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Browse Classroom Capsules and Notes

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Displaying 201 - 210 of 225

Proof Without Words: Exponential Inequalities

Two visual proofs of some exponential inequalities are presented.

Descartes Tangent Lines

Given any positive integer \(n\), find polynomial curves that have exactly \(n\) tangent lines which intersect the curve exactly once.

Partial Fractions by Substitution
Use substitution to calculate partial fraction expansions
Finding Curves with Computable Arc Length

The author describes a method for identifying curves where the arc length is easy to compute by symbolic integration.

Arc Length and Pythagorean Triples

A family of curves whose lengths are closely related to Pythagorean Triples and therefore rational

An Area Approach to the Second Derivative
Use area to describe the second derivative of a function
The Convergence Behavior of \(f_{\alpha}(x)=(1+\frac{1}{x})^{x+\alpha}\)
Descriptions of the behavior of a family of functions
Proof Without Words: Steiner's Problem on the Number \(e\)

For what positive number \(x\) is the \(x\)-th root of \( x\) the greatest?

The Relationship between Hyperbolic and Exponential Functions
The article attempts to explain why exponential functions were chosen to define hyperbolic functions.
Applications of Transformations to Numerical Integration

The authors discuss the evaluation of certain improper integrals by first transforming them.

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