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Thursday, June 5, 2014

BQ #7: Origin of the Difference Quotient

What is the difference quotient?
The difference quotient is the formula in order to deduce the slope of a line that touches a curve on the graph. It allows us to be able to find any slope of any line at any point. It essentially is very similar to the slope formula where the main formula is the change of y divided by the change of x. The main difference here is that the difference quotient, we label the change in y as f(x) while in the slope formula, we see that we instead use the change of y as the y-axis.

Slope Formula
Slope Formula
In order to understand the difference quotient, we must explain how it relates to the slope formula. If we use what we learned earlier, we would change this equation to the change of y over the change of x. The issue with this equation is that it is only applicable to straight, non-curved lines. In order to find a curved one, we must use another formula, which is where the difference quotient comes into play.

Difference Quotient

Where this equation comes from is where a line passes through two main points at A and B, which is
 (x, f(x)) as well as (x+h, f(x+h)). This is essentially also called the secant line.
                                                              graphs of function f with secant line
                                                     http://www.analyzemath.com/calculus/Differentiation/difference_quotient.html
Furthermore the change of x is basically h. This is where we get delta x from. To not be mistaken, the difference quotient is not used only for curved lines. In fact, we can also use it for straight lines as well. It is therefore a paramount concept that will most definitely be a boon in calculus. To actually find the slope of a line given this formula, we use the difference quotient in order to find the derivative. We do not directly find the slope from the difference quotient. In fact, it is a multiple step process that involves us to find the derivative. After we do so, we input the value of x into the resulting derivative to get our slope. The tangent line is merely what the secant line is trying to get closer to as we solve for the derivative. We take the limit as a point and another one move closer to one another, which is how we find our tangent line. The difference quotient, all in all, is a method we use to find the slope of any line.

References:
http://www.analyzemath.com/calculus/Differentiation/difference_quotient.html
http://math.about.com/od/allaboutslope/ss/Find-Slope-With-Formula-JW.htm
http://www.tc3.edu/instruct/sbrown/calc/ln021.htm