# 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

##### Proof without Words: A Sum and Product of Three Tangents

The author presents a visual proof that the sum of the tangents of the angles in an acute triangle is equal to the product of the tangents.

##### A Quick Proof that the Least Squares Formulas Give a Local Minimum

Use the second derivative test to establish that the regression parameters give a minimum.

##### The Chain Rule for Matrix Exponential Functions

If $M(t)$ is a matrix of differentiable functions, the chain rule applies to $\exp(M)$ if $M$ and $M'$ commute.

##### A Property Possessed by Every Differentiable Function

If $f$  is differentiable at $x$, an interesting relationship exists between the tangent and two secant lines.

##### Combinatorial Proofs via Flagpole Arrangements

Guy wires to support flag poles give pictorial representations that lead to combinatorial proofs.

##### Rounding Errors to Knock Your Stocks Off

The author describes how small truncation errors propagated dramatically on the Vancouver stock exchange.