Unit V Big Question

Explain in detail where the formula for the difference quotient comes from.

The picture above is just to refresh your memory on how to evaluate the difference quotient. (and for me to reference to the formula to instead of typing it out)

What is the difference quotient? The difference quotient is just a fancy way of writing out the slope of an equation. And in calculus it is used when introducing a concept called the derivative.

Because the difference quotient is just another way of saying slope, we know that this formula must relate to the slope formula:

Moreover, we know that the slope formula needs two coordinate points on the graph, this is much like a secant line. A secant line looks like this: (in red)

Using this information, lets figure out how the difference quotient formula came to be!
Watch this video up to minute 6:15 to see how the slope formula turns into the difference quotient formula.

As you can see, the difference quotient comes from using two points to find slope, and these two points on the graph form to make a secant line. But what about tangent lines? Well, we know that tangent lines only touch the graph once, so how to we find the equation of the tangent line of a graph?
First off, the slopes of tangent lines are referred to as derivatives.
To find the derivative, do the difference quotient formula normally, but take one more step, find the limit as "h" approaches 0
Here's an example: (watch starting at 2:30 to the end of the video)

Sources: First picture-http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/sld002.htm
Second Picture-http://math.about.com/od/allaboutslope/ss/Find-Slope-With-Formula-JW.htm
Third Picture-http://clas.sa.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm
Difference Quotient Video- http://youtu.be/1mlkc3Pfxu4
Derivative video- http://youtu.be/vzDYOHETFlo