Thread: Explaining validity of the Difference Quotient.

I have a question on my homework that I'm on a bit stumped on, I mean, I know how to use the difference quotient easy-breezy, but this really stumped me because it asks me WHY its a valid formula.

If anyone could help me out and point me in the right direction I'd really appreciate it!

Question:

The slope of the tangent to a curve y=f(x) at a point P(a,f(a)) is *insert difference quotient here as 'h' approaches zero*.

Using words and a diagram, explain the validity of the equation.

I'm assuming you're talking about the limit definition of a derivative.

This is a weird problem because I don't know how in depth or rigorous your teacher wants of an answer. I would point out the derivation of the terms in the formula.

Assume that P(a,f(a)) exists and that f(x) is a function that is continuous in the neighborhood around x=a. If we want to approximate the slope of the tangent line at x=a, then we can use a second point very close to "a" an estimate. This is where "a+h" comes from. Then you should fill in how the formula uses the simple formula for the slope of a line and where the terms come from. Now I would say that as we take "a+h" to be close to "a", the approximation becomes better and better. This is where the limit part comes in.

I hope that helps some. I don't know how much you already knew but I can do another way if you need.