MAT 3238 Vector Calculus 15.4 Tangent Planes.

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Presentation transcript:

MAT 3238 Vector Calculus 15.4 Tangent Planes

Homework Both written and WA HW due Monday.

Tangent Lines vs T. Planes

How to find Tangent Planes? Two ingredients A point 𝑎,𝑏,𝑐 A normal vector 𝑛 1 , 𝑛 2 , 𝑛 3

Recall: The Geometric Meanings of Partial Derivatives 𝑓 𝑥 and 𝑓 𝑦

Geometric Meanings (15.3)

Geometric Meanings (15.3)

Normal Vector at (𝑎, 𝑏, 𝑓(𝑎,𝑏))

Normal Vector at (𝑎, 𝑏, 𝑓(𝑎,𝑏)) Similarly,

Normal Vector at (𝑎, 𝑏, 𝑓(𝑎,𝑏)) The equation of the tangent plane is

Example 1 Find the tangent plane to the surface 𝑧= 𝑥 2 +2 𝑦 2 at (1, 2, 9).

Linear Approximations: 2D Difficult to compute Easy to compute

Linear Approximations: 3D Difficult to compute Easy to compute

Differentials You may/will see these in another class. This is the best time to see them for the first time because we can explain them.

Differentials