Chapter 12, Functions of Several Variables Video Solutions, Calculus Multivariable

Section 1

Functions of Two Variables

03:16

Problem 1

Which of the points $P=(1,2,1)$ and $Q=(2,0,0)$ is closest to the origin?

Kayla R.

Numerade Educator

02:58

Problem 2

Which two of the three points $P_{1}=(1,2,3), P_{2}=$ (3,2,1) and $P_{3}=(1,1,0)$ are closest to each other?

Kayla R.

Numerade Educator

04:21

Problem 3

Which of the points $P_{1}=(-3,2,15), P_{2}=(0,-10,0)$ $P_{3}=(-6,5,3)$ and $P_{4}=(-4,2,7)$ is closest to $P=$ (6,0,4)$?$

Kayla R.

Numerade Educator

02:41

Problem 4

Which of the points $A=(1.3,-2.7,0), B=(0.9,0,3.2)$ $C=(2.5,0.1,-0.3)$ is closest to the $y z$ -plane? Which one lies on the $x z$ -plane? Which one is farthest from the $x y$ -plane?

Kayla R.

Numerade Educator

02:05

You are at the point $(3,1,1),$ standing upright and facing the $y z$ -plane. You walk 2 units forward, turn left, and walk another 2 units. What is your final position? From the point of view of an observer looking at the coordinate system in Figure 12.2 on page $654,$ are you in front of or behind the $y z$ -plane? To the left or to the right of the $x z$ -plane? Above or below the $x y$ -plane?

Carson M.

Numerade Educator

01:42

Problem 6

On a set of $x, y$ and $z$ axes oriented as in Figure 12.5 on page $654,$ draw a straight line through the origin, lying in the $y z$ -plane and such that if you move along the line with your $y$ -coordinate increasing, your $z$ -coordinate is increasing.

Kayla R.

Numerade Educator

02:26

Problem 7

What is the midpoint of the line segment joining the points (-1,3,9) and (5,6,-3)$?$

Kayla R.

Numerade Educator

01:02

Problem 8

Which of $(1)-(1 V)$ lie on the graph of the equation?
I. (2,2,4)
II. (-1,1,0)
III. (-3,-2,-1)
IV. (-2,-2,4)
$$z=4$$

Kayla R.

Numerade Educator

02:57

Problem 9

Which of $(1)-(1 V)$ lie on the graph of the equation?
I. (2,2,4)
II. (-1,1,0)
III. (-3,-2,-1)
IV. (-2,-2,4)
$$x+y+z=0$$

Kayla R.

Numerade Educator

03:43

Problem 10

Which of $(1)-(1 V)$ lie on the graph of the equation?
I. (2,2,4)
II. (-1,1,0)
III. (-3,-2,-1)
IV. (-2,-2,4)
$$x^{2}+y^{2}+z^{2}=14$$

Kayla R.

Numerade Educator

03:16

Problem 11

Which of $(1)-(1 V)$ lie on the graph of the equation?
I. (2,2,4)
II. (-1,1,0)
III. (-3,-2,-1)
IV. (-2,-2,4)
$$x-y=0$$

Kayla R.

Numerade Educator

01:52

Problem 12

Sketch graphs of the equations in 3 -space.
$$z=4$$

Kayla R.

Numerade Educator

01:29

Problem 13

Sketch graphs of the equations in 3 -space.
$$x=-3$$

Kayla R.

Numerade Educator

02:06

Problem 14

Sketch graphs of the equations in 3 -space.
$$y=1$$

Kayla R.

Numerade Educator

03:57

Problem 15

Sketch graphs of the equations in 3 -space.
$z=2$ and $y=4$

Kayla R.

Numerade Educator

01:43

Problem 16

With the z-axis vertical, a sphere has center (2,3,7) and lowest point $(2,3,-1) .$ What is the highest point on the sphere?

Kayla R.

Numerade Educator

02:18

Problem 17

Find an equation of the sphere with radius 5 centered at the origin.

Kayla R.

Numerade Educator

02:08

Problem 18

Find the equation of the sphere with radius 2 and centered at (1,0,0)

Kayla R.

Numerade Educator

00:41

Problem 19

Find the equation of the vertical plane perpendicular to the $y$ -axis and through the point (2,3,4)

Prashant B.

Numerade Educator

03:27

Problem 20

Refer to the map in Figure 12.1 on page 652.
Give the range of daily high temperatures for:
(a) Pennsylvania
(b) North Dakota
(c) California

Kayla R.

