Finding a Tangent Plane on a Surface Examples 3

Recall from the Finding a Tangent Plane on a Surface page that if $z = f(x, y)$ is a two variable real-valued function, then the equation of the tangent plane at a point $P(x_0, y_0)$ on the surface is given by the following formula:

(1)

\begin{align} \quad z - z_0 = f_x (x_0, y_0) (x - x_0) + f_y (x_0, y_0) \end{align}

We will now look at some more examples of finding tangent planes to surfaces.

Example 1

Find the equation of the tangent plane on the paraboloid $f(x, y) = 3x^2 + 3y^2$ at the point $(1, 1, 6)$.

We first compute the partial derivatives of $f$:

(2)

\begin{align} \quad f_x (x, y) = 6x \quad \quad , \quad \quad f_y (x, y) = 6y \end{align}

Therefore we have that the equation of this tangent plane is given by:

(3)

\begin{align} \quad z - 6 = f_x (1, 1) (x - 1) + f_y (1, 1) (y - 1) \\ \quad z - 6 = 6(x - 1) + 6(y - 1) \end{align}

Example 2

Find the equation of the tangent plane on the hyperboloid $f(x, y) = \frac{1}{3}x^2 - y^2$ at the point $(3, 1, 2)$.

We first compute the partial derivatives of $f$:

(4)

\begin{align} \quad f_x (x, y) = \frac{2}{3} x \quad , \quad f_y (x, y) = -2y \end{align}

Therefore the equation of this tangent plane is given by:

(5)

\begin{align} \quad z - 2 = f_x (3, 1) (x - 3) + f_y (3, 1) (y - 1) \\ \quad z - 2 = 2(x - 3) - 2(y - 1) \end{align}