Example+Post+(equation+of+tangent)

Here I will provide an example of what I would consider to be a suitable post on this wiki. We will probably cover more than this most days, so your post will likely be longer. Keep in mind that your audience is students in your class and other similar classes. is a Mathematica file showing how I achieved these graphs using the "free form" input. Make sure and save this file to a disk and open it with Mathematica(found in the computer labs in Searles).

We say that a line is **secant** to a curve if they locally intersect at two points (illustrated below).
 * Definition:**

We say that a line is **tangent** to a curve if they locally intersect at one point (illustrated below).
 * Definition:**

math $

Suppose our goal is to find the equation of a tangent line to the curve $y=f(x)$ at the point $(a,f(a))$. The trickiest part of the calculation is to find the slope of the tangent line. We will start by taking the slope of the secant line through the points $(a,f(a))$ and $(a+h,f(a+h))$. Notice that this quantity is

$

math

Notice here how the inline math mode makes this page look a little disjointed. Because of how the inline math is achieved it is tricky and not really worth it to make all of the fonts match. If anyone wants to know how to fix this, let me know. If anyone finds a simple way to do this, also let me know.

math \frac{f(a+h)-f(a)}{h} math

Now if we take the limit as h approaches 0, assuming this limit exists, we have the slope of the tangent line.

The **derivative** of a function f at a is
 * Definition:**

math f^{'}(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}, math provided this limit exists. Furthermore, the quantity is the slope of the line tangent to the curve y=f(x) at x=a.

Now as part of this post you would provide several examples covering as many different types of problems as possible. I'll just post one here as an example
 * Examples:**


 * Example 1:** Find the equation of the line tangent to the curve

math f(x)=x^2 math

at x=1.


 * Solution:** The slope of the tangent line is given by the derivative so we have

math m=f^{'}(1)=\lim_{h\to 0}\frac{f(1+h)-f(1)}{h} math

math =\lim_{h\to 0}\frac{(1+h)^2-1}{h}=\lim_{h\to 0}\frac{h^2+2h}{h}=2 math

So the slope of the tangent line is 2 and we are traveling through the point (1,1). Using the point-slope form of the line, we have an equation

math y-1=2(x-1) math