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Applied Calculus for the MLSS A Brief Approach

Soo T. Tan

Chapter 8

Calculus of Several Variables - all with Video Answers

Educators

Section 1

Functions of Several Variables

01:04

Problem 1

Let $f(x, y)=2 x+3 y-4 .$ Compute $f(0,0), f(1,0), f(0,1), f(1,2),$ and $f(2,-1)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 2

Let $g(x, y)=2 x^{2}-y^{2} .$ Compute $g(1,2), g(2,1), g(1,1), g(-1,1),$ and $g(2,-1)$

Tyler Moulton
Tyler Moulton
Numerade Educator
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Problem 3

Let $f(x, y)=x^{2}+2 x y-x+3 .$ Compute $f(1,2), f(2,1), f(-1,2),$ and $f(2,-1)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:05

Problem 4

Let $h(x, y)=(x+y) /(x-y) .$ Compute $h(0,1), h(-1,1), h(2,1),$ and $h(\pi,-\pi)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:14

Problem 5

Let $g(s, t)=3 s \sqrt{t}+t \sqrt{s}+2 .$ Compute $g(1,2), g(2,1), g(0,4),$ and $g(4,9)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:18

Problem 6

Let $f(x, y)=x y e^{x^{2}+y^{2}} .$ Compute $f(0,0), f(0,1), f(1,1),$ and $f(-1,-1)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:12

Problem 7

Let $h(s, t)=s \ln t-t \ln s .$ Compute $h(1, e), h(e, 1),$ and $h(e, e)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:30

Problem 8

Let $f(u, v)=\left(u^{2}+v^{2}\right) e^{u w^{2}} .$ Compute $f(0,1), f(-1,-1), f(a, b),$ and $f(b, a)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 9

Let $g(r, s, t)=r e^{s / t} .$ Compute $g(1,1,1), g(1,0,1),$ and $g(-1,-1,-1)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:11

Problem 10

Let $g(u, v, w)=\left(u e^{v w}+v e^{u w}+w e^{u v}\right) /\left(u^{2}+v^{2}+w^{2}\right) .$ Compute $g(1,2,3)$ and $g(3,2,1)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:07

Problem 11

Find the domain of the function.
$$f(x, y)=2 x+3 y$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:10

Problem 12

Find the domain of the function.
$$g(x, y, z)=x^{2}+y^{2}+z^{2}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 13

Find the domain of the function.
$$h(u, v)=\frac{u v}{u-v}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:11

Problem 14

Find the domain of the function.
$$f(s, t)=\sqrt{s^{2}+t^{2}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:25

Problem 15

Find the domain of the function.
$$g(r, s)=\sqrt{r s}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:07

Problem 16

Find the domain of the function.
$$f(x, y)=e^{-x y}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:15

Problem 17

Find the domain of the function.
$$h(x, y)=1 n(x+y-5)$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:27

Problem 18

Find the domain of the function.
$$h(u, v)=\sqrt{4-u^{2}-v^{2}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:14

Problem 19

Sketch the level curves of the function corresponding to each value of $z$.
$$f(x, y)=2 x+3 y ; z=-2,-1,0,1,2$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:09

Problem 20

Sketch the level curves of the function corresponding to each value of $z$.
$$f(x, y)=-x^{2}+y ; z=-2,-1,0,1,2$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:08

Problem 21

Sketch the level curves of the function corresponding to each value of $z$.
$$f(x, y)=2 x^{2}+y ; z=-2,-1,0,1,2$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:47

Problem 22

Sketch the level curves of the function corresponding to each value of $z$.
$$f(x, y)=x y ; z=-4,-2,2,4$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:20

Problem 23

Sketch the level curves of the function corresponding to each value of $z$.
$$f(x, y)=\sqrt{16-x^{2}-y^{2}} ; z=0,1,2,3,4$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:14

Problem 24

Sketch the level curves of the function corresponding to each value of $z$.
$$f(x, y)=e^{x}-y ; z=-2,-1,0,1,2$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:09

Problem 25

Find an equation of the level curve of $f(x, y)=\sqrt{x^{2}+y^{2}}$ that contains the point (3,4)

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 26

Find an equation of the level surface of $f(x, y, z)=2 x^{2}+3 y^{2}-z$ that contains the
point (-1,2,-3)

Tyler Moulton
Tyler Moulton
Numerade Educator
01:56

Problem 27

Match the graph of the surface with one of the contour maps labeled (a) and (b).
$$f(x, y)=x+y^{2}$$

Aman Gupta
Aman Gupta
Numerade Educator
02:14

Problem 28

Match the graph of the surface with one of the contour maps labeled (a) and (b).
$$f(x, y)=e^{1-2 x^{2}-4 y^{2}}$$

Aman Gupta
Aman Gupta
Numerade Educator
01:26

Problem 29

Can two level curves of a function $f$ of two variables $x$ and $y$ intersect? Explain.

