Find the linearization of the function f(x,y) = \sqrt{100 - 4 x^2 - 2 y^2} at the point (-4, 4).
L(x,y) =
Use the linear approximation to estimate the value of f(-4.1, 4.1) =

Question

Find the linearization of the function  f(x,y) = \sqrt{100 - 4 x^2 - 2 y^2} at the point (-4, 4). 
L(x,y) =  
Use the linear approximation to estimate the value of  f(-4.1, 4.1) =

anonymous · Answer

$$
\Delta L = m_x\Delta x + m_y\Delta y
$$

anonymous · Answer

I believe we start with this.

anonymous · Answer

Then we can expand it a bit...

anonymous · Answer

$$
L(x,y) - L(x_0,y_0) = m_x(x-x_0)+m_y(y-y_0)
$$

anonymous · Answer

Now $\langle m_x,m_y\rangle$ is just the gradient.

anonymous · Answer

Basically $\langle m_x,m_y\rangle = \langle f_x(x_0,y_0), f_y(x_0,y_0)\rangle $

anonymous · Answer

$$
L(x,y) - L(x_0,y_0) = f_x(x_0,y_0)\cdot (x-x_0)+f_y(x_0,y_0)\cdot (y-y_0)
$$

anonymous · Answer

So, find $f_x$ and $f_y$.   Remember that in this case $(x_0,y_0) = (-4,4)$ and $(x,y) = (-4.1,4.1)$.  That accounts for everything.

anonymous · Answer

Remember that the single variable version is: $$
\Delta y \approx  dy = y'dx
$$In this case we are using: $$
\Delta z \approx dz = z_xdx+z_ydy
$$