Question: Find the quadratic polynomial that best approximates the unit circle centered at $(0,0)$ near the point, P $(0,1)$ (i.e. the osculating parabola).
Thoughts/Attempts:Right now, we are right now on the section on Taylor Series. So, I think this question is related to using Taylor Series of a higher order. Like, when we use the first derivative to approximate the tangent, now we use the the first and second derivative to approximate the quadratic polynomial (osculating parabola) (at least this what I think we do something like this).
I'm honestly confused which one to use as the equation of a circle is $x^2+y^2=1$ . I'm also confused whether I should be taking derivatives or partial derivatives. Also, I'm somewhat confused whether $(x_0,y_0)$ is $(0,1)$? Or is it $(0,0)$? I'm pretty sure its $(0,1)$ as we are taking it near that point, but I just want to clarify. Sorry for what may seem like simple questions. Thank you.