A cute proof that makes e natural

Summary

For the full article covering many properties of, including history and comparison with existing methods of teaching: PDF from arXiv. A video explanation will be posted here shortly.This webpage pulls out the part of the article which uses Pre-Calculus language to explain what is so natural about, while intuitively connecting the following two important properties:The slope of the tangent line toat the pointis just. (In Calculus language:is its own derivative.)The expressionapproachesasgrows.Key conceptual starting pointGeometrically, there really is only one exponential function curve shape, because all exponential function curves(with positive real bases) are just horizontal stretches of each other. This is exactly like how all ellipses are just stretches of each other (and for the same reason).For example,, stretched horizontally by a factor of, is.Geometrically, since stretching is a continuous process, exactly one of these horizontally stretched exponential curves has the property that its tangent line at its-intercept has the particularly nice and natural slope of.We defineto be the unique positive real base corresponding to that curve.EasyapproximationLet's find a number whose exponential curve has tangent slopeat the-axis. For this, we take the curveand estimate what factor to horizontally stretch it. To start, we must estimate the slope of the tangent line toat its-intercept. But how? Does that need Calculus? No! Algebra is enough!Consider a very-nearby point on the curve:, whereis tiny but not zero. The slope of lineisUseto approximate that tangent slope:Thus a horizontal stretch by a factor ofwill make the tangent slope. Sohas a tangent slope of.Therefore,is close to. This is pretty good, because actually.Beautiful tangent slopes everywhereThe same method derives the slope of the tangent line toat any point. Consider a very-nearby point on the curve:, whereis tiny but not zero. The slope of lineisThe bracket is the slope of the line throughand, so asshrin...