Q: Suppose $f(x, y) = \sqrt{9 - x^2 - y^2}$ models the maximum safe dosage of a medication based on two variables $x$ and $y$. What does the boundary $x^2 + y^2 = 9$ represent?

The maximum allowable combined value of $x$ and $y$ for safe dosage.

Q: Which of the following best describes the graph of $f(x, y) = \sqrt{9 - x^2 - y^2}$?

The upper hemisphere of a sphere of radius 3 centered at the origin

Question 1

Which of the following best describes the domain of the function $f(x, y) = \frac{x^2 + y^2}{x - 1}$?

Accepted Answer

All real numbers $(x, y)$ except where $x = 1$

Question 2

Given the function $g(x, y) = \ln(y^2 - x)$, what is the domain of $g$?

Accepted Answer

All $(x, y)$ such that $y^2 - x > 0$

Question 3

For the function $f(x, y) = \sqrt{9 - x^2 - y^2}$, which of the following describes its range?

Accepted Answer

$0 \leq f(x, y) \leq 3$

Question 4

Which of the following is the equation for a level curve of the function $f(x, y)$?

Accepted Answer

$f(x, y) = k$

Question 5

A function of the form $f(x, y) = ax + by + c$ represents which geometric object?

Accepted Answer

A plane

Question 6

Suppose $f(x, y) = \sqrt{9 - x^2 - y^2}$. What is the domain of $f$?

Accepted Answer

All $(x, y)$ such that $x^2 + y^2 \leq 9$

Question 7

Which of the following statements about the range of a function of two variables is correct?

Accepted Answer

The range is the set of all possible output values $f(x, y)$ for $(x, y)$ in the domain.

Question 8

If $f(x, y) = ax + by + c$, which equation describes the graph of $f$ in three-dimensional space?

Accepted Answer

$z = ax + by + c$

Question 9

Which of the following is a correct interpretation of a level curve for a function $f(x, y)$ in a real-world context?

Accepted Answer

A contour line on a topographic map representing points of equal elevation.

Question 10

Given $f(x, y) = \frac{x^2 + y^2}{x - 1}$, what is the value of $f(3, 2)$?

Accepted Answer

$\frac{13}{2}$

Question 11

For the function $g(x, y) = \ln(y^2 - x)$, which of the following points is in the domain?

Accepted Answer

$(1, 2)$

Question 12

Which of the following best describes the set of level curves for $f(x, y) = x^2 + y^2$?

Accepted Answer

Concentric circles centered at the origin

Question 13

If $f(x, y) = ax + by + c$ and $a, b, c$ are real constants, which of the following is true about the graph of $f$?

Accepted Answer

It is a plane in three-dimensional space.

Question 14

Which of the following is NOT a possible domain for a function of two variables?

Accepted Answer

All $(x, y)$ such that $x^2 + y^2 < 0$

Question 15

Suppose $f(x, y) = \sqrt{9 - x^2 - y^2}$ models the maximum safe dosage of a medication based on two variables $x$ and $y$. What does the boundary $x^2 + y^2 = 9$ represent?

Accepted Answer

The maximum allowable combined value of $x$ and $y$ for safe dosage.

Question 16

Which of the following is true about the function $f(x, y) = \frac{x^2 + y^2}{x - 1}$?

Accepted Answer

It is undefined when $x = 1$.

Question 17

If $f(x, y) = x^2 + y^2$, what is the level curve for $k = 4$?

Accepted Answer

$x^2 + y^2 = 4$

Question 18

Which of the following functions has a domain that is all real numbers $(x, y)$?

Accepted Answer

$f(x, y) = ax + by + c$

Question 19

Which of the following best describes the graph of $f(x, y) = \sqrt{9 - x^2 - y^2}$?

Accepted Answer

The upper hemisphere of a sphere of radius 3 centered at the origin

Section 4.1 - Functions of Several Variables

Study Guide - Practice Questions

Study Guide - Flashcards