Browse Classroom Capsules and Notes

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Convergence-Divergence of \( p\) -Series

This article is a discussion of convergence and divergence of \(p\)-series without using the Integral Test.

Investigating Possible Boundaries Between Convergence and Divergence

Once students master the Integral Test, it is useful to show that there cannot be a series on the boundary between convergence and divergence.

On the Indeterminate Form \(0^0\)

This article gives an explanation of why most textbook examples of the form \(0^0\) have a limit of 1.

On the Work to Fill a Water Tank

This article describes an alternate way to motivate the integral set up for work done in raising water in a tank.

Off on a Tangent

The authors investigate families of curves whose tangent lines at a fixed \(y\)-coordinate go through the origin.

A Dozen Minima for a Parabola

Except at the vertex, the normal to a parabola at \(P\) intersects it again at a point \(Q\). There are many interesting minimization problems generated by the line segment \(PQ\).

Column Integration and Series Representations

The author explains why column integration (also called tabular integration by parts) works.

Constrained Optimization with Implicit Differentiation

Optimization of \(f(x,y)\), given the constraint \(g(x,y)=0\), can be done using implicit differentiation on both \(f(x,y)\) and \(g(x,y)=0\).

Visual Proof of Two Integrals

Two indefinite integrals are found by analysis of areas.

Maximizing the Area of a Quadrilateral

Given the lengths for four sides, the quadrilateral of maximum area is cyclic, i.e., its vertices lie on a circle.