Tangent Planes and Linear Approximation — Mathematics Knowledge System

Overview

Just as a tangent LINE gives the best linear approximation to a single-variable function at a point, a tangent PLANE gives the best linear approximation to a surface z = f(x,y) at a point. The tangent plane touches the surface at one point and matches both partial derivatives there, hugging the surface closely nearby. This lets us replace a complicated surface with a simple flat plane for small movements away from the point of tangency — the linear approximation (or linearization) L(x,y), which underlies error estimation, optimization algorithms, and the very definition of differentiability in several variables.

Intuition

Standing on a smooth hilltop and looking at just your immediate surroundings, the ground looks essentially flat — that flat approximation IS the tangent plane. Formally, it is built from the same information a tangent line uses (a point and a slope), except now there are two independent 'slopes' (the partial derivatives in the x and y directions), so the tangent object needs to be a plane, not a line, to capture both. Near the point of tangency, the plane and the surface are nearly indistinguishable, which is exactly why the linear approximation is useful: it turns a hard nonlinear computation into simple arithmetic with only a small, controllable error.

Formal Definition

Definition

If f has continuous partial derivatives at (a,b), the tangent plane to z=f(x,y) at the point (a, b, f(a,b)) is:

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

Tangent plane equation

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

Same expression as the tangent plane, viewed as a function L(x,y) approximating f(x,y)

Linearization (linear approximation)

dz = f_x(a,b)\, dx + f_y(a,b)\, dy

The change in the tangent plane's height for small changes dx, dy — used to estimate propagated error

Total differential

Notation

Notation	Meaning
$L(x,y)$	Linearization of f at (a,b): the tangent-plane formula treated as an approximating function
$dz = f_x\,dx+f_y\,dy$	Total differential — the exact change along the tangent plane for small dx, dy
$\Delta z = f(a+\Delta x, b+\Delta y) - f(a,b)$	Actual change in f; dz approximates Δz for small Δx, Δy

Derivation

The tangent plane is derived by requiring it to be a plane through (a,b,f(a,b)) that matches both partial derivatives of f there, since any plane through that point has the form z = f(a,b)+A(x-a)+B(y-b):

z = f(a,b) + A(x-a) + B(y-b)

General equation of a (non-vertical) plane through the point (a,b,f(a,b))

\frac{\partial z}{\partial x} = A \quad \text{must equal} \quad f_x(a,b), \qquad \frac{\partial z}{\partial y}=B \quad \text{must equal} \quad f_y(a,b)

For the plane to be tangent, its slopes in the x- and y-directions must match f's partial derivatives exactly at (a,b)

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

Substituting A=f_x(a,b), B=f_y(a,b) gives the tangent plane formula

Properties

Normal vector to the tangent plane

\mathbf{n} = (f_x(a,b),\, f_y(a,b),\, -1) \text{ is normal to the tangent plane (and to the surface) at } (a,b,f(a,b))

Linearization is exact at the base point

L(a,b) = f(a,b) \text{ always — the tangent plane touches the surface exactly at the point of tangency}

Error estimate via differentials

\Delta z \approx dz \text{ for small } dx, dy \text{; the approximation error shrinks faster than } dx, dy \text{ themselves}

Theorems

Theorem 1: Differentiability implies linear approximability

\text{If } f_x, f_y \text{ are continuous near } (a,b)\text{, then } f(x,y) \approx L(x,y) \text{ with error} \to 0 \text{ faster than } \sqrt{(x-a)^2+(y-b)^2} \to 0

Applications

Propagated error analysis uses the total differential dz = f_x dx + f_y dy to estimate how small measurement errors in inputs (e.g. radius and height) affect a computed output (e.g. volume).

3D Visualization

Worked Examples

Compute f(1,2) and the partial derivatives.
$f(1,2)=1+4=5, \qquad f_x=2x \implies f_x(1,2)=2, \qquad f_y=2y \implies f_y(1,2)=4$
Substitute into the tangent plane formula.
$z = 5 + 2(x-1) + 4(y-2)$
Simplify: 5 + 2x - 2 + 4y - 8 = 2x + 4y - 5.
$z = 2x+4y-5$

Answer: z = 2x + 4y − 5

Practice Problems

Difficulty 5/10

Find the tangent plane to f(x,y) = xy at the point (2, 3).

Difficulty 6/10

Use linear approximation to estimate √(4.02² + 2.97²) using f(x,y)=√(x²+y²) linearized at (4,3) (note f(4,3)=5).

Difficulty 7/10

Ohm's law gives power P = I²R. If a measured current I=2 A and resistance R=8 Ω each have possible error ±0.05, estimate the maximum error in computed power using the total differential.

Common Mistakes

Common Mistake

Confusing the tangent plane formula with a first-order Taylor polynomial in only one variable, forgetting the second partial-derivative term entirely.

The tangent plane MUST include both f_x(a,b)(x-a) AND f_y(a,b)(y-b) — omitting either term collapses the plane into a line and loses information about one direction of the surface.

Common Mistake

Believing the linear approximation is exact for any size of Δx, Δy.

L(x,y) ≈ f(x,y) is only accurate for SMALL Δx, Δy near (a,b); the approximation error grows (often quadratically) as you move farther from the point of tangency.

Common Mistake

Computing the normal vector to the tangent plane as (f_x, f_y, 1) instead of (f_x, f_y, -1).

Writing the surface as F(x,y,z)=f(x,y)-z=0 and taking ∇F gives (f_x, f_y, -1) — the sign on the z-component is negative, not positive.

Quiz

The tangent plane to z=f(x,y) at (a,b) is given by:

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

The total differential dz = f_x dx + f_y dy is used to:

Compute the exact change in f for any dx, dyEstimate the change in f for SMALL dx, dyFind critical points of fCompute the second derivative of f

Flashcards

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Summary

The tangent plane z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) is the best flat approximation to a surface at a point.
The linearization L(x,y) uses this same formula to approximate f(x,y) for points near (a,b).
The total differential dz = f_x dx + f_y dy estimates the change in f for small changes dx, dy, underlying propagated-error analysis.
The normal vector to the tangent plane is (f_x, f_y, -1), used for surface normals in graphics and optimization.
Accuracy of the linear approximation degrades as you move farther from the point of tangency.

References

BookStewart, J. Calculus: Early Transcendentals, 8th ed. Ch. 14.4.
WebsiteWikipedia — Linear approximation
WebsiteWikipedia — Tangent plane (Differentiable surface)