11.88 min read

Visualizing the Local Linear Approximation

The local linear approximation of a scalar-valued function $f(x,y)$ has a beautiful geometric interpretation: it is the tangent plane to the surface $z = f(x,y)$ at the point of approximation.

The tangent plane at $(a, b, f(a,b))$ has the equation $z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ . This plane touches the surface at exactly the point $(a, b, f(a,b))$ , and near this point the surface is well approximated by the plane.

The tangent plane diagram helps build intuition: moving in the $x$ -direction, the plane rises with slope $f_x$ ; moving in the $y$ -direction, it rises with slope $f_y$ . The LLA says the actual surface has the same slope in both coordinate directions — and in all other directions as well.

Formal View

Definition 11.9 — Tangent Plane

The tangent plane to

z = f(x,y)

at the point

(a, b)

is the graph of the local linear approximation:

z = f(a, b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)

This plane is the unique plane through $(a, b, f(a,b))$ that has the same partial derivatives in $x$ and $y$ as $f$ at $(a,b)$ .

Example 11.3 — Tangent Plane to a Paraboloid

For

f(x,y) = x^2 + y^2

, the tangent plane at

(1, 1)

z = 2 + 2(x-1) + 2(y-1) = 2x + 2y - 2

. At

(0,0)

it is

z = 0

(horizontal), reflecting the minimum of the paraboloid.

Interactive Visualization

Local Linear Approximation

Why This Matters

Tangent planes are the multivariable analog of tangent lines and underlie virtually all first-order optimization.

Gradient descent follows the tangent plane approximation to step toward a minimum
Computer graphics: normal vectors to surfaces are perpendicular to tangent planes
Numerical methods: Newton's method uses tangent hyperplanes to solve systems of equations

Learning Resources

Tangent Planes — Visual Intuition

3Blue1Brown

Beautiful visual explanation of local linear approximation as tangent planes.

14 min

Tangent Planes and Normal Lines

Khan Academy

Computing tangent planes and relating them to gradients.

12 min

Quiz

Question 1

The tangent plane to $z = f(x,y)$ at $(a,b)$ is the graph of which function?

Question 2

At a local minimum of $f(x,y)$ , the tangent plane is horizontal (parallel to the $xy$ -plane).

Common Mistakes

Confusing the tangent plane equation with the surface itself — the plane equals the surface only at the point of tangency.
Forgetting the constant term when writing the tangent plane equation.
Using the tangent plane too far from the point of tangency where the approximation error becomes large.