# How do you find the linearization of the function z=xsqrt(y) at the point (-7, 64)?

The linear function that best aproximates $z = x \sqrt{y}$ at $\left(- 7 , 64\right)$ is $z = - 56 + 8 \left(x + 7\right) - \frac{7}{16} \left(y - 64\right) = 28 + 8 x - \frac{7}{16} y$.

To get this result, we must first notice that $z$ is a function of the two variables $x$ and $y$. Let's write $z = f \left(x , y\right)$. So, the best linear approximation ${L}_{{r}_{0}} \left(x , y\right)$ of $f$ at ${r}_{0} = \left({x}_{0} , {y}_{0}\right) = \left(- 7 , 64\right)$ is given by

${L}_{{r}_{0}} \left(x , y\right) = f \left({x}_{0} , {y}_{0}\right) + \vec{\nabla} f \left({x}_{0} , {y}_{0}\right) \cdot \left(\left(x , y\right) - \left({x}_{0} , {y}_{0}\right)\right)$

Where $\vec{\nabla} f$ is the gradient of $f$ and $\cdot$ is the dot product.

Geometrically, this linear approxiamtion is the tangent plane of $f$ at ${r}_{0}$. The deduction of this equation is very similar to the deduction of the equation for the tangent line of a real function at a point, with the gradient $\vec{\nabla} f$ playing the role of the derivative.

Now we need to calculate the components of the equations for the linear aproximation. $f \left({x}_{0} , {y}_{0}\right)$ is simply the value of the function at $\left({x}_{0} , {y}_{0}\right)$:

$f \left({x}_{0} , {y}_{0}\right) = f \left(- 7 , 64\right) = - 7 \times \sqrt{64} = - 56$

The gradient $\vec{\nabla} f \left(x , y\right)$ of $f$ is given by the expression

$\vec{\nabla} f \left(x , y\right) = \left(\frac{\partial f}{\partial x} , \frac{\partial f}{\partial y}\right) = \left(\sqrt{y} , \frac{x}{2 \sqrt{y}}\right)$

So, $\vec{\nabla} f \left({x}_{0} , {y}_{0}\right) = \left(\sqrt{64} , - \frac{7}{2 \sqrt{64}}\right) = \left(8 , - \frac{7}{16}\right)$

Finally, we have:

${L}_{{r}_{0}} \left(x , y\right) = - 56 + \left(8 , - \frac{7}{16}\right) \cdot \left(\left(x , y\right) - \left(- 7 , 64\right)\right) =$
$= - 56 + \left(8 , - \frac{7}{16}\right) \cdot \left(x + 7 , y - 64\right) =$
$= - 56 + 8 \left(x - 7\right) - \frac{7}{16} \left(y - 64\right) = 28 + 8 x - \frac{7}{16} y$