Browse Classroom Capsules and Notes

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Means to an End

The limit of the geometric mean of the first \(n\) integers raised to the real positive power \(s\), divided by their arithmetic mean is shown to be \((s+1)/e^s\). An elementary derivation of...

Proof Without Words: Logarithm of a Number and its Reciprocal

A visual representation of the title is presented.

The Product and Quotient Rules Revisited

Mathematical elegance is illustrated by rewriting the product and quotient rules of basic calculus in strikingly parallel forms. Applications are given for which these forms suggest meaning....

Limit Interchange and L'Hôpital's Rule

The author provides a somewhat recreational application of limit interchange and L'Hopital's rule. The goal is to hint at the importance of limit/sum interchange in Analysis without,...

Derivative Sign Patterns

Motivated by the observation that the derivatives of \(e^x\) are all positive and the derivatives of \(e^{-x}\) alternate sign, the author asks what kinds of ``sign patterns" are possible...

Walking with a Slower Friend

Fay and Sam go for a walk. Sam walks along the left side of the street while Fay, who walks faster, starts with Sam but walks to a point on the right side of the street and then returns to...

Variations of the Sliding Ladder Problem

This article takes another look at the sliding ladder problem that students meet in the study of related rates in calculus. Three variations of the problem are analyzed using elementary...

The Right Theta

The formula \( \theta = \arctan(y/x) \) gives the angle associated with a point \( (x,y) \) in the plane, valid for \( \mid \theta \mid < \pi/2 \). This capsule presents a formula...

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.

Riemann Sums and the Exponential Function

Every standard calculus textbook contains the derivations for the definite integral of \(x\) and \(x^2\) using Riemann sums \(\ldots\)