BQ#7 - The Difference Quotient

The Difference Quotient

Explain in detail where the formula for the difference quotient comes from. Include all appropriate terminology (secant line, tangent line, h/delta x, etc). 

A: The formula for the difference quotient comes from way back in our middle school ages. We have learned about it in our rise over run method. The slope formula. Now it's time to talk about how to evaluate the difference quotient formula. A common expression from calculus is the difference quotient. It is used to when introducing a concept called the derivative. The expression for the difference quotient is commonly given by: f(x+h)-f(x)/h. Plug wherever there's an "x" and plug in "x+h" then you want to subtract the original equation then divide everything by "h." Evaluating the limit of the difference quotient is also known as finding the slope of the tangent line to a graph (the derivative!) We also learn about secant lines and tangent lines (these aren't things that you may think you know, they do not correspond to geometry). Secant lines: touches the graph twice. Tangent lines: touches the graph once. The only thing we do differently now is evaluate the limit as "h" approaches "0" to get the slope of the tangent line instead of the secant line (this is called the derivative)! In Math Analysis, we find the derivatives of linear, quadratic, higher powered binomials, radical and rational functions using the "limit definition of the derivative". We have been doing this for quite some time now or all year - now all we have to do is evaluate the limit as "h" approaches "0" (using direct substitution) to get the "derivative." In concepts 2,3,4, we took the derivatives and applied more things to it. We are not just doing the difference quotient, we are taking the limit as "h" approaches "0" and we call this f'(x), and we read it as "f prime of x". (http://www.coastal.edu/mathcenter/HelpPages/Difference%20Quotient/sld002.htm)