MATH1011 MULTIVARIABLE CALCULUS Lecture Review

A set of flashcards summarizing key concepts from the multivariable calculus lecture notes, including definitions, properties of functions, and theorems related to limits and vector-valued functions.

What is a function?

A function f is a rule that assigns to each element x in a set D exactly one element f(x) in a set E.

What is the domain of a function?

The domain is the set D from which the function f assigns values.

What are intervals in the context of functions?

Intervals specify subsets of real numbers, such as (a, b), [c, d], or D = (−∞, ∞).

What is the range of a function?

The range is the subset of E consisting of all possible values of f(x) as x varies throughout the domain.

What is a vector-valued function?

A vector-valued function is a function that takes a real number and assigns a vector in Rn.

What is the definition of the dot product for vectors u and v?

The dot product is defined as u · v = u1v1 + u2v2 + u3v3.

How do you define a level curve for functions of two variables?

A level curve is the set of points (x, y) satisfying f(x, y) = k, where k is a real constant.

What is the formula for the limit of a scalar-valued function as x approaches c?

limx→c f(x) = L if f converges to L as x approaches c.

What does it mean if a function diverges to infinity?

It means that the function values f(x) get arbitrarily large as x approaches a.

What is L'Hospital's rule?

If limx→a f(x) = 0 and limx→a g(x) = 0, then limx→a f(x)/g(x) = limx→a f'(x)/g'(x), if the limit exists.

Define a function of two variables.

A function of two variables assigns a real number f(x, y) to each pair (x, y) in its domain D.

What is the standard notation for limits at infinity?

We write limx→∞ f(x) = L if f(x) approaches L as x tends to infinity.

What is the Squeeze Theorem?

If g(x) ≤ f(x) ≤ h(x) and limx→a g(x) = limx→a h(x) = L, then limx→a f(x) = L.