Question 1

Given the function $P(t) = \frac{1}{1 + Ae^{-kt}}$, at what value of $P$ does the maximum value of $\frac{dP}{dt}$ occur?

Accepted Answer

$P = \frac{1}{2}$

Question 2

For the function $f(x, y) = x^2 + y^2$, what is the equation of the contour line at height $c$?

Accepted Answer

$x^2 + y^2 = c$

Question 3

Which of the following best describes the surface $g(x, y) = x^2 - y^2$?

Accepted Answer

A saddle-shaped surface

Question 4

Suppose you have the point $A = (1, -1, 0)$. Which coordinate is closest to the $xz$-plane?

Accepted Answer

The $y$-coordinate

Question 5

What is the distance between the points $A = (1, -1, 0)$ and $B = (4, 2, 0)$ in 3D space?

Accepted Answer

$\sqrt{(4-1)^2 + (2-(-1))^2 + (0-0)^2} = \sqrt{9 + 9} = \sqrt{18}$

Question 6

Which equation represents a plane parallel to $z=0$ but three units above it?

Accepted Answer

$z = 3$

Question 7

For the function $f(x, y) = \sqrt{x^2 + y^2}$, what is the radius of the contour at height $c$?

Accepted Answer

$c$

Question 8

Which of the following is a function of two variables?

Accepted Answer

Area of a rectangle as a function of length and width

Question 9

If $f(x, y) = 2x + 3y + 1$, what is the equation of the contour line at $c$?

Accepted Answer

$2x + 3y + 1 = c$

Question 10

On a topographic map, what do closely spaced contour lines indicate?

Accepted Answer

A steep slope

Question 11

For the function $f(x, y) = x^2 + y^2$, what is the radius of the contour corresponding to $c = 9$?

Accepted Answer

$3$

Question 12

Which of the following best describes a contour line?

Accepted Answer

A curve connecting points of equal value of a function

Question 13

If you fix $x = 2$ in $f(x, y) = x^2 + y^2$, what is the resulting cross-section as a function of $y$?

Accepted Answer

$f(2, y) = 4 + y^2$

Question 14

Which of the following is true about the graph of $z = x^2 + y^2$?

Accepted Answer

It is a paraboloid opening upward

Question 15

For the function $g(x, y) = x^2 - y^2$, what is the equation of the cross-section when $x = 1$?

Accepted Answer

$g(1, y) = 1 - y^2$

Question 16

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

Accepted Answer

BMI as a function of height and weight

Question 17

What is the equation for the contour line of $f(x, y) = \sqrt{x^2 + y^2}$ at height $c$?

Accepted Answer

$x^2 + y^2 = c^2$

Question 18

Which of the following statements about contour lines is correct?

Accepted Answer

Contour lines are obtained by taking horizontal slices of a surface

Question 19

For the function $f(x, y) = 2x + 3y + 1$, what is the contour line when $c = 4$?

Accepted Answer

$2x + 3y + 1 = 4$

Functions of Several Variables: Graphs, Surfaces, and Contour Diagrams

Study Guide - Practice Questions