# How do you find the linear approximation of f(x,y) = sqrt(53-9x^2-y^2) at (2,-1)?

Feb 10, 2015

The linear approximation to a function $f$ of two variables (at a point) is the equation of the tangent plane to the surface (at that point).

The equation of that tangent plane depends on the slope in each direction; the partial derivatives ${f}_{x}$ and ${f}_{y}$. If the surface is

$z = f \left(x , y\right)$, then the tangent plane at $\left({x}_{0} , {y}_{0}\right)$ is

$z = f \left({x}_{0} , {y}_{0}\right) + {f}_{x} \left({x}_{0} , {y}_{0}\right) \left(x - {x}_{0}\right) + {f}_{y} \left({x}_{0} , {y}_{0}\right) \left(y - {y}_{0}\right)$.

You need to take the partials of your $f \left(x , y\right)$, evaluate them at $\left(2 , - 1\right)$, and simplify the equation.

• Hint: If you get stuck, look for previous examples of derivatives of a function to a power. Use the Chain Rule to do the job.*

dansmath to the (partial) rescue!