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Browse Classroom Capsules and Notes

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A mathematician and his dog go out for a run. On their return, the mathematician wonders how much further he ran than the dog. His analysis uses parametric curves and vector calculus to obtain...
Several different ways of averaging class sizes and population densities are presented from different perspectives, and the possible relations among them are discussed.

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

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

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 authors provide a condition for a function to have nested \(n\)-th degree Taylor polynomials with varying centers, which can approximate the function visually.

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.