# 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

##### A Bug Problem

This capsule discusses rates of change related to a vessel obtained as a surface of revolution. The question is about the rate a bug must move on the surface as liquid is being poured into the solid.

##### Permutations and Coin-Tossing Sequences

The author provides a combinatorial explanation of a strikingresult from Herbert Wilf`s book “generatingfunctionology” , equating the likelyhood of certain permutations with the likelyhood of $$n/2$$ heads in $$n$$ tosses of a coin.

##### A Short Proof of Turan' s Theorem

This note shows that graphs with $$n$$ vertices containing no complete graph with $$r$$ vertices, have no more than $$(r - 2)n^2/(2r - 2)$$ edges , for $$r \geq 2$$.

##### Inverse Conjugacies and Reversing Symmetry Groups

Let $$G$$ be a group and $$C(a)$$ be the centralizer of $$a\in G$$.  The author studies the properties of the skew centralizer $$B(a)=\{x\in G : xa=a^{-1}x \}$$ and the reversing symmetry group $$E(a)=B(a)\cup C(a)$$ of $$a$$.  Many properties provide nice exercises for an introductive course of abstract algebra.  The author also shows the dynamical origin and applications of such algebraic structures, which arise naturally from the ergodic theory of measure-preserving transformations.

##### Using Continuity Induction

Here is a technique for proving the fundamental theorems of analysis that provides a unified way to pass from local properties to global properties on the real line, just as ordinary induction passes from local implication (if true for $$k$$, the theorem is true for $$k + 1$$) to a global conclusion in the natural numbers. The author demonstrates this method by proving the Intermediate Value Theorem and the Heine-Borel Theorem.

##### Proof without Words: Recursion

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