Numerade Educator

01:09

Problem 21

Sketch a possible graph of the predicted high temperature $T$ on a line north-south through Topeka.

Carson M.

Numerade Educator

01:08

Problem 22

Refer to the map in Figure 12.1 on page 652.
Sketch possible graphs of the predicted high temperature on a north-south line and an east-west line through Boise.

Carson M.

Numerade Educator

02:45

Problem 23

Refer to Table 12.1 on page 653 where $w \text { is a person's weight (in } 1 \text { bs })$, and $h$ their height (in inches).
Compute a table of values of BMI, with $h$ fixed at 60 inches and $w$ between 120 and 200 lbs at intervals of 20

Kayla R.

Numerade Educator

03:37

Problem 24

Refer to Table 12.1 on page 653 where $w \text { is a person's weight (in } 1 \text { bs })$, and $h$ their height (in inches).
Medical evidence suggests that BMI values between 18.5 and 24.9 are healthy values. $^{2}$ Estimate the range of weights that are considered healthy for a woman who is 6 feet tall.

Kayla R.

Numerade Educator

03:13

Problem 25

Refer to Table 12.1 on page 653 where $w \text { is a person's weight (in } 1 \text { bs })$, and $h$ their height (in inches).
Estimate the BMI of a man who weighs 90 kilograms and is 1.9 meters tall.

Kayla R.

Numerade Educator

01:04

Problem 26

Refer to Table 12.1 on page 653 where $w \text { is a person's weight (in } 1 \text { bs })$, and $h$ their height (in inches).
The temperature adjusted for wind chill is a temperature which tells you how cold it feels, as a result of the combination of wind and temperature. $^{3}$ See Table 12.2

Temperature adjusted for wind chill $\left(^{\circ} \mathrm{F}\right)$ as a function of wind speed and temperature
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline & 35 & 30 & 25 & 20 & 15 & 10 & 5 & 0 \\
\hline 5 & 31 & 25 & 19 & 13 & 7 & 1 & -5 & -11 \\
\hline 10 & 27 & 21 & 15 & 9 & 3 & -4 & -10 & -16 \\
\hline 15 & 25 & 19 & 13 & 6 & 0 & -7 & -13 & -19 \\
\hline 20 & 24 & 17 & 11 & 4 & -2 & -9 & -15 & -22 \\
\hline 25 & 23 & 16 & 9 & 3 & -4 & -11 & -17 & -24 \\
\hline
\end{array}$$
(a) If the temperature is $0^{\circ} \mathrm{F}$ and the wind speed is 15 mph, how cold does it feel?
(b) If the temperature is $35^{\circ} \mathrm{F}$, what wind speed makes it feel like $24^{\circ} \mathrm{F} ?$
(c) If the temperature is $25^{\circ} \mathrm{F},$ what wind speed makes it feel like $12^{\circ} \mathrm{F} ?$
(d) If the wind is blowing at 20 mph, what temperature feels like $0^{\circ} \mathrm{F} ?$

Carson M.

Numerade Educator

03:28

Problem 27

Use Table 12.2 to make tables with the given properties.
Temperature adjusted for wind chill $\left(^{\circ} \mathrm{F}\right)$ as a ficnction of wind speed and temperature
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline & 35 & 30 & 25 & 20 & 15 & 10 & 5 & 0 \\
\hline 5 & 31 & 25 & 19 & 13 & 7 & 1 & -5 & -11 \\
\hline 10 & 27 & 21 & 15 & 9 & 3 & -4 & -10 & -16 \\
\hline 15 & 25 & 19 & 13 & 6 & 0 & -7 & -13 & -19 \\
\hline 20 & 24 & 17 & 11 & 4 & -2 & -9 & -15 & -22 \\
\hline 25 & 23 & 16 & 9 & 3 & -4 & -11 & -17 & -24 \\
\hline
\end{array}$$
The temperature adjusted for wind chill as a function of wind speed for temperatures of $20^{\circ} \mathrm{F}$ and $0^{\circ} \mathrm{F}$

Kayla R.

Numerade Educator

03:48

Problem 28

Kayla R.