Aman Gupta
Aman Gupta
Numerade Educator
01:27

Problem 30

A level set of $f$ is the set $S=\{(x, y) | f(x, y)=k\},$ where $k$ is in the range of $f .$ Let $$f(x, y)=\left\{\begin{array}{ll}
0 & \text { if } x^{2}+y^{2} < 1 \\
x^{2}+y^{2}-1 & \text { if } x^{2}+y^{2} \geq 1
\end{array}\right.$$
Sketch the level set of $f$ for $k=0$ and 3

Aman Gupta
Aman Gupta
Numerade Educator
01:09

Problem 31

The volume of a cylindrical tank of radius $r$ and height $h$ is given by
$$V=f(r, h)=\pi r^{2} h$$
Find the volume of a cylindrical tank of radius $1.5 \mathrm{ft}$ and height $4 \mathrm{ft}$.

Tyler Moulton
Tyler Moulton
Numerade Educator
01:09

Problem 32

The IQ (intelligence quotient) of a person whose mental age is $m$ years and whose chronological age is $c$ years is defined as $$f(m, c)=\frac{100 m}{c}$$
What is the IQ of a 9 -year-old child who has a mental age of 13.5 years?

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 33

The current P/E ratio (price-to-earnings ratio) of a stock is defined as
$$R=\frac{P}{E}$$
where $P$ is the current market price per share of the stock and $E$ is the earnings per share for the most recent 12 -month period.
a. What is the domain of the function $R$
b. The earnings per share of IBM Corporation for 2011 were $\$ 13.09$, and its price per share on March 19,2012 was $\$ 205.56 .$ What was the $\mathrm{P} / \mathrm{E}$ ratio of IBM at that time?

Tyler Moulton
Tyler Moulton
Numerade Educator
02:38

Problem 34

The current dividend yield for common stock is calculated by using the formula
$$Y=\frac{D}{P}$$
where $D$ is the most recent full-year dividend and $P$ is the current share price (both measured in dollars).
a. What is the domain of the function $Y ?$
b. The annualized dividend of IBM Corporation for the year 2011 was $\$ 3,$ and its price per share was $\$ 205.56$ on March $19,2012 .$ What was the current dividend yield for the common stock of IBM at that time?

William Semus
William Semus
Numerade Educator
01:38

Problem 35

The body mass index (BMI) is used to identify, evaluate, and treat overweight and obese adults. The BMI value for an adult of weight $w$ (in kilograms) and height $h$ (in meters) is defined to be
$$
M=f(w, h)=\frac{w}{h^{2}}
$$
According to federal guidelines, an adult is overweight if he or she has a BMI value greater than 25 but less than 30 and is "obese" if the value is greater than or equal to 30.
a. What is the BMI of an adult who weighs in at $80 \mathrm{kg}$ and stands $1.8 \mathrm{m}$ tall?
b. What is the maximum weight for an adult of height $1.8 \mathrm{m},$ who is not classified
as overweight or obese?

Aman Gupta
Aman Gupta
Numerade Educator
01:04

Problem 36

Poiseuille's Law states that the resistance $R$ of blood flowing in a blood vessel of length $l$ and radius $r$ is given by
$$
R=f(l, r)=\frac{k l}{r^{4}}
$$
where $k$ is a constant that depends on the viscosity of blood. What is the resistance, in terms of $k,$ of blood flowing through an arteriole $4 \mathrm{cm}$ long and of radius $0.1 \mathrm{cm} ?$

Aman Gupta
Aman Gupta
Numerade Educator
02:16

Problem 37

Acrosonic manufactures a bookshelf loudspeaker system that may be bought fully assembled or in a kit. The costs in labor and material incurred in manufacturing a fully assembled system and a kit are $\$ 200$ and $\$ 120$, respectively. In addition, Acrosonic has fixed costs of
$\$ 20,000 /$ month.
a. Write the monthly cost function $C$ for Acrosonic in terms of the number of fully assembled systems $x$ and the number of kits $y$ manufactured.
b. What is the domain of $C ?$
c. What is the total cost incurred in manufacturing 1000 fully assembled systems and 200 kits in a month?

