Generally, my question is about the relationship between how the slopes of two line segments (defined as segments connecting points along a curve) change in relation to each other as a common end point changes along the curve.
As an example, consider the function $f(x)=x^2$, and three points $a=1$, $b=2$ and $c=3$. plugging each of these in gives $f(a)=1$, $f(b)=4$ and $f(c)=9$ of course. Thus, the line segment connecting points $f(a)$ and $f(c)$ would be $4$, and the slope of the line segment connecting $f(b)$ and $f(c)$ would be $5$. We can then consider the ratio (the relationship of interest between the two slopes) to be $4/5$. I'm curious about the optimal method for calculating how this ratio changes as $c$ is moved away from $3$. While optimally I'd love an expression general enough so that $c$ can either increase or decrease, it would never decrease all the way to $b$ (so $c$ would always be strictly greater than 2).
Again, the above is a relatively simple example for the sake of explanation- the more general the answer, the better!!
Thank you!!
(PS- I wasn't sure what the most appropriate tags would be- I'd be glad to change them if recommended!)
In the general case, you are looking for $$\frac{f(b)-f(a)}{b-a}\cdot \frac{c-b}{f(c)-f(b)}$$ which we can't say much about in the general case unless we know some properties about $f$. However, for $f(x) = x^2$, we get $$\frac{b^2-a^2}{b-a}\cdot \frac{c-b}{c^2-b^2} = \frac{(b-a)(b+a)}{b-a}\cdot\frac{c-b}{(c-b)(c+b)} = \frac{b+a}{c+b}$$