Numerade Educator

05:14

Problem 29

Refer to Table 12.3 which contains values of beef consumption $C$ (in pounds per week per household) as a function of household income, $I$ (in thousands of dollars per year), and the price of beef, $p$ (in dollars per pound). Values of $p$ are shown across the top, values of $I$ are down the left side, and corresponding values of beef consumption $C=f(I, p)$ are given in the table. $^{4}$
Quantity of beef bought (pounds/householdiweek)
$$\begin{array}{|c|c|c|c|c|}
\hline & 3.00 & 3.50 & 4.00 & 4.50 \\
\hline 20 & 2.65 & 2.59 & 2.51 & 2.43 \\
\hline 40 & 4.14 & 4.05 & 3.94 & 3.88 \\
\hline 60 & 5.11 & 5.00 & 4.97 & 4.84 \\
\hline 80 & 5.35 & 5.29 & 5.19 & 5.07 \\
\hline 100 & 5.79 & 5.77 & 5.60 & 5.53 \\
\hline
\end{array}$$
Give tables for beef consumption as a function of $p$ with $I$ fixed at $I=20$ and $I=100 .$ Give tables for beef consumption as a function of $I,$ with $p$ fixed at $p=3.00$ and $p=4.00 .$ Comment on what you see in the tables.

Kayla R.

Numerade Educator

05:35

Problem 30

Refer to Table 12.3 which contains values of beef consumption $C$ (in pounds per week per household) as a function of household income, $I$ (in thousands of dollars per year), and the price of beef, $p$ (in dollars per pound). Values of $p$ are shown across the top, values of $I$ are down the left side, and corresponding values of beef consumption $C=f(I, p)$ are given in the table. $^{4}$
Quantity of beef bought (pounds/householdiweek)
$$\begin{array}{|c|c|c|c|c|}
\hline & 3.00 & 3.50 & 4.00 & 4.50 \\
\hline 20 & 2.65 & 2.59 & 2.51 & 2.43 \\
\hline 40 & 4.14 & 4.05 & 3.94 & 3.88 \\
\hline 60 & 5.11 & 5.00 & 4.97 & 4.84 \\
\hline 80 & 5.35 & 5.29 & 5.19 & 5.07 \\
\hline 100 & 5.79 & 5.77 & 5.60 & 5.53 \\
\hline
\end{array}$$
Make a table of the proportion, $P$, of household income spent on beef per week as a function of price and income. (Note that $P$ is the fraction of income spent on beef. $)$

Kayla R.

Numerade Educator

03:01

Problem 31

Refer to Table 12.3 which contains values of beef consumption $C$ (in pounds per week per household) as a function of household income, $I$ (in thousands of dollars per year), and the price of beef, $p$ (in dollars per pound). Values of $p$ are shown across the top, values of $I$ are down the left side, and corresponding values of beef consumption $C=f(I, p)$ are given in the table. $^{4}$
Quantity of beef bought (pounds/householdiweek)
$$\begin{array}{|c|c|c|c|c|}
\hline & 3.00 & 3.50 & 4.00 & 4.50 \\
\hline 20 & 2.65 & 2.59 & 2.51 & 2.43 \\
\hline 40 & 4.14 & 4.05 & 3.94 & 3.88 \\
\hline 60 & 5.11 & 5.00 & 4.97 & 4.84 \\
\hline 80 & 5.35 & 5.29 & 5.19 & 5.07 \\
\hline 100 & 5.79 & 5.77 & 5.60 & 5.53 \\
\hline
\end{array}$$
How does beef consumption vary as a function of household income if the price of beef is held constant?

Kayla R.

Numerade Educator

01:58

Problem 32

A person's body mass index (BMI) is a function of their weight $W$ (in $\mathrm{kg}$ ) and height $H$ (in $\mathrm{m}$ ) given by $B(W, H)=W / H^{2}$.
What is the BMI of a $1.72 \mathrm{m}$ tall man weighing $72 \mathrm{kg} ?$

Kayla R.

Numerade Educator

03:04

Problem 33

A person's body mass index (BMI) is a function of their weight $W$ (in $\mathrm{kg}$ ) and height $H$ (in $\mathrm{m}$ ) given by $B(W, H)=W / H^{2}$.
A $1.58 \mathrm{m}$ tall woman has a BMI of $23.2 .$ What is her weight?

Kayla R.

Numerade Educator

03:43

Problem 34

A person's body mass index (BMI) is a function of their weight $W$ (in $\mathrm{kg}$ ) and height $H$ (in $\mathrm{m}$ ) given by $B(W, H)=W / H^{2}$.
With a BMI less than $18.5,$ a person is considered underweight. What is the possible range of weights for an underweight person $1.58 \mathrm{m}$ tall?

Kayla R.

Numerade Educator

01:10

Problem 35

A person's body mass index (BMI) is a function of their weight $W$ (in $\mathrm{kg}$ ) and height $H$ (in $\mathrm{m}$ ) given by $B(W, H)=W / H^{2}$.
For weight $w$ in 1 bs and height $h$ in inches, a persons BMI is approximated using the formula $f(w, h)=$ $703 w / h^{2} .$ Check this approximation by converting the formula $B(W, H)$

Carson M.

