Nov 11 - Exponential Functions

Yesterday I applied the Fundamental Theorem of Calculus to "assign" an antiderivative to y = 1/x. When we studied the properties of the antiderivative, it became apparent that it was a logarithm, thus ln(x) is defined to be the integral of 1/t on the interval 1 to x.

Today we use this definition of ln(x) to arrive at a definition of the number e. We will define e as the bound of the region that yields an area between y = 1/x, y = 0 (the x-axis), x = 1 and x = e equal to 1. Thus e is the number such that ln(e) = 1.

The most significant consequence of this is that it turns out if y = e^x, then dy/dx = e^x. It's weird and it's wonderful. And it brings a whole new feel to finding derivatives.

(Note too that it means that e^x is also it's own _antiderivative_, but we'll save that for next Tuesday...)

Notes from today are posted below.
p1

p2