---
title: "Tangent Planes and Linear Approximation"
description: "Replace a surface near a point by the plane its partials define, and use that plane to estimate values and propagate small errors."
canonical: https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m
source: https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m.md
retrieved: 2026-09-12
---

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# Tangent Planes and Linear Approximation

Replace a surface near a point by the plane its partials define, and use that plane to estimate values and propagate small errors.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m

## Ready when they can

- Write the equation of the tangent plane at a point on a surface
- Estimate a function value near a known point with the linear approximation
- Give a function that has both partials at a point but no tangent plane

## Lesson: The plane that hugs a surface

The tangent plane is the flat surface that best hugs a smooth graph near a point. Its equation is z = f + fx(x minus a) + fy(y minus b). For z = x squared + y squared at (1, 2): height 5, fx = 2, fy = 4, so z = 5 + 2(x minus 1) + 4(y minus 2).

**Example.** Dropping the z gives the linear approximation L, a quick estimator for nearby values. With L above: L(1.1, 1.9) = 5 + 0.2 minus 0.4 = 4.8, and L(0.9, 2.1) = 5 minus 0.2 + 0.4 = 5.2. The rhythm is always evaluate, differentiate, assemble.

Each partial tilts the plane in its own direction, and (fx, fy, minus 1) points straight out of it. Here that normal is (2, 4, minus 1). If you can read the normal, you can check your plane in one glance.

**Tip.** A subtle point: partials alone do not guarantee the plane fits. Wild functions can have both slopes on the axis slices but no true tangent. Only trust the plane on a smooth surface near its base point.

**Recap.** Build the plane from height plus two slopes, use it for nearby estimates, and trust it only when the surface is smooth.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Tangents and Normals to a Curve](https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp)
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9)

## Opens up

- [The Multivariable Chain Rule](https://lightmysky.com/learn/mathematics/the-multivariable-chain-rule-mt_CHSHdNxPmJ)