Numerade Educator

04:08

Problem 36

A person's body mass index (BMI) is a function of their weight $W$ (in $\mathrm{kg}$ ) and height $H$ (in $\mathrm{m}$ ) given by $B(W, H)=W / H^{2}$.
A car rental company charges $\$ 40$ a day and 15 cents a mile for its cars.
(a) Write a formula for the cost, $C$, of renting a car as
a function, $f,$ of the number of days, $d,$ and the number of miles driven, $m$
(b) If $C=f(d, m),$ find $f(5,300)$ and interpret it.

Kayla R.

Numerade Educator

03:35

Problem 37

A cable company charges $100$ for a monthly subscription to its services and $5$ for each special feature movie that a subscriber chooses to watch.
(a) Write a formula for the monthly revenue, $R$ in dollars, earned by the cable company as a function of $s,$ the number of monthly subscribers it serves, and $m,$ the total number of special feature movies that its subscribers view.
(b) If $R=f(s, m),$ find $f(1000,5000)$ and interpret it in terms of revenue.

Kayla R.

Numerade Educator

03:08

Problem 38

The gravitational force, $F$ newtons, exerted on an object by the earth depends on the object's mass, $m$ kilograms, and its distance, $r$ meters, from the center of the earth, so $F=f(m, r) .$ Interpret the following statement in terms of gravitation: $\quad f(100,7000000) \approx 820$

Kayla R.

Numerade Educator

04:52

Problem 39

A heating element is attached to the center point of a metal rod at time $t=0 .$ Let $H=f(d, t)$ represent the temperature in "C of a point $d$ cm from the center after $t$ minutes.
(a) Interpret the statement $f(2,5)=24$ in terms of temperature.
(b) If $d$ is held constant, is $H$ an increasing or a decreasing function of $t ?$ Why?
(c) If $t$ is held constant, is $H$ an increasing or a decreasing function of $d$ ? Why?

Kayla R.

Numerade Educator

02:57

Problem 40

The pressure, $P$ atmospheres, of 10 moles of nitrogen gas in a steel cylinder depends on the temperature of the gas, $T$ Kelvin, and the volume of the cylinder, $V$ liters, so $P=f(T, V) .$ Interpret the following statement in terms of pressure: $f(300,5)=49.2$

Kayla R.

Numerade Educator

04:06

Problem 41

The monthly payment, $m$ dollars, for a 30 -year fixed rate mortgage is a function of the total amount borrowed, $P$ dollars, and the annual interest rate, $r \% .$ In other words, $m=f(\boldsymbol{P}, \boldsymbol{r})$
(a) Interpret the following statement in the context of monthly payment: $f(300,000,5)=1610.46$
(b) If $P$ is held constant, is $m$ an increasing or a decreasing function of $r ?$ Why?
(c) If $r$ is held constant, is $m$ an increasing or a decreasing function of $P ?$ Why?

Kayla R.

Numerade Educator

02:29

Problem 42

Consider the acceleration due to gravity, $g,$ at a distance
$h$ from the center of a planet of mass $m$
(a) If $m$ is held constant, is $g$ an increasing or decreasing function of $h ?$ Why?
(b) If $h$ is held constant, is $g$ an increasing or decreasing function of $m ?$ Why?

Kayla R.

Numerade Educator

04:23

Problem 43

A cube is located such that its top four corners have the coordinates (-1,-2,2),(-1,3,2),(4,-2,2) and
$(4,3,2) .$ Give the coordinates of the center of the cube.

Kayla R.

Numerade Educator

01:41

Problem 44

Describe the set of points whose distance from the $x$ axis is 2

Prashansha K.

Numerade Educator

Problem 45

Describe the set of points whose distance from the $x$ axis equals the distance from the $y z$ -plane.

Check back soon!

01:50

Problem 46

Find the point on the $x$ -axis closest to the point (3,2,1)

Kayla R.

Numerade Educator

03:01

Problem 47

Does the line parallel to the $y$ -axis through the point
(2,1,4) intersect the plane $y=5 ?$ If so, where?

Kayla R.

Numerade Educator

01:07

Problem 48

Find a formula for the shortest distance between a point $(a, b, c)$ and the $y$ -axis.

Carson M.

