I have forgot the exact method how to approximate a function at given points and I need your help to recall it (sure I will show how I think about it, because my question would not be just only problem posting, I will show my effort).
Suppose that our function is given by following two variable form, $f(x,y)=\sqrt{x^3+y^3}$, and we want to approximate the value of this functon at the point $(2.02,1.97)$. As I remember I should use the total differential method (generally tangent line's geometrical interpretation is as best approximation of function at given points), so I used
$Df(x,y) = \frac{3x^2}{\sqrt{x^3+y^3}}dx + \frac{3y^2}{\sqrt{x^3+y^3}}dy.$
I took the exact value of $(x,y)$ as $x = 2$ and $y = 1$ and consequently $dx = .02$ and $dy = .97$. So if I put this variable I would get the required answer, right? Or is there another method?
Thanks a lot.