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)