Directional derivatives. Gradient vectors and tangent planes. Maxima, minima and saddle points

1. Mathematical Analysis 2

Lecture 7 - Directional derivatives. Gradient
vectors and tangent planes. Maxima, minima and
saddle points.
Akbota Myrzakul
a.myrzakul@astanait.edu.kz
Astana IT University

2. Previous Lecture’s Summary

• Multivariable functions
• Limits of multivariable functions
• Continuity of multivariable functions
• Partial differentiation

3. Lecture overview

• Differentiability of multivariable functions
• Differentials & linear approximations
• Tangent plane
• Extrema of multivariable functions
• Directional Derivatives
• Gradients

4. Differentiability of multivariable functions

Definition:
Definition:

5. Differentiability of multivariable functions

Example: Prove that