Flashcards: Multivariable Functions and Optimization — 11 cards

What does a function of two variables assign to each ordered pair (x,y)?

A unique real number f(x,y).

Function of Two Variables - Definition

Assigns a real number to each (x,y) in its domain.

What is an n-tuple in the context of functions of several variables?

An ordered set of n elements used as input.

Domain of Function

Subset of ℝ² where function is defined.

What is the domain of a multivariable function?

The subset of ℝ² where the function is defined.

Limit in Multivariable

Approaches L along every path to (x₀,y₀).

Why is the domain of $f_1(x,y)=\frac{-x^2+y^3-5}{x-1}$ all pairs with $x \ne 1$?

Because the denominator cannot be zero.

Continuity

Function value equals limit at (x₀,y₀).

Partial Derivative

Derivative treating one variable as constant.

Total Differential

dz = f'_x dx + f'_y dy.

Second Partial Derivatives

Partial derivatives of partial derivatives.