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

How is the Difference Quotient derived ?

Throughout this entire year, we have practiced and memorized the difference quotient through song, and rhythm. In order to understand it, one must know that it is derived from the slope of a secant line, that touches two points on the function.  

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0413.JPG

A secant line is when it touches two points of the functions graphed. touching two points we know that the slope is not steep, but is negative or positive, depending on where it points. 

http://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Secant_line_and_curve_basic.svg/450px-Secant_line_and_curve_basic.svg.png

A tangent line is when it touches the graph only ONCE! This shows that the graph is steep, and or the slope is equal to 0, that means the line is horizontal. 

http://mathinsight.org/media/image/image/neuron_firing_rate_kink_tangent_line.png