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.).

First of all, we must remember that the difference quotient is just [f(x +h) - f(x]/h. But how is it derived?

Let's say we have a graph like the one below.

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist)

Our goal would be to find the slope of tangent line at a certain point. A tangent line that is a line that touches the graph only once.  A secant line would be a line that touches the graph at two points. Logically, we would then try to find the slope of the secant line using the slope formula: m = (y2 - y1)/(x2 - x1). Since we don't have any concrete numbers, let's use our first point as (x, f(x)). Then. we can pick another point that is farther than x and let's call it (x + h, f(x + h)) because we moved h units away from x. Sometimes, h might also be referred as delta x because we had two x values and the change in between the two x values would be the same as moving h units away.

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist)

We simply plug in the variables into the slope formula to get m = (f(x+h) - f(x))/((x + h) - x). This simplifies to the the much familiar difference quotient: (f(x + h) - f(x))/h. 

To find the slope of the tangent line to a graph at a certain point, we must find the limit as h approaches 0. That is because as h decreases the slope of the secant line becomes increasingly similar to that of the tangent line. To do this, we simply substitute 0 into h, and solve.

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist)

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist)

(http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist)

Therefore, the derivative is simply just the slope of the tangent line while the difference quotient is the mathematical definition of the derivative.

(http://en.wikipedia.org/wiki/Numerical_differentiation)

References:
http://www.sophia.org/tutorials/unit-v-concept-1--2?cid=embedplaylist
http://en.wikipedia.org/wiki/Numerical_differentiation