Tangent Planes and Linear Approximations

Tangent Planes and Linearization | Multivariable Calculus

In this video, we explore tangent planes and their applications in multivariable calculus. We begin by reviewing the equation of a plane and the gradient vector, which lays the foundation for understanding tangent planes. You’ll learn how to derive the equation of a tangent plane and compute the normal line to a surface at a given point.

Through detailed examples, we find tangent planes and normal lines, determine points where the tangent plane is parallel to a given plane, and adapt the tangent plane equation for explicitly defined surfaces. Additionally, we cover linearization and tangent plane approximation for functions of two variables, demonstrating how these concepts provide useful approximations.

This video features step-by-step examples to help you confidently calculate tangent planes, normal lines, and linearizations, with practical applications for solving multivariable calculus problems. Whether you’re learning to compute tangent planes or apply them to approximations, this lesson has you covered!

Chapters
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0:00 Reviewing Plane Equations and the Gradient Vector
3:34 Defining the Tangent Plane Equation
5:17 Example #1: Finding the Tangent Plane to a Surface at a Point
8:02 Defining the Normal Line to a Surface
9:00 Example #2: Finding the Normal Line to a Surface at a Point
9:59 Example #3: Finding Both the Tangent Plane and the Normal Line
11:07 Example #4: Points Where the Tangent Plane is Parallel to a Given Plane
15:22 Example #5: Tangent Plane to a Surface at a Specific Point
17:38 Tangent Plane Equation for an Explicitly Defined Surface
19:08 Linearization and Tangent Plane Approximation for Two-Variable Functions
21:23 Example #6: Linearization of a Two-Variable Function
23:52 Linearization and Tangent Plane Approximation for Three-Variable Functions
24:11 Example #7: Linearization of a Three-Variable Function