Browse Classroom Capsules and Notes

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The Arithmetic-Geometric Mean Inequality and the Constant \(e\).

In this note the authors present an elementary proof that the inequalities \( (1+1/n)^n < e \leq (1+1/(m-1))^m \) hold for \(n>0\) and \(m>1\), using only the arithmetic-geometric mean...

This capsule discusses the mathematics of a real-life experience with dogs playing fetch into the water. The leap-off point chosen by the dog turned out to ones that will either minimize the...

Partial Fraction Decomposition by Division

This capsule introduces a straightfoward way of calculating partial fraction decomposition for functions of specific forms by long division. The obtained partial fractions can be easily...

Do Dogs Know Calculus of Variation

This article generalizes a problem of dogs fetching a ball in the water and presents it as a problem of calculus of variation.

An Elegant Mode for Determining the Mode

A technique is introduced for the purpose of finding the relative maximum values of the probability density function. The crux is to find the critical numbers of the exponential of the...

Controlling the Discrepancy in Marginal Analysis Calculus

An example in business calculus is used to show a short-cut to compute the discrepancy between the difference of values and the differential for certain class of functions.

Another Look at Some p-series

Certain p-series are the focus of this capsule. This project comes with scenarios to help students "visualize" the convergence or divergence of the p-series.

Complementary Coffee Cups

This capsule started with two coffee cups that are complementary, i.e., their profiles fit together. The author then explores the condition in which the two cups, obtained as solids of...

The Existence of Multiplicative Inverse

This capsule discusses a way to show each non-zero element of certain rings have a multiplicative inverse. The approach is to set up a system of linear equations and the solution is the...

Proof Without Words: The Triple-Angle Formulas for Sine and Cosine.

A visualization of the triple angle formulas for sines and cosines is presented.