BQ #7: Unit V- Derivatives and the Area Problem

1. Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc).
- Well to begin, the difference quotient is as follows, below.

http://images.tutorvista.com/cms/images/39/difference-quotient-formula.png

The formula helps when you need to find the slope of a tangent line at a given function. This formula was derived from the visual of a graph.

http://www.teacherschoice.com.au/images/derivative_secant_1.gif

http://www.zweigmedia.com/ThirdEdSite/summarypic/cs3_1.gif

The picture on the left goes to show the visual and we see a secant line(a lone on the graph that touches twice) that in the x-values we have x and x+h with the distance of h between them. For the first y-value we have f(x) and that makes logical sense because the shift to the right was x and the shift up was call f of x. So then the second point which is x+h has a y-value of f(x+h). 

http://webpages.charter.net/mwhitneyshhs/calculus/tangent_

lines/tngt-slope-eq001.jpg

Then we would want to find the slope of those two points which is (x,f(x)) and (x+h,f(x+h)) so that we would be able to find the distance between them. So we would then have to use the slope formula which is the following picture to the left. So we would take our ordered pairs and we would plug them into this formula to get, f(x+h)-f(x)/ x+h-x and is you simply it we would be left with f(x+h)-f(x)/h which matches the original formula that we have.

The video below is great help in putting what I said to the test. It would help tremendously to view it to see it worked out.

 https://www.youtube.com/watch?v=iMaJDAV7as0