Browse Classroom Capsules and Notes

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Do Estimates of an Integral Really Improve as \(n\) Increases

The author discusses necessary conditions for montonicity of various integral approximation methods as the number of points increases.

Oscillating Sawtooth Functions

The author describes examples of sawtooth functions which are derivatives but are not continuous.

The Trapezoidal Rule (for Increasing Functions)

This is a "Proof Without Words," giving a visual representation of the calculation for the Trapezoidal Rule.

Variations on a Theme of Newton

The author uses the computer algebra system Maple to introduce a number of interesting investigations based on Newton's Method for finding roots of an equation.

Farmer Ted Goes Natural

The author investigates properties of "almost squares," rectangles with integer sides and area \(A\) such that the the perimeter is least among all rectangles with integer sides and...

Using Less Calculus in Teaching Calculus: An Historical Approach

The author uses topics from the history of mathematics to advance the notion that geometry and algebra have a pedagogical role to play in solving some problems routinely solved by "brute...

Two Methods for Approximating \(\pi\)

The author summarizes a variety of ways to approximate \(\pi\).

Just an Average Integral

The antiderivative of the cube of the secant function is the average (arithmetic mean) of the derivative and antiderivatives of the secant function. That is, what are the solutions of \(\int y...

Parametric Integration Techniques

A definite integral involving a parameter is an equation in that parameter. Differentiating both sides can yield new integrals.

Proof without Words: Differentiated Geometric Series

The author presents a visual proof of the formula for the derivative of the geometric series.