Aman Gupta
Aman Gupta
Numerade Educator
06:31

Problem 38

Bell Motors manufactures two models of electric cars. The demand equations giving the relationship between the
unit price, $p$ and $q$, and the number of cars demanded per year, $x$ and $y,$ of Models $S 1$ and S2 are
$$
p=60,000-4 x-2 y
$$
and
$$
q=50,000-2 x-4 y
$$
respectively.
a. What is the total yearly revenue function $R(x, y) ?$
b. What is the domain of $R ?$ Sketch the domain $R$.
c. Is the point (3000,2000) in the domain of $R ?$ Interpret your result.
Hint: Show that $x=3000$ and $y=2000$ satisfy the system of inequalities obtained in part (b).
d. What is the total revenue realized by Bell Motors if it sells 3000 Model S1s and
2000 Model S2s?

Aman Gupta
Aman Gupta
Numerade Educator
02:53

Problem 39

Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week of its rolltop desks in the finished and unfinished versions are $x$ and
$y$ units when the corresponding unit prices are
$$
\begin{array}{l}
p=200-\frac{1}{5} x-\frac{1}{10} y \\
q=160-\frac{1}{10} x-\frac{1}{4} y
\end{array}
$$
dollars, respectively.
a. What is the weekly total revenue function $R(x, y) ?$
b. Find the domain of the function $R$.

Aman Gupta
Aman Gupta
Numerade Educator
03:00

Problem 40

For the total revenue function $R(x, y)$ of Exercise $39,$ compute $R(100,60)$ and $R(60,100) .$ Interpret your results.

Aman Gupta
Aman Gupta
Numerade Educator
03:28

Problem 41

Weston Publishing publishes a deluxe edition and a standard edition of its Spanish-English dictionary. Weston's management estimates that the number of deluxe editions demanded is $x$
copies/day and the number of standard editions demanded is $y$ copies/day when the unit prices are
$$
\begin{array}{l}
p=20-0.005 x-0.001 y \\
q=15-0.001 x-0.003 y
\end{array}
$$
dollars, respectively.
a. Find the daily total revenue function $R(x, y)$
b. Find the domain of the function $R$

Aman Gupta
Aman Gupta
Numerade Educator
03:04

Problem 42

For the total revenue function $R(x, y)$ of Exercise $41,$ compute $R(300,200)$ and $R(200,300) .$ Interpret your results.

Aman Gupta
Aman Gupta
Numerade Educator
01:08

Problem 43

The volume of a certain mass of gas is related to its pressure and temperature by the formula
$$
V=\frac{30.9 T}{P}
$$
where the volume $V$ is measured in liters, the temperature $T$ is measured in kelvins (obtained by adding 273 to the Celsius temperature), and the pressure $P$ is measured in millimeters of mercury pressure. V
b. Calculate the volume of the gas at standard temperature and pressure-that is, when $T=273 \mathrm{K}$ and $P=760 \mathrm{mm}$ of mercury.

Aman Gupta
Aman Gupta
Numerade Educator
01:38

Problem 44

An empirical formula by E. F. Dubois relates the surface area $S$ of a human body (in square meters) to its weight $W$ (in kilograms) and its height $H$ (in centimeters). The formula, given by $$S=0.007184 W^{0.425} H^{0.725}$$
is used by physiologists in metabolism studies.
a. Find the domain of the function $S$.
b. What is the surface area of a human body that weighs $70 \mathrm{kg}$ and has a height of
$178 \mathrm{cm} ?$

Aman Gupta
Aman Gupta
Numerade Educator
01:17

Problem 45

A formula for estimating the weight of a trout (from measurements) is
$$
W=\frac{L G^{2}}{800}
$$
where $L$ is its length and $G$ is its girth (the distance around the body of the fish at its largest point , both measured in inches. The weight of the fish $W$ is in pounds.
a. What is the domain of the function $W$ ?
b. Sue caught a trout and measured its length to be 20 in. and its girth to be 12 in. What is its approximate weight?

Aman Gupta
Aman Gupta
Numerade Educator
03:17

Problem 46

Refer to Exercise $45 .$ A trout caught by Ashley is $20 \%$ longer and has a girth that is $10 \%$ shorter than the one caught by Jane.
a. Whose catch is heavier?
b. By how much does the weight of Ashley's catch differ from the weight of Jane's catch?

