Suppose that $f ( x , y ) = x ^ { 2 } e ^ { y }$ Find an equation for the tangent plane to f at the point (3, 0).
A) $z = - 9 + 6 x + 27 y$
B) $z = 9 + 6 x + 27 y$
C) $z = - 9 + 6 x - 27 y$
D) $z = - 9 - 6 x + 27 y$

Question

Quizplus · Accepted Answer

The equation of the tangent plane to a surface $z = f(x, y)$ at the point $(x_0, y_0, z_0)$ is given by $$z - z_0 = f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0)$$So, for the given function $f(x, y) = x^2 e^y$, we have$$f_x(x, y) = 2xe^y, \quad f_y(x, y) = x^2 e^y$$and the point $(x_0, y_0, z_0) = (3, 0, 9)$. Therefore, the equation of the tangent plane is$$z - 9 = 2(3)e^0 (x - 3) + (3)^2 e^0 (y - 0)$$Simplifying, we get$$z - 9 = 6(x - 3) + 27y$$$$z = -9 + 6x + 27y$$

Suppose That f(x,y)=x2eyf ( x , y ) = x ^ { 2 } e ^ { y }f(x,y)=x2ey

Suppose That $f ( x , y ) = x ^ { 2 } e ^ { y }$