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Browse Classroom Capsules and Notes

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Using a simple trigonometric limit, the author provides an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix.

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...

The author provides a simple context and some history in which actuarial computations take place.

In a classic pouring problem, given two jugs with capacities \(m\) and \(n\) pints, where \(m\) and \(n\) are relatively prime integers, and an...

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...

Consider the sum of \(n\) random real numbers, uniformly distributed in the unit interval. Although the expected value of this sum is \(n/2\), the value of \(n\) for which...

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...

In Sam Loyd's classical Courier Problem, a courier goes around an army while both travel at constant speeds. If the army travels its length during the time the courier makes his trip, how...

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 result \( \frac{1}{n} + \frac{1}{n^2}  + \frac{1}{n^3} + \cdots = \frac{1}{n-1} \) is illustrated for \( n = 9 \) and the remark is made that a similar construction shows the result...

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