SECTION

CONCEPTS

METHODS

14.1 The Partial Derivative

• Definition of partial derivative

• Notations for partial derivatives

14.2 Computing Partial Derivatives Algebraically

• Basic derivative rules

• Holding variables constant

14.3 Local Linearity

• Local linearity
• Linear approximation

• Finding a tangent plane
• f(x,y) is approximately equal to L(x,y) near (a,b)

14.4 Gradients and Directional Derivatives in the Plane

• Definition of directional derivative
• Definition of gradient for f(x,y)

• Converting to a unit vector
• Calculating directional derivative by gradient times the unit vector
• Gradient properties: direction of greatest increase, magnitude of greatest increase, perpendicular to contour

14.5 Gradients and Directional Derivatives in Space

• Definition of gradient for f(x,y,z)

• Same list of properties
• Finding a tangent plane to a level surface

14.6 The Chain Rule

• Composition of multivariable functions

• Drawing a diagram of dependencies among variables
• Writing a chain rule
• Calculating with a chain rule

14.7 Second-Order Partial Derivatives

• Definition of second-order derivatives

• Notations for second-order partials
• Calculating second-order partials
• Equality of mixed partials
• Hessian matrix

14.8 Differentiability

• Is the local linearization the tangent plane?

• Having partial derivatives is not enough to guarantee differentiability.

• Meaning of local extrema
• Saddle points

• Finding critical points
• Recognizing extrema on contour plots
• Classifying critical points by the determinant of the Hessian matrix (i.e. the second derivative test)

• Meaning of global extrema
• Closed and bounded regions
• Extreme Value Theorem

• Finding extrema by critical points
• Checking boundary points

• Objective function
• Constraint
• Lagrange multiplier

• Setting up the Lagrange system of equations
• Solving the Lagrange system
• Determining global extrema