Find an equation of the tangent plant to the surface $z = \ln ( 2 x + y )$ at the point $( - 1,3,0 )$ .
A) $2 x + y - z = 1$

B) $x + 2 y + z = 5$
C) $x - 2 y + z = 7$
D) $3 x + 2 y + z = 3$
E) $2 x + 3 y + z = 7$
F) $x + 2 y + 3 z = 5$

G) $3 x + y - z = 0$

H) $x + 3 y + 2 z = 8$

Question

Find an equation of the tangent plant to the surface $z = \ln ( 2 x + y )$ at the point $( - 1,3,0 )$ .
A) $2 x + y - z = 1$

B) $x + 2 y + z = 5$
C) $x - 2 y + z = 7$ 
D) $3 x + 2 y + z = 3$ 
E) $2 x + 3 y + z = 7$ 
F) $x + 2 y + 3 z = 5$

G) $3 x + y - z = 0$

H) $x + 3 y + 2 z = 8$

Quizplus · Accepted Answer

To find the equation of the tangent plane to the surface $z = \ln(2x + y)$ at the point $(-1, 3, 0)$, we first need to calculate the partial derivatives of $z$ with respect to $x$ and $y$ at the given point. The partial derivative of $z$ with respect to $x$ is $\frac{\partial z}{\partial x} = \frac{2}{2x + y}$, and with respect to $y$ is $\frac{\partial z}{\partial y} = \frac{1}{2x + y}$. Evaluating these at the point $(-1, 3)$ gives $\frac{\partial z}{\partial x} = 2$ and $\frac{\partial z}{\partial y} = 1$. The equation of the tangent plane is then given by $z - z_0 = \frac{\partial z}{\partial x}(x - x_0) + \frac{\partial z}{\partial y}(y - y_0)$, which simplifies to $z = 2(x + 1) + (y - 3)$. Rearranging gives $2x + y - z = 1$.

Find an Equation of the Tangent Plant to the Surface z=ln⁡(2x+y)z = \ln ( 2 x + y )z=ln(2x+y)

Find an Equation of the Tangent Plant to the Surface $z = \ln ( 2 x + y )$