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

Nov 10 - Natural logarithm

We begin working with logarithmic functions today, starting with differentiation. The highlight of class today is seeing the "new" definition of ln(x). We give it an integral definition and see how log properties are consequences of calculus ideas. It's a brand new world!

We will rarely, if ever, worry about logs with other bases. Generally, we use the base change formula to turn another based logarithm into ln x. For example, log x = ln(x) / ln(10).

Notes from today are below. Tonight's assignment will be mostly practice finding derivatives that involve ln(x).