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Browse Classroom Capsules and Notes

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Displaying 191 - 200 of 225

Proof without Words: Arithmetic Progressions with Sum Equal to the Square of the Number of Terms
A visual verification of the fact that the sum of the integers from n to 3n-2 is (2n-1)2 in the case that n = 4.
Math Bite: Once in a While, Differentiation Is Multiplicative
The author finds and discusses functions that obey the "erroneous product rule".
Proof without Words: The Cosine of a Difference

The authors present a visualization of the formula for the cosine of the difference of angles.

Names of Functions: The Problems of Trying for Precision
Mathematics notation is discussed, focusing on the importance, inconsistencies, eccentricities and abbreviations.
A Differentiation Test for Absolute Convergence
The author presents an easy absolute convergence test for series based solely on differentiation, with examples.
Note on the Evaluation of \(I = \int_{0}^{x} \frac{1}{1+t^{2^n}} dt\)
The authors develop a general formula for integrals of \(1/(1 + t^m)\), where \(m\) is an integral power of \(2\).
Inequalities of the Form \( f(g(x)) \geq f(x)\)

The author gives two applications of a method for finding a function \(g\) such that \(f(g(x)) \geq f(x)\).

Proof Without Words: Alternating Sums of Squares of Odd Numbers

A visual solution for an alternating sum of odd squares is presented.

Proof Without Words: Alternating Sum of an Even Number of Triangular Numbers
The author provides a visual solution for the alternating sum of an even number of triangular numbers.
On Antiderivatives of the Zero Function

The author discusses an intuitive direct proof of the fact that functions with zero derivative must be constant, which turns into a rigorous proof by simply invoking the completeness of the...

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