# Slope from the Graph of Difference Quotients

1. Nov 19, 2007

### veravarya

1. The problem statement, all variables and given/known data
I have 12 x,y data points (x is distance, y is elevation) divided into three groups (sets). From those three sets, I created 3 cubic polynomials of exact fit, and now I am supposed to find their difference quotients, graph the difference quotients, and use the graph to figure out where the slope is greatest. h is supposed to equal 0.1 so that it's small enough to be almost tangent instead of secant (0.1 is small compared to the data I'm working with, which is in the thousands). I figured out the three difference quotients, but they are all quadratic functions and I don't know how to figure out the slope of those and I don't know understand what the graph should look like.

2. Relevant equations
difference quotient $= \frac{f(x+h) - f(x)}{h}$
generic cubic polynomial $= ax^3 + bx^2 + cx + d$
diff. quot. from gen. cub. poly. $= ah^2 + 3ax^2 + 3ahx +2bx + bh + c$

3. The attempt at a solution
I tried using the X (distance) data points as inputs into their difference quotient equations and got their respective Y-values, but the scatter plot I made just looks like three different sharp U-shaped pieces. I usually have a very good grasp of things like this but I just don't understand what to do.

2. Nov 19, 2007

### veravarya

More Specifically...

In summary, I really need someone to tell me this:

How do I find the numerical rate of a difference quotient that is quadratic?

3. Nov 19, 2007

### veravarya

Solved!

After pondering about it for a while and letting my subconscious consider it for a couple hours while I did other things, my brain figured it out on its own. Amazing. I get it now.