$$ pints, where $$p$$ is an integer, $$0 < p < m+n$$. This capsule uses properties of least residues to show that there are two distinct pouring sequences to achieve the desired result. The more efficient sequence can be determined by solving a linear congruence.

##### Walking with a Slower Friend

Fay and Sam go for a walk. Sam walks along the left side of the street while Fay, who walks faster, starts with Sam but walks to a point on the right side of the street and then returns to meet Sam to complete one segment of their journey. The authors determine Fay’s optimal path minimizing segment length, and thus maximizing the number of times they meet during the walk. Two solutions are given: one uses derivatives; the other uses only continuity.

##### Rings on $$R^2$$

How can we define multiplication of ordered pairs of real numbers to make $$R^2$$ into a ring, or field? In this article, the author derives conditions for certain types of rings and extends the results to $$R^3$$.

##### Proof without Words: Recursion

The author presents a visual proof of the closed form of a recursively defined sequence: $$a_2 = 3$$ and $$a_n = 2 a_{n-1} + 1$$

##### Approaches to the Formula for the $$n$$th Fibonacci Number

In this capsule, proofs of the equivalence of the two definitions of the Fibonacci numbers are discussed. This helps the undergraduate view mathematics as a unified whole with a variety of techniques.