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Browse Classroom Capsules and Notes

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Displaying 11 - 20 of 225

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...

A visual representation of the title is presented.

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....

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,...

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...

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...

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 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...

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.

An inductive proof is presented for the bounds of the remainder of Taylor expansion. This result, with Darboux's theorem, implies the classical formula for the remainder.

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