Tangent Planes to Level Surfaces Examples 2

Tangent Planes to Level Surfaces Examples 2

Recall from the Tangent Planes to Level Surfaces page that if we can represent a surface $S$ as a level surface of a three variable real-valued function $f(x, y, z) = k$, then the equation of the tangent plane at a point $P(x_0, y_0, z_0)$ on the surface can be calculated with the following formula:

(1)
\begin{align} \quad \quad \frac{\partial}{\partial x} f(x_0, y_0, z_0) (x - x_0) + \frac{\partial}{\partial y} f(x_0, y_0, z_0) (y - y_0) + \frac{\partial}{\partial x} f(x_0, y_0, z_0) (z - z_0) = 0 \end{align}

Recall that this formula is especially useful when the variable $z$ is not isolated in the equation of the surface. We are now going to look at some examples of calculating tangent planes using the formula above.

Example 1

Find the equation of the tangent plane to the sphere $x^2 + y^2 + z^2 = 4$ at the point $(1, 1, \sqrt{2})$.

Let $f(x, y, z) = x^2 + y^2 + z^2 = 4$. Then we have that the partial derivatives of $f$ are:

(2)
\begin{align} \quad \frac{\partial f}{\partial x} = 2x \quad , \quad \frac{\partial f}{\partial y} = 2y \quad , \quad \frac{\partial f}{\partial z} = 2z \end{align}

The partial derivatives evaluated at the point $(1, 1, \sqrt{2})$ is:

(3)
\begin{align} \quad \frac{\partial}{\partial x} f(1, 1, \sqrt{2}) = 2 \quad , \quad \frac{\partial}{\partial y} f(1, 1, \sqrt{2}) = 2 \quad , \quad \frac{\partial}{\partial z} f(1, 1, \sqrt{2}) = 2 \sqrt{2} \end{align}

Therefore the equation of the tangent plane at $(1, 1, \sqrt{2})$ is given by:

(4)
\begin{align} \quad 2 (x - 1) + 2 (y - 1) + 2\sqrt{2} ( z - \sqrt{2} ) = 0 \end{align}