Tangent Planes
Approximating a complex surface with a simple flat plane.
Introduction
In 1D calculus, we used tangent lines to approximate curves. In 3D calculus, we use tangent planes to approximate surfaces.
A tangent plane touches the surface at exactly one point and perfectly matches the slope of the surface in both the and directions.
Geometric Meaning
Interactive: Tangent Plane to a Paraboloid
The red line is the tangent line in the x-direction. The plane contains this line (and the y-tangent line).
Equation of the Plane
The Formula
The equation of the tangent plane to at is:
Notice how similar this is to point-slope form: . Here we just have two "slopes", one for and one for .
Worked Examples
Example 1: Finding Tangent Plane Equation
Find the tangent plane to at .
1. Calculate Partial Derivatives:
2. Plug into Formula:
3. Simplify:
Example 2: Linear Approximation
Find the linear approximation of at .
1. Evaluate Function:
.
2. Evaluate Partials:
3. Form Approximation:
.
Example 3: Using Approximation
Use the linearization from Example 2 to estimate .
We need to evaluate .
Plug in:
Actual value: .
Approximation is accurate to 3 decimal places!
Practice Quiz
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