Po-Shen Loh | A cute proof that makes e natural

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 point

Geometrically, 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.

Easyapproximation

Let’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 everywhere

The 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 lineis

The bracket is the slope of the line throughand, so asshrinks, the bracket becomes the slope of the tangent toat the-intercept. That miraculously cleans up to justby our definition of. (And that is precisely why we built the definition this way.)

So, the slope of the tangent atis just.

Rephrased in Calculus language:is its own derivative. This is perhaps the single most important property of, because all of the Calculus facts stemming fromcan be deduced from this fact.

Compound interest limit

Pre-Calculus usually teaches a different definition of, as the limit of the expressionwhich arises from continuously compounded interest. To reconcile the approaches, we now visually prove thatapproaches the same number we defined.

Sinceis the inverse function offor any base, using our basewe getWe used base(instead of, say,) because it now conveniently suffices to show that the expression in the exponent tends toasgrows. That expression rearranges into a slope calculation!That’s the slope of the line through the pointon the curveand another point very nearby on the curve. Asgrows, that tends to the slope of the tangent line at. We are done as soon as we prove that slope is(which is also a natural objective to seek).

To that end, sinceis the inverse function of, their graphs are reflections over the line.

Both of the following lines have slope:

• the tangent line tothroughby definition of; and
• the line.

So, they are parallel, making this nice reflection:

Therefore, the slope of the tangent line toatis indeed, completing the proof!

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