Aman Gupta
Aman Gupta
Numerade Educator
01:20

Problem 47

Suppose the output of a certain country is given by
$$
f(x, y)=100 x^{3 / 5} y^{2 / 5}
$$
billion dollars if $x$ billion dollars are spent on labor and $y$ billion dollars are spent on
capital. Find the output if the country spent $\ 32$ billion on labor and $\ 243$ billion on capital.

Aman Gupta
Aman Gupta
Numerade Educator
03:09

Problem 48

Economists have found that the output of a finished product, $f(x, y),$ is sometimes described by the function
$$
f(x, y)=a x^{b} y^{1-b}
$$
where $x$ stands for the amount of money expended on labor, $y$ stands for the amount expended on capital, and $a$ and $b$ are positive constants with $0 < b < 1$
b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by $r \%,$ then the output is also increased by $r \%$

Aman Gupta
Aman Gupta
Numerade Educator
02:00

Problem 49

A study of arson for profit was conducted by a team of paid civilian experts and police detectives appointed by the mayor of a large city. It was found that the number of suspicious fires in that city in 2016 was very closely related to the concentration of tenants in the city's public housing and to the level of reinvestment in the area in conventional mortgages by the ten largest banks. In fact, the number of fires was closely approximated by the formula
$$N(x, y)=\frac{100\left(1000+0.03 x^{2} y\right)^{1 / 2}}{(5+0.2 y)^{2}} \quad(0 \leq x \leq 150 ; 5 \leq y \leq 35)$$
where $x$ denotes the number of persons per census tract and $y$ denotes the level of
reinvestment in the area in cents per dollar deposited. Using this formula, estimate the total number of suspicious fires in the districts of the city where the concentration of public housing tenants was $100 /$ census tract and the level of
reinvestment was 20 cents/dollar deposited.

Aman Gupta
Aman Gupta
Numerade Educator
01:17

Problem 50

If a principal of $P$ dollars is deposited in an account earning interest at the rate of $r /$ year compounded continuously, then the accumulated amount at the end of $t$ years is given by $$A=f(P, r, t)=P e^{r t}$$
dollars. Find the accumulated amount at the end of 3 years if a sum of $\ 10,000$ is deposited in an account earning interest at the rate of $6 \% /$ year.

Aman Gupta
Aman Gupta
Numerade Educator
03:53

Problem 51

The monthly payment that amortizes a loan of $A$ dollars in $t$ years when the interest rate is $r /$ year, compounded monthly, is given by $$P=f(A, r, t)=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]}$$
a. What is the monthly payment for a home mortgage of $\ 300,000$ that will be amortized over 30 years with an interest rate of $4 \% /$ year? An interest rate of $6 \% /$ year?
b. Find the monthly payment for a home mortgage of $\ 300,000$ that will be amortized over 20 years with an interest rate of $6 \% /$ year.

Aman Gupta
Aman Gupta
Numerade Educator
03:53

Problem 52

Suppose a home buyer secures a bank loan of $A$ dollars to purchase a house. If the interest rate charged is $r /$ year compounded monthly and the loan is to be amortized in $t$ years, then the principal repayment at the end of
$i$ months is given by $$\begin{aligned}
B &=f(A, r, t, i) \\
&=A\left[\frac{\left(1+\frac{r}{12}\right)^{i}-1}{\left(1+\frac{r}{12}\right)^{12 t}-1}\right] \quad(0 \leq i \leq 12 t)
\end{aligned}$$
Suppose the Blakelys borrow a sum of $\ 280,000$ from a bank to help finance the purchase of a house and the bank charges interest at a rate of $6 \% /$ year. If the Blakelys agree to repay the loan in equal installments over 30 years, how much will they owe the bank after the 60th payment (5 years)? The 240th payment (20 years)?

Aman Gupta
Aman Gupta
Numerade Educator
01:26

Problem 53

The Wilson lot-size formula in economics states that the optimal quantity $Q$ of goods for a store to order is given by
$$
Q=f(C, N, h)=\sqrt{\frac{2 C N}{h}}
$$
where $C$ is the cost of placing an order, $N$ is the number of items the store sells per week, and $h$ is the weekly holding cost for each item. Find the most economical quantity of 10-speed bicycles to order if it costs the store $\ 20$ to place an order, $\ 5$ to hold a bicycle for a week, and the store expects to sell 40 bicycles/week.

