# 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 Waiting-Time Surprise

Consider the sum of $$n$$ random real numbers, uniformly distributed in the unit interval. Although the expected value of this sum is $$n/2$$, the value of $$n$$ for which this sum first exceeds a given target value $$t$$ is expected to be more than $$2t$$, by an amount that is asymptotically constant.

##### Pairs of Equal Surface Functions

The authors describe how to generate many pairs of smooth functions having the property that slices of the two corresponding surfaces of revolution have equal surface areas.

##### Chaos and Love Affairs

Love affairs are modeled, amusingly, using chaotic dynamical systems.

##### Proof without Words: A 2 x 2 Determinant Is the Area of a Parallelogram

The author presents a visual proof that the determinant of a 2 by 2 matrix equals the area of the corresponding parallelogram.

##### The Euler-Maclaurin Formula and Sums of Powers

Bounds are obtained for the sums of powers of the first $$n$$ integers.

##### Linear Algebra in the Financial World

Linear algebra is used to study financial trading strategies and expectations. Financial conditions are examined via matrix equations, using rank, column space, and null space arguments.