Find the equation of the tangent plane to the given surface at the specified point. $\backslash$
Select the correct answer.
$z + 4 = x e ^ { y } \cos z , ( 4,0,0 )$
A) $x + y - z = 4$
B) $x + 4 y + z = 4$
C) $4 x + y - z = 4$
D) $x + y - 4 z = 4$
E) $x + 4 y - z = 4$

Question

Quizplus · Accepted Answer

The tangent plane to a surface $z = f(x, y)$ at a point $(x_0, y_0, z_0)$ is given by the formula: $z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$. Here, $f(x, y) = x e^y \cos z - 4$. At the point (4,0,0), the partial derivatives are $f_x = e^y \cos z$, $f_y = x e^y \cos z$, and $f_z = -x e^y \sin z$. Evaluating these at (4,0,0) gives $f_x = 1$, $f_y = 4$, and $f_z = 0$. The equation of the tangent plane is $1(x - 4) + 4(y - 0) + 0(z - 0) = 0$, simplifying to $x + 4y - 4 = 0$, or $x + 4y - z = 4$.

Find the Equation of the Tangent Plane to the Given \\backslash\

Find the Equation of the Tangent Plane to the Given $\backslash$