The Natural Exponential Function

Calculus 1 · Axiom Academy

LESSON The Natural Exponential Function Discovering e : the one base whose exponential curve has a slope exactly equal to its own height at every point. 1. Trapping e Between 2^x and 3^x Watch the slope of a^x at the starting point x=0 , where every one of these curves passes through height 1 . The tangent's steepness there is exactly . For 2^x the tangent is too shallow ( ); for 3^x it is too steep ( ). The perfect base e is trapped in between — the one whose tangent at x=0 has slope exactly 1 . The slope of a^x at x=0 is — it equals 1 only when a=e . 2. Tuning the Base Until the Slope Hits 1 Now sweep the base a from 2 upward and keep your eye on the tangent to a^x at x=0 . As a grows, that tangent rotates upward and its slope climbs along the dial below. The animation stops the instant the slope reads exactly 1.000 — and the base sitting there is . That is the definition of e : the base whose curve leaves height 1 at a rate. The tangent at x=0 is flatter than : slope . The tangent at x=0 rises at exactly : slope . The tangent at x=0 is steeper than : slope . The needle tracks ; e is precisely where it lands on 1 . Why this is the same e as compound interest Compounding 1 at 100\% continuously, the balance after one year is _ n (1+ 1n )^n=e . The sequence 2,\,2.25,\,2.4414,\,2.6130,\, creeps up to the very base our slope dial selects: e 2.71828 . 3. Slope Equals Height — Everywhere on e^x

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