8.4 Tangent Planes

We can find a tangent plane at any point on the graph of a function \( z \depends (x, y) \). Like tangent lines, the tangent plane will help us better understand the shapes of graphs.

Definition.

Suppose \( z \mathrel{\overset{f}{\leftarrow}} (x, y) \) is a function. The tangent to \( f \) is the planar function \( z_T \depends (x_T, y_T) \) described by the equation

\[ \begin{align} z_T - z &= \operatorname{\overset{\mathit{x}}{\slope}} z \mult (x_T - x) + \operatorname{\overset{\mathit{y}}{\slope}} z \mult (y_T - y) \end{align} \]

where

\[ \begin{align} &x : \Number, & &y : \Number, & &z : \Number, \\ &\operatorname{\overset{\mathit{x}}{\slope}} z : \Number, & &\operatorname{\overset{\mathit{y}}{\slope}} z : \Number & & \end{align} \]

are constants found by localizing at a point \( p : \Point_{(x, y)} \) called the center.

The tangent plane is the plane that best matches the graph of \( f \) near the center \( p \).

Example.

We'll find the tangent plane for the bowl

\[ \begin{align} z &\depends (x, y) \\ z &= x^2 + y^2 \end{align} \]

centered at the point \( p = (1, -1) \). We start by taking partials.

\[ \begin{align} \operatorname{\overset{\mathit{x}}{\slope}} z &= 2 x & \operatorname{\overset{\mathit{y}}{\slope}} z &= 2 y \end{align} \]

Localizing at the point \( p = (1, -1) \), we find constants:

\[ \begin{align} &x = 1, & &y = -1, & &z = 2, \\ &\operatorname{\overset{\mathit{x}}{\slope}} z = 2, & &\operatorname{\overset{\mathit{y}}{\slope}} z = -2. & & \end{align} \]

This is everything we need to write down an equation for the tangent plane!

\[ \begin{align} z_T - 2 &= 2 \mult (x_T - 1) - 2 \mult (y_T + 1) \end{align} \]

Here's a sketch of what the tangent plane from the example actually looks like.

In each vertical slice the tangent plane shows up as a tangent line!

We see the \( \{ {y = -1} \} \)-slice when viewing the bowl from the front, and we see the \( \{ {x = 1} \} \)-slice when viewing the bowl from the right.