# 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.

## Featured Items

##### Bounds on the Perimeter of an Ellipse via Minkowski Sums

The author uses Minkowski sums to estimate the length of the perimeter of an ellipse.

##### Why is the Sum of Independent Normal Random Variables Normal?

The authors give two proofs, one geometric and one algebraic, to provide insight into why the sum of independent normal. random variables must be normal.

##### Monotonic Convergence to $e$ via the Arithmetic-Geometric Mean

Hansheng Yang and Heng Yang used the arithmetic-geometric mean inequality to prove that the sequence $[1+1/n]^n$ is monotonic increasing converging to $e$ whereas $[1+1/n]^{n+1}$ is monotonic decreasing converging to $e$. The author provides a simpler proof using the same technique.

##### Factoring Quartic Polynomials: A Lost Art

An old and forgotten method for determining whether a quartic polynomial over the rationals is reducible is revived.

##### Proof Without Words: The Area of a Right Triangle

A visual determination of the area of a right triangle is given using an inscribed circle.

##### The Locus of the Focus of a Rolling Parabola

The authors show that the locus of the focus of a parabola rolling on the $x$-axis is a catenary.