Question 1

Which of the following best describes the domain of the function $$f(x, y) = \sqrt{y - x^2$}$$?

Accepted Answer

All points $$(x, y)$$ such that $$y \geq x^2$.

Question 2

Given the function $$z = f(x, y) = 100 - x^2$ - y^2$, what is the equation of the level curve for z = 51$?

Accepted Answer

$$x^2$$ + $$y^2 = 49$

Question 3

Which of the following statements about open and closed sets is TRUE?

Accepted Answer

A set is open if it contains only interior points and no boundary points.

Question 4

Consider the region $$R = \{(x, y) \mid x^2$ + y^2$ \u003c 1\}. $$Which of the following is TRUE?

Accepted Answer

The region is open and bounded.

Question 5

Which of the following is a real-world example of a function of three variables?

Accepted Answer

The temperature at a point in a room as a function of $$x$$, $$y$$, and $$z$$ coordinates.

Question 6

Suppose $$f(x, y, z) = x^2$ + y^2$ + z^2$. What is the geometric description of the level surface f(x$$, y, z) = 4$$?

Accepted Answer

A sphere of radius $$2$$ centered at the origin.

Question 7

Which of the following best describes a boundary point of a region $$R in $$the $$xy-$$plane?

Accepted Answer

A point where every disk centered at that point contains both points inside and outside $$R.$$

Question 8

Which of the following is the range of the function $$z = \sin(xy)$$ for all real $$x$$ and $$y$$?

Accepted Answer

$$[-1, 1]$$

Question 9

A region in the plane is said to be bounded if:

Accepted Answer

It lies inside a disk of finite radius.

Question 10

Which of the following is NOT a possible domain for the function $$w = \frac{1}{x^2$ + y^2$ + z^2$}$$?

Accepted Answer

All points $$(x, y, z) in $$space.

Question 11

Which of the following best describes a level curve of a function $$f(x, y)$$?

Accepted Answer

A set of points $$(x, y) in $$the domain where $$f(x, y) = c$$ for some constant $$c.$$

Question 12

Which of the following is TRUE about the graph of a function $$z = f(x, y)$$?

Accepted Answer

It is a surface in three-dimensional space.

Question 13

Suppose $$f(x, y) = \frac{1}{xy}. $$What is the domain of $$f$$?

Accepted Answer

All $$(x, y)$$ except where $$x = 0 or$$ $$y = 0.$$

Question 14

Which of the following is an example of a function of more than three variables?

Accepted Answer

The temperature $$T at a $$point $$(x, y, z) at $$time $$t.$$

Question 15

Which of the following statements about level surfaces is correct?

Accepted Answer

A level surface of $$f(x, y, z) is a $$set of points where $$f(x, y, z) = c$$ for some constant $$c.$$

Question 16

Which of the following best describes the range of the function $$w = x^2$ + y^2$ + z^2$ for all real x$$, y, z$$?

Accepted Answer

$$[0, \infty)$$

Question 17

A region in space is said to be unbounded if:

Accepted Answer

It does not lie inside any ball of finite radius.

Question 18

Which of the following is a correct interpretation of a contour curve for $$z = f(x, y)$$?

Accepted Answer

It is the intersection of the surface $$z = f(x, y)$$ with the plane $$z = c.$$

Functions of Several Variables and Their Domains: Partial Derivatives Chapter 14.1 Study Guide

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