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https://youtu.be/4Q37iOyBq44 (5:32 time)

In this video professor refers to why is $E_1 \approx E^2_0$ implying there is somewhat geometric interpretation in this case. It feels natural to me, but when I tried to write this down, I understood that I can't produce logic steps to this conclusion.

My guess is: because we use tangent line as approximation of our quadratic function, it only natural that (in this case), we have to land close to our desired $x$ and if we take $|x-x_0|$ as our $\Delta x$ on this interval it will grow as square compare to growing from $x-\Delta x $ to $x$. So if our tangent line approximation is good enough we will land close to $x$, therefore our $y_1 = x_1^2; y_1 \approx \sqrt(y_0) , y_0 = x_0^2$

Could someone please explain this to me?

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