applied partial differentiation

Summary

Directional derivative and gradient vector

Tangent plane to level surface

Maximizing/minimizing

First derivative of local maximum/minimum

Second derivative test

Concept

Directional derivative

• vectorization of increments

Gradient

• vector-valued functions

normal to a given level curve/surface

Maximizing/minimizing directional derivative

maximised when points in the direction of the gradient vector

Local maximum and minimum

• largest/smallest value for all points within a disk centered at the point

Absolute maximum and minimum

• largest/smallest value over the entire domain

Critical/stationary point

Saddle point

• is a critical point
• increases in one direction and decreases in another

Application

Directional derivative

Tangnent plane

Finding critical points