# Consider x =h(y,z) as a parametrized surface in the natural way. Write the equation of the...

## Question:

Consider {eq}x =h(y,z) {/eq} as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point {eq}(5,\ 5,\ 4) {/eq} given that {eq}\left. \frac{\partial h}{\partial y}\right|_ {(5,4)}=4 \text{ and } \left. \frac{\partial h}{\partial z}\right|_ {(5,4)}=4 {/eq}.

## Tangent Planes

The tangent plane to a surface {eq}\displaystyle x=h(y,z), {/eq} at a point {eq}\displaystyle (x_0,y_0,z_0) {/eq},

is given by {eq}\displaystyle x=x_0+ h_y(y_0,z_0)(y-y_0)+ h_z(y_0,z_0)(z-z_0) {/eq} where

{eq}\displaystyle h_y, h_z {/eq} are the partial derivatives of {eq}\displaystyle h(y,z). {/eq}

The vector {eq}\displaystyle \left\langle 1,h_y(y,z), h_z(y,z)\right\rangle {/eq} is the normal vector to the surface {eq}\displaystyle x=h(y,z). {/eq}

The tangent plane to the surface {eq}\displaystyle x=h(y,z), {/eq} at a point {eq}\displaystyle (x_0,y_0,z_0)=(5, 5, 4) {/eq} knowing that {eq}\displaystyle \left. \frac{\partial h}{\partial y}\right|_ {(5,4)}=4 \text{ and } \left. \frac{\partial h}{\partial z}\right|_ {(5,4)}=4 {/eq}

is given by

{eq}\displaystyle \begin{align}x&=x_0+ \left. \frac{\partial h}{\partial y}\right|_ {(y_0,z_0)}(y-y_0)+ \left. \frac{\partial h}{\partial z}\right|_ {(y_0,z_0)}(z-z_0)\\ x&=5+ \left. \frac{\partial h}{\partial y}\right|_ {(5,4)}(y-5)+ \left. \frac{\partial h}{\partial z}\right|_ {(5,4)}(z-4)\\ x&=5+ 4(y-5)+ 4(z-4)\ \implies \boxed{\text{ the tangent plane is }x=-31+ 4y+ 4z}. \end{align} {/eq} 