# Browse Classroom Capsules and Notes

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Displaying 1171 - 1180 of 1211

Several different ways of averaging class sizes and population densities are presented from different perspectives, and the possible relations among them are discussed.

The author provides a characterization of $n$-dimensional quadratic functions in terms of tangent planes.

The authors derive an upper bound of the expected range of random points chosen from the unit interval according to any distribution.

Following his senior seminars, the author discusses Fermat's last theorem for rational and irrational exponents, in which the rational solutions are characterized.

The authors use inequalities to solve optimization problems without resorting to calculus, as illustrated by four common examples in calculus.

The author finds Riemann sums that equal exactly the definite integrals for polynomials and negative-integer power functions.

The authors model a real traffic problem by using the fundamental theorem of calculus.

The author proves visually four chain inequalities among five common means: harmonic, geometric, arithmetic, root square, and contraharmonic.

The authors show that the locus of the focus of a parabola rolling on the $x$-axis is a catenary.

The author characterizes the positive integers that are the products of two pairs of factors such that the sum of one pair equals the difference of the other.