Find the equation of the plane tangent to the surface given by the given function at the specified point $P$ $f ( x , y ) = x ^ { 3 } - x ^ { 2 } y , P = ( 1,2 )$
A) $z = - 1 - y - 2$
B) $z = - x - y + 7$
C) $z = - x - y + 2$
D) $z = - x - y$

Question

Quizplus · Accepted Answer

The equation of the tangent plane to a surface $f(x, y)$ at the point $P(x_0, y_0)$ is given by $$z - f(x_0, y_0) = f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0)$$Given $f(x, y) = x^3 - x^2 y$ and $P(1, 2)$, we have$$f_x(x, y) = 3x^2 - 2xy$$$$f_y(x, y) = -x^2$$So, at the point $P(1, 2)$, we have$$f_x(1, 2) = 3(1)^2 - 2(1)(2) = -1$$$$f_y(1, 2) = -(1)^2 = -1$$Therefore, the equation of the tangent plane is$$z - (1^3 - 1^2 \cdot 2) = -1(x - 1) - 1(y - 2)$$$$z - 1 = -x + 1 - y + 2$$$$z = -x - y + 2$$

Find the Equation of the Plane Tangent to the Surface PPP

Find the Equation of the Plane Tangent to the Surface $P$