How does "The error in linear approximation" make sense?

Question

How does the following theorem make sense? I understand that the greater the magnitude of $f''(a)$ and $∣x-a∣$ are, the greater the error should be, but how do we actually get a formula? What is the intuition?

Thanks.

Answer 1

They are comparing it a parabola with constant concavity M. They have said that if the function is less concave then a parabola with concavity M, the error must be less than that parabola. You can draw a few examples and convince yourself of that.

Answer 2

Partial integration gives $$f(x)=f(a)+\int_a^x f'(t)\>dt=f(a)+ f'(t)(t-x)\biggr|_{t=a}^x-\int_a^x f''(t)(t-x)\>dt\ ,$$ so that we obtain $$\bigl|f(x)-f(a)-f'(a)(x-a)\bigr|\leq M\int_a^x|t-x|\>|dt|={M\over2}(x-a)^2\ .$$