3.5 Tangent Planes and Linear Approximations

Tangent Planes and Linear Approximations

Tangent planes and linear approximations let you replace a complicated surface with a simple flat one near a specific point. This is the multivariable version of the same idea from Calc I, where you used the tangent line to approximate a curve. Here, partial derivatives define a tangent plane that hugs the surface at a given point, and you can use that plane to estimate nearby function values without plugging into the original (often messy) function.

Tangent Planes and Linear Approximations

Tangent plane equation for surfaces

A tangent plane touches a surface at a single point and lies flat against it there, just like a tangent line touches a curve. It's the best possible flat surface to approximate the shape near that point.

To write the equation, you need two things:

• The point of tangency $(x_0, y_0, z_0)$, where $z_0 = f(x_0, y_0)$
• The partial derivatives $f_x(x_0, y_0)$ and $f_y(x_0, y_0)$, which capture how the surface slopes in the $x$- and $y$-directions

The tangent plane equation is:

$z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$

Notice the structure: each partial derivative multiplies the displacement in its corresponding variable. If both partial derivatives are zero, the tangent plane is horizontal.

Steps to find a tangent plane:

1. Identify the point $(x_0, y_0)$ and compute $z_0 = f(x_0, y_0)$.
2. Find $f_x(x, y)$ and $f_y(x, y)$ using standard differentiation rules (power rule, chain rule, etc.).
3. Evaluate both partial derivatives at $(x_0, y_0)$.
4. Plug everything into the tangent plane equation above.

For example, if $f(x,y) = x^2 + y^2$ at the point $(1, 1, 2)$, you get $f_x = 2x = 2$ and $f_y = 2y = 2$, so the tangent plane is $z - 2 = 2(x - 1) + 2(y - 1)$.

Function approximation using tangent planes

The tangent plane doubles as a tool for estimating function values. Because the plane hugs the surface near the point of tangency, you can plug a nearby $(x, y)$ into the plane equation instead of the original function and get a reasonable estimate.

Rearranging the tangent plane equation gives the linearization (or linear approximation):

$f(x, y) \approx f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$

This is often written as $L(x, y)$ and called the linearization of $f$ at $(x_0, y_0)$.

The approximation works best when $(x, y)$ is close to $(x_0, y_0)$. As you move farther away, the surface curves away from the plane and the estimate gets worse. Think of it like zooming in on a topographic map: up close, the terrain looks flat, but zoom out and the hills become obvious.

Normal lines to surfaces

The normal line is perpendicular to the tangent plane at the point of tangency. It points straight "out" from the surface, like a flagpole sticking out of a hillside.

The direction of the normal line comes directly from the tangent plane equation. For a surface $z = f(x,y)$, the direction vector of the normal line is:

$\langle f_x(x_0, y_0),\; f_y(x_0, y_0),\; -1 \rangle$

This vector is normal to the tangent plane because the plane has the form $f_x(x - x_0) + f_y(y - y_0) - (z - z_0) = 0$, and the coefficients of $x, y, z$ in a plane equation give its normal vector.

The parametric equations of the normal line are:

$x = x_0 + t\,f_x(x_0, y_0)$ $y = y_0 + t\,f_y(x_0, y_0)$ $z = z_0 - t$

where $t$ is the parameter. At $t = 0$ you're at the point of tangency; as $t$ varies, you move along the line in both directions.

Steps to find the normal line:

1. Compute $f_x$ and $f_y$ and evaluate them at $(x_0, y_0)$.
2. Form the direction vector $\langle f_x, f_y, -1 \rangle$.
3. Write the parametric equations using the point $(x_0, y_0, z_0)$ and the direction vector.

Linear approximations for multivariable functions

Everything above extends naturally to functions of three or more variables. You can't visualize a tangent "plane" in 4D, but the algebra works the same way.

For a function $f(x, y, z)$, the linear approximation at a base point $(x_0, y_0, z_0)$ is:

$f(x, y, z) \approx f(x_0, y_0, z_0) + f_x(x_0, y_0, z_0)(x - x_0) + f_y(x_0, y_0, z_0)(y - y_0) + f_z(x_0, y_0, z_0)(z - z_0)$

Each partial derivative captures the rate of change along one coordinate axis, and the full approximation combines all of them. The pattern generalizes to any number of variables: one term per variable, each involving that variable's partial derivative.

Steps for a three-variable linear approximation:

1. Choose the base point $(x_0, y_0, z_0)$ and compute $f(x_0, y_0, z_0)$.
2. Find $f_x$, $f_y$, and $f_z$, then evaluate each at the base point.
3. Substitute into the formula above.
4. Plug in the point you want to estimate.

As with two variables, accuracy drops off as you move farther from the base point. The linear approximation captures only first-order behavior; it misses curvature entirely. For better accuracy far from the base point, you'd need higher-order (quadratic) approximations, which show up later with Taylor series in several variables.