Answer
$z = 1/4x+y-2$
Work Step by Step
Question:
$z = x / y^{2}$, $(-4,2,-1)$
The equation of the tangent plane to the surface is:
$z - z_{0} = f_{x}(x_{0},y_{0})(x - x_{0}) + f_{y}(x_{0},y_{0})(y - y_{0})$
$f_{x} = (y^{2}*1 - x * 0) / ((y^{2})^{2}$ = $1/y_{2}$ (use the quotient rule)
so $f_{x}(-4,2) = 1/4$
$f_{y} = (y^{2}*0 - x * 2y) / ((y^{2})^{2}$ = $(-2xy)/y_{4}$ (use the quotient rule)
so $f_{y}(-4,2) = 1$
So the equation is:
$z + 1 = 1/4*(x+4) + 1(y-2)$
And the final answer is:
$z = 1/4x+y-2$
