Section+3.14

Section 3.14
 * Analysis of f(x)=e^x and g(x)=ln x**
 * BTW I MADE A PAGE CALLED CHAPTER 14 WHICH I HAVE NO IDEA HOW TO DELETE :( PLS HELP !!!**

In section 3.14 we explored the equations f(x)=media type="custom" key="11814124" and g(x)=ln x and how they relate to each other.

In class we did an exercise with the slopes of ln x and media type="custom" key="11814120". We used the method of finding the average rate of change, or the slope of the secant line at two very close together points. Using this method we came up with two tables showing the slopes at certain points.

= = || 1.28 ||
 * X || F(x) || Slope of F(x) ||
 * 2 || media type="custom" key="11814104" || 7.39 ||
 * 3 || media type="custom" key="11814086" || 20.09 ||
 * 1 || media type="custom" key="11814090" || 2.72 ||
 * 1/2 || media type="custom" key="11814032" || 1.64 ||
 * 1/4 || media type="custom" key="11814034"

What we can take away from these tables is that the slope of ln x at a certain point is 1/x. and that the Slope of the function e^x at a certain point is simply e^x.
 * X || F(x) || Slope of F(x) ||
 * 2 || [[image:http://latex.codecogs.com/gif.latex?%5Cln%202 width="27" height="14"]] || 1/2 ||
 * 3 || [[image:http://latex.codecogs.com/gif.latex?%5Cln%203]] || 1/3 ||
 * 1 || [[image:http://latex.codecogs.com/gif.latex?%5Cln%201]] || 1 ||
 * 1/2 || [[image:http://latex.codecogs.com/gif.latex?%5Cln%201/2]] || 2 ||
 * 1/4 || [[image:http://latex.codecogs.com/gif.latex?%5Cln%201/4]] || 4 ||


 * Theorem 3.13 : The tangent to the graph of y= ln x at the point (a, ln a) has slope of 1/a**
 * Theorem 3.14 : The tangent to the graph of y=media type="custom" key="11814134" at the point (a, media type="custom" key="11814326") has slope media type="custom" key="11814324"**

Its also important to remember that e^x and ln x are inverse functions. this means that -> Graphing the two functions will also allow us to see some interesting properties of the two. First off the domain of the graph of media type="custom" key="11814138" is (-∞, ∞) and the range is (0, ∞). Therefore the domain of ln x is (0, ∞) and the range is (-∞, ∞). The two functions also appear to be reflected over the line y= x.
 * IMPORTANT NOTE!!!!! THIS IS WHY MATHEMATICIANS (such as Mrs. Tyson) FIND media type="custom" key="11814128" TO BE SO AWESOME!!!! ITS SLOPE IS BALLER!!!!**

Your Fellow Admin Daniel Lian