$y= f(x_0) + f '(x_0)(x-x_0) +\frac{1}{2}f ''(x_0)(x-x_0)^2$
What is the intuition behind the quadratic term. I found the linear term quite straightforward because $\Delta y/\Delta x$ approximates $f '(x_0)$. But what about the $f ''(x_0)(x-x_0)^2$?
Why is there a $(x-x_0)^2$ term? What is the proof for it?
As far as intuition goes, if you differentiate both sides you get $$\frac{dy}{dx}=f’(x_0)+f’’(x_0)(x-x_0)$$ So the quadratic term in $y$ is precisely in the form needed for the linear term in $dy/dx$ to make sense.