Find an equation of the tangent plant to the surface $z = \frac { x } { y + 1 }$ at the point $\left( 1,4 , \frac { 1 } { 5 } \right)$ .
A) $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y + \frac { 4 } { 25 }$

B) $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$
C) $z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$
D)
$z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$
E) $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$
F) $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y - \frac { 4 } { 25 }$

G) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y - \frac { 4 } { 25 }$

H) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y + \frac { 4 } { 25 }$

Question

Find an equation of the tangent plant to the surface $z = \frac { x } { y + 1 }$ at the point $\left( 1,4 , \frac { 1 } { 5 } \right)$ .
A) $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y + \frac { 4 } { 25 }$ 
 
B) $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$ 
C) $z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y - \frac { 4 } { 25 }$ 
D)
$z = \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$ 
E) $z = - \frac { 1 } { 5 } x - \frac { 1 } { 25 } y + \frac { 4 } { 25 }$
F) $z = \frac { 1 } { 5 } x + \frac { 1 } { 25 } y - \frac { 4 } { 25 }$

G) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y - \frac { 4 } { 25 }$

H) $z = \frac { 1 } { 25 } x - \frac { 1 } { 5 } y + \frac { 4 } { 25 }$

Quizplus · Accepted Answer

To find the equation of the tangent plane to the surface $z = \frac{x}{y+1}$ at the point $(1,4,\frac{1}{5})$, we first need to calculate the partial derivatives of $z$ with respect to $x$ and $y$, and then use the point-slope form of the plane.The partial derivative of $z$ with respect to $x$ is $\frac{\partial z}{\partial x} = \frac{1}{y+1}$, and at the point $(1,4,\frac{1}{5})$, it equals $\frac{1}{5}$.The partial derivative of $z$ with respect to $y$ is $\frac{\partial z}{\partial y} = -\frac{x}{(y+1)^2}$, and at the point $(1,4,\frac{1}{5})$, it equals $-\frac{1}{25}$.Using the point-slope form of the plane, $z - z_0 = \frac{\partial z}{\partial x}(x - x_0) + \frac{\partial z}{\partial y}(y - y_0)$, we substitute the values to get:$z - \frac{1}{5} = \frac{1}{5}(x - 1) - \frac{1}{25}(y - 4)$Simplifying, we get the equation of the tangent plane as $z = \frac{1}{5}x - \frac{1}{25}y + \frac{4}{25}$, which matches choice D.

Find an Equation of the Tangent Plant to the Surface z=xy+1z = \frac { x } { y + 1 }z=y+1x​

Find an Equation of the Tangent Plant to the Surface $z = \frac { x } { y + 1 }$