Derivatives (12 videos)
Power Series/Euler's Great Formula
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A special power series is e^x = 1 + x + x^2 / 2! + x^3 / 3! + … + every x^n / n!
The series continues forever but for any x it adds up to the number e^x
If you multiply each x^n / n! by the nth derivative of f(x) at x = 0, the series adds to f(x)
This is a TAYLOR SERIES. Of course all those derivatives are 1 for e^x.
Two great series are cos x = 1  x^2 / 2! + x^4 / 4! … and sin x = x  x^3 / 3! ….
cosine has even powers, sine has odd powers, both have alternating plus/minus signs
Fermat saw magic using i^2 = 1 Then e^ix exactly matches cos x + i sin x.
Professor Strang’s Calculus textbook (1st edition, 1991) is freely available here.
Subtitles are provided through the generous assistance of Jimmy Ren.
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As Taught In:  Spring 2010 
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Undergraduate
