I want to prove that the line passing through $(a,b)$ and $(c,d)$ is the best linear approximation of any function passing through those 2 points, as $a$ approaches $c$.
I'm not sure how to define "best" rigorously, but my hope is that the average value of the line over $[a,c]$ will be as close as possible to that of the function. By average value, I am referring to the Mean Value Theorem of integrals.
Thus, as $|a-b|$ becomes smaller and smaller, the average value of the line should approach that of the function.
It seems pretty obvious visually. I'm not sure what field this is, or how to approach this proof at all. I just happened to come across this in unrelated research. I don't necessarily need a full proof, I just want the first couple steps or so so that I can know the right approach.
I do not mean the tangent line or Taylor polynomial. I mean simply the line passing through $(a,b)$ and $(c,d)$. I don't want the tangent line, as then I can't apply the Mean Value Theorem.
For your function f(x) passing through the point (a,b) we have what is called the Taylor polynomial namely ,
$$ P(x) = f(a)+ f'(a)(x-a) + f''(a)(x-a)^2/2 + ... +f^{k}(a)(x-a)^k/(k!)$$
Now if you let $x=c$ you get $$ P(c) = f(a)+ f'(a)(c-a) + f''(a)(c-a)^2/2 + ... +f^{k}(a)(c-a)^k/(k!)$$
Now if you let c approaches to $a$, the higher powers of $(c-a)$ approach zero and you get the tangent line $$ f(a) + f'(a)(c-a)$$
Thus you are approximating $f(c)$ with $$b+f'(a)(c-a)$$
The straight line going through $(a,b)$ and $(c,d)$ is $$y=b+ m (c-a)$$ where $m$ is the slope of the line.
Thus you see the straight line approximation which is indeed the tangent line approximation to the curve is good for smooth curves as far as the approximation is for points very close to $x=a$