Find an equation of the tangent plant to the surface $z = e ^ { x + y }$ at the point $( 0,0,1 )$ .
A) $z = x + y + 1$

B) $z = e x + e y - 2 e + 1$
C) $z = 2 x + 2 y + 1$
D)$z = 2 e x + 2 e y - 4 e + 1$
E) $z = 4 x + 4 y - 7$
F) $z = e ^ { 2 } x + e ^ { 2 } y - 2 e ^ { 2 } + 1$

G) $z = 2 e ^ { 2 } x + 2 e ^ { 2 } y - 4 e ^ { 2 } + 1$

H) $z = 4 e x + 4 e y - 8 e + 1$

Question

Find an equation of the tangent plant to the surface $z = e ^ { x + y }$ at the point $( 0,0,1 )$ .
A) $z = x + y + 1$

B) $z = e x + e y - 2 e + 1$ 
C) $z = 2 x + 2 y + 1$
D)$z = 2 e x + 2 e y - 4 e + 1$ 
E) $z = 4 x + 4 y - 7$ 
F) $z = e ^ { 2 } x + e ^ { 2 } y - 2 e ^ { 2 } + 1$

G) $z = 2 e ^ { 2 } x + 2 e ^ { 2 } y - 4 e ^ { 2 } + 1$

H) $z = 4 e x + 4 e y - 8 e + 1$

Quizplus · Accepted Answer

The equation of the tangent plane to the surface $z = f(x, y)$ at a point $(x_0, y_0, z_0)$ is given by $z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$, where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x$ and $y$, respectively. For $z = e^{x+y}$ at the point $(0,0,1)$, $f_x = e^{x+y}$ and $f_y = e^{x+y}$. At $(0,0)$, both $f_x$ and $f_y$ equal $e^0 = 1$. Thus, the equation becomes $z - 1 = 1(x - 0) + 1(y - 0)$, simplifying to $z = x + y + 1$.

Find an Equation of the Tangent Plant to the Surface z=ex+yz = e ^ { x + y }z=ex+y

Find an Equation of the Tangent Plant to the Surface $z = e ^ { x + y }$