Fun Topics in Elementary Calculus

The ideas of calculus appear in elemenary mathematics. On this page, I collect some of the ones I've thought about. Enjoy!

Elementary introduction to the logarithm rule

The logarithm rule \[ \log(st) = \log(s) + \log(t) \] comes from the geometry of the hyperbola \(xy = 1\). Let \(A(a,b)\) denote the area between \(a\) and \(b\) under the curve \(y = 1/x\) and consider the animation here.

In the first half, the blue area is scaled horizontally by a factor of \(1/3\) and \(1/x\) changes to \(1/(3x)\). Then the blue area is scaled vertically by a factor of \(3\) and \(1/(3x)\) changes to \(3/(3x)\), which is the same as \(1/x\). The total scaling of the blue area area is the product of the two factors, \(3 * 1/3 = 1\) ! (excitement)

[the logarithm rule]

The area under \(1/x\) satisfies the logarithm rule.

tags: fun-topics

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