So I was given this question:
Use the linear approximation to $f(x, y) = \sqrt{\frac {x+5} {y+2}}$ at $(2, 3)$ to estimate $\sqrt \frac {7.1} {4.9}$.
I was able to get the actual value by just doing the $\sqrt{7.1 / 4.9}$ which was $1.203735$. But, I tried getting the estimated value, and by extension the percent error, but ultimately failed. Can someone please help?
You can use Taylor's formula
$$ \sqrt{7.1/4.9} = f(2+0.1,3-0.1)\approx f(2,3) + 0.1 f'_x(2,3)-0.1 f'_y(2,3) $$
Of course, to use this formula you must know or be able to estimate $f(2,3), f'_x(2,3)$ and $f'_y(2,3)$.