How would I use linear approximation to estimate $\sqrt{3.90^2 + 3.02^2}$? I want to use the linear approximation to estimate $\sqrt{3.90^2 + 3.02^2}$ but I am stuck.
Would I first replace the numbers with $x$ and $y$ respectively then get the partial differentiation for $x$ and $y$?
I would appreciate any help. Thanks.
 A: Hint $\#1$: We have that $(x,y)=(3.90,3.02)$. Find $(x_0,y_0)$ such that $\sqrt{x_0^2+y_0^2}$ is simple enough for you to compute, and $|x-x_0|$ and $|y-y_0|$ is close enough.
Hint $\#2$: $(3,4,5)$ is a Pythagorean triplet.
A: Try to scale your problem so that you only have something about $\sqrt{1+\epsilon}$ with $\epsilon$ small. Than taylor gives you 
$$\sqrt{1+\epsilon}\approx 1+\frac{\epsilon}{2}$$
Just for you to check, $3^2+4^2=5^2$ so your result should be something about $5$.
This gives you the idea to make the following
\begin{align*}
\sqrt{3.90^2+3.02^2}&=\sqrt{\frac{25}{25} (3.90^2+3.02^2)}\\
&=5\sqrt{0.6084+ 0.364816} \\
&=5\cdot \sqrt{0.973216} \\
&=5\cdot \sqrt{1- 0.026784}\\
&\approx 5\cdot (1- \frac{0.026784}{2})\\
&=5 \cdot 0.986608\\
&=4.93304
\end{align*}
Mathematica gives me as result for the original computation 
$$\sqrt{3.9^2+3.02^2}=4.93259$$
which is pretty good I think.
A: Use taylor approximation:
$$ f(x, y) \approx f(4, 3) + \frac{\partial f}{\partial x}(4, 3) (x - 4) + \frac{\partial f}{\partial y}(4, 3) (y - 3).$$
This gives me $\sqrt{3.90^2 + 3.02^2} \approx 5 - 17/250 = 4.932$.
