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).

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG

                                   

To find our derivative, which is the slope of all tangent lines, we use the difference quotient. How is the difference quotient derived? The first point is (x,f(x)). The second point is (x+h), f(x+h). The distance across is h. We then plug it into our slope formula and get (f(x+h)-f(x)/h). The h cancels out when we simplify so it concludes to be the difference quotient, which is f(x+h)-f(x)/h. H is also known as delta x. The difference quotient is used to find the possible slopes on any graph. A tangent line touches the graph once. On the other hand, a secant line touches the graph twice. To find the slope, we find the limit as h approaches 0. We do this by simply plugging in 0 for h. We then get a derivative, which we then can use to find values and the slope.

http://clas.sa.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm

http://clas.sa.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm

https://www.youtube.com/watch?v=iMaJDAV7as0
Here is an application of my explanation. 



Sources: 
https://www.youtube.com/watch?v=iMaJDAV7as0
http://clas.sa.ucsb.edu/staff/lee/Secant,%20Tangent,%20and%20Derivatives.htm
http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG