# Browse Classroom Capsules and Notes

You can filter the list below by selecting a subject category from the drop-down list below, for example by selecting 'One-Variable Calculus'. Then click the 'APPLY' button directly, or select a subcategory to further refine your results.

Displaying 71 - 80 of 1209

In this article a classroom "trick" involving square arrangements of natural numbers is used to motivate a discussion of a special class of matrices. In particular, a basis is...

In this note, the Cauchy-Schwarz inequality is derived from Pearson's inequality. It follows that the two are equivalent.

This note is on the digital (floating-point) representation in various arithmetic bases of the reciprocal of an integer $$m$$. An algorithm is given to change the representation of $$1/m... 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. Some results in plane geometry can be proved easily using 3-dimensional vectors. This article explores several such examples. Two of Euler's familiar identities, the sum of reciprocals of squares of integers and the infinite product expression of sine function, are proved again. Every standard calculus textbook contains the derivations for the definite integral of \(x$$ and $$x^2$$ using Riemann sums $$\ldots$$

The classical Buffon needle problem is to find the probability that a needle of length $$n$$ when dropped on a floor made of boards of width $$b$$ will cross a crack between the boards.

Geometrical notions are abundant in calculus, where one learns how problems involving them can be addressed via the derivative or integral.

There are few random processes more avidly watched than the state lottery drawings in which six numbered balls are chosen from a set of forty-four.