Multivariable Calculus Review Flashcards

Flashcards covering key concepts from multivariable calculus review topics, including functions, partial derivatives, critical points, the second derivative test, and Lagrange multipliers.

Multivariable Function

A function with multiple input variables.

Partial Derivative

The derivative of a multivariable function with respect to one variable, holding the other variables constant.

∂f/∂x vs fx

Different notations representing the partial derivative of f with respect to x.

Clairaut’s Theorem

States that if the second partial derivatives are continuous, then fxy = fyx.

Critical Points

Points where all first partial derivatives are equal to zero or do not exist. These are potential locations for local maxima, local minima, or saddle points.

Second Derivative Test

A test using second partial derivatives to classify critical points as local maxima, local minima, or saddle points.

Discriminant (D(x, y))

In the context of the second derivative test, D(x, y) = (fxx)(fyy) - (fxy)^2.

Local Maximum (Second Derivative Test)

If D > 0 and fxx < 0 at a critical point, then f has a local maximum.

Local Minimum (Second Derivative Test)

If D > 0 and fxx > 0 at a critical point, then f has a local minimum.

Saddle Point (Second Derivative Test)

If D < 0 at a critical point, then f has a saddle point (neither max nor min).

Method of Lagrange Multipliers

A method for finding the maxima and minima of a function subject to a constraint.

Objective Function

The function you are trying to maximize or minimize in an optimization problem.

Constraint Function

An equation that limits the possible values of the variables in an optimization problem.

Auxiliary/Lagrange Function

F(x, y, λ) = f(x, y) - λg(x, y) where f(x, y) is the objective function, g(x, y) is the constraint function, and λ is the Lagrange multiplier.