1. MAT 3238 Vector Calculus
15.4 Tangent Planes

2. Homework
Both written and WA HW due Monday.

3. Tangent Lines vs T. Planes

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

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

6. Geometric Meanings (15.3)

7. Geometric Meanings (15.3)

8. Normal Vector at (𝑎, 𝑏, 𝑓(𝑎,𝑏))

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

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

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

12. Linear Approximations: 2D
Difficult to compute
Easy to compute

13. Linear Approximations: 3D
Difficult to compute
Easy to compute

14. 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.

15. Differentials