Browse Classroom Capsules and Notes

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Calculus Measures Tank Capacity and Avoids Oil Spills

"Imagine a fuel tank in the shape of a horizontal cylinder, with length \(l\) and circular cross section with diameter \(d \ldots\)"

Disks, Shells and Integrals of Inverse Functions

Most calculus students take it for granted that the shell and disk methods for computing the volume of a solid of revolution must always give the same result.

An Optimization Oddity

"This problem also fits quite nicely in the area of convex geometry, from which we draw the following proof..."

A Balloon Experiment in the Classroom

Hands-on activities using balloons for a differential equations class

A Generalization of the Mean Value Theorem for Integrals

A generalization of the Mean Value Theorem for Integrals in terms of Riemann sums

Some Subtleties in L'Hôpital's Rule

This is a discusion of the importance of the existence of limits in L'Hôpital's Rule.

An Alternative for Certain Partial Fractions

An alternative to partial fractions for certain integral calculations

The Derivatives of \(\sin x\) and \(\cos x\)

Simple proofs of the formulas for the derivatives of \(\sin x\) and \(\cos x\) when \(x\) is acute

Queries

Examples of an elementary transcendental function having three different kinds of asymptotes \(x-a-0, y-b=0, y=mx+c\) and defined by one single equation in its domain of definition

Another Look at \(x^{1/x}\)

Using the Arithmetic-Geometric Means Inequality and the Mean Value Theorem to show that the maximum value of \(x^{1/x}\) occurs at \(x = e\)