Numerade Educator

04:30

Problem 49

Find the equations of planes that just touch the sphere $(x-2)^{2}+(y-3)^{2}+(z-3)^{2}=16$ and are parallel to
(a) The $x y$ -plane
(b) The $y z$ -plane
(c) The $x z$ -plane

Kayla R.

Numerade Educator

05:26

Problem 50

Find an equation of the largest sphere contained in the cube determined by the planes $x=2, x=6 ; y=5, y=$
$9 ;$ and $z=-1, z=3$

Kayla R.

Numerade Educator

05:11

Problem 51

A cube has edges parallel to the axes. One corner is at $A=(5,1,2)$ and the corner at the other end of the longest diagonal through $A$ is $B=(12,7,4)$
(a) What are the coordinates of the other three vertices
on the bottom face?
(b) What are the coordinates of the other three vertices
on the top face?

Kayla R.

Numerade Educator

01:16

Problem 52

An equilateral triangle is standing vertically with a vertex above the $x y$ -plane and its two other vertices at (7,0,0) and $(9,0,0) .$ What is its highest point?

Carson M.

Numerade Educator

05:25

Problem 53

(a) Find the midpoint of the line segment joining $A=$ (1,5,7) to $B=(5,13,19)$
(b) Find the point one quarter of the way along the line segment from $A$ to $B$
(c) Find the point one quarter of the way along the line segment from $B$ to $A$

Kayla R.

Numerade Educator

02:49

Problem 54

Explain what is wrong with the statement.
In 3 -space, $y=1$ is a line parallel to the $x$ -axis.

Kayla R.

Numerade Educator

01:46

Problem 55

Explain what is wrong with the statement.
The $x y$ -plane has equation $x y=0$

Kayla R.

Numerade Educator

02:55

Problem 56

Explain what is wrong with the statement.
The distance from (2,3,4) to the $x$ -axis is 2

Kayla R.

Numerade Educator

01:03

Problem 57

Give an example of:
A formula for a function $f(x, y)$ that is increasing in $x$ and decreasing in $y$

Carson M.

Numerade Educator

03:02

Problem 58

Give an example of:
A point in 3 -space with all its coordinates negative and farther from the $x z$ -plane than from the plane $z=-5$

Kayla R.

Numerade Educator

01:05

Problem 59

Are the statements true or false? Give reasons for your answer.
If $f(x, y)$ is a function of two variables defined for all $x$ and $y,$ then $f(10, y)$ is a function of one variable.

Carson M.

Numerade Educator

01:51

Problem 60

Are the statements true or false? Give reasons for your answer.
The volume $V$ of a box of height $h$ and square base of side length $s$ is a function of $h$ and $s$

Kayla R.

Numerade Educator

02:00

Problem 61

Are the statements true or false? Give reasons for your answer.
If $H=f(t, d)$ is the function giving the water temperature $H^{\circ} \mathrm{C}$ of a lake at time $t$ hours after midnight and depth $d$ meters, then $t$ is a function of $d$ and $H$

Kayla R.

Numerade Educator

01:35

Problem 62

Are the statements true or false? Give reasons for your answer.
A table for a function $f(x, y)$ cannot have any values of
$f$ appearing twice.

Kayla R.

Numerade Educator

01:05

Problem 63

Are the statements true or false? Give reasons for your answer.
If $f(x)$ and $g(y)$ are both functions of a single variable then the product $f(x) \cdot g(y)$ is a function of two variables.

Carson M.

Numerade Educator

01:14

Problem 64

Are the statements true or false? Give reasons for your answer.
The point (1,2,3) lies above the plane $z=2$

Kayla R.

Numerade Educator

02:35

Problem 65

Are the statements true or false? Give reasons for your answer.
The graph of the equation $z=2$ is a plane parallel to the $x z$ -plane.

Kayla R.

Numerade Educator

02:07

Problem 66

Are the statements true or false? Give reasons for your answer.
The points (1,0,1) and (0,-1,1) are the same distance from the origin.

Kayla R.

Numerade Educator

01:36

Problem 67

Are the statements true or false? Give reasons for your answer.
The point (2,-1,3) lies on the graph of the sphere $(x-2)^{2}+(y+1)^{2}+(z-3)^{2}=25$

Kayla R.

Numerade Educator

01:45

Problem 68

Are the statements true or false? Give reasons for your answer.
There is only one point in the $y z$ -plane that is a distance
3 from the point (3,0,0)

Kayla R.

Numerade Educator

02:44

Problem 69

Are the statements true or false? Give reasons for your answer.
There is only one point in the $y z$ -plane that is a distance
5 from the point (3,0,0)

Kayla R.

Numerade Educator

02:05