Aman Gupta
Aman Gupta
Numerade Educator
02:33

Problem 54

The power output (in watts) of a certain brand of wind turbine generators is estimated to be
$$
P=f(R, V)=0.772 R^{2} V^{3}
$$
where $R$ is the radius (in meters) of a rotor blade and $V$ is the wind speed (in meters per second). Estimate the power output of a model of these generators if its radius is $30 \mathrm{m}$ and the wind speed is $16 \mathrm{m} / \mathrm{s}$.

William Semus
William Semus
Numerade Educator
03:34

Problem 55

Drafted by an international committee in $1989,$ the rules for the new International America's Cup Class (IACC) include a formula that governs the basic yacht dimensions. The formula
$$
f(L, S, D) \leq 42
$$
where
$$
f(L, S, D)=\frac{L+1.25 S^{1 / 2}-9.80 D^{1 / 3}}{0.388}
$$
balances the rated length $L$ (in meters), the rated sail area $S$ (in square meters), and the displacement $D$ (in cubic meters). All changes in the basic dimensions are tradeoffs. For example, if you want to pick up speed by increasing the sail area, you must pay for it by decreasing the length or increasing the displacement, both of which slow down the boat. Show that Yacht $A$ of rated length $20.95 \mathrm{m},$ rated sail area $277.3 \mathrm{m}^{2},$ and displacement $17.56 \mathrm{m}^{3}$ and the longer and heavier Yacht $B$ with $L=21.87, S=311.78,$ and $D=22.48$ both satisfy the formula.

Aman Gupta
Aman Gupta
Numerade Educator
03:26

Problem 56

A centrifuge is a machine designed for the specific purpose of subjecting materials to a sustained centrifugal force. The actual amount of centrifugal force, $F$, expressed in dynes (1 gram of force $=980$ dynes) is given by
$$F=f(M, S, R)=\frac{\pi^{2} S^{2} M R}{900}$$
where $S$ is in revolutions per minute (rpm), $M$ is in grams, and $R$ is in centimeters.
Show that an object revolving at the rate of $600 \mathrm{rpm}$ in a circle with radius of $10 \mathrm{cm}$
generates a centrifugal force that is approximately 40 times gravity.

- -
- -
Numerade Educator
03:56

Problem 57

According to the ideal gas law, the volume $V$ of an ideal gas is related to its pressure $P$ and temperature $T$ by the formula $$V=\frac{k T}{P}$$
where $k$ is a positive constant. Describe the level curves of $V$ and give a physical interpretation of your result.

William Semus
William Semus
Numerade Educator
01:58

Problem 58

Let $f(x, y)$ denote the output of a country. If $x$ units of its resources are spent on labor and $y$ units are spent on capital, then the level curve of $f$ with equation $f(x, y)=k,$ where $k$ denotes a constant output, is an isoquant. Each point $(x, y)$ on the isoquant corresponds to a level of investment in labor, $x,$ and investment in capital, $y,$ that results in the same level of output $k$. Suppose the output of a certain country is
$$f(x, y)=100 x^{3 / 4} y^{1 / 4}$$
billion dollars when $x$ billion dollars are spent on labor and $y$ billion dollars are spent on capital.
a. Find the output if the country spends $\$ 81$ billion on labor and $\$ 16$ billion on capital.
b. Find an equation giving the relationship between $x$ and $y$ if the output remains constant at the level found in part (a).
c. Complete the following table for the equation found in part (b).
$$\begin{array}{llllll}
x & 50 & 60 & 70 & 80 & 90 \\
\hline y & & & & \\
\hline
\end{array}$$
d. Use the results of part $(\underline{c})$ to sketch the isoquant of $f$ corresponding to the constant output found in part (b).

James Kiss
James Kiss
Numerade Educator
01:59

Problem 59

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why, or give an example to show why it is false.
If $h$ is a function of $x$ and $y,$ then there are functions $f$ and $g$ of one variable such that $$h(x, y)=f(x)+g(y)$$

Manisha Sarker
Manisha Sarker
Numerade Educator
02:16

Problem 60

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why, or give an example to show why it is false.
If $f$ is a function of $x$ and $y$ and $a$ is a real number, then
$$
f(a x, a y)=a f(x, y)
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
02:24

Problem 61

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why, or give an example to show why it is false.
The domain of $f(x, y)=1 /\left(x^{2}-y^{2}\right)$ is $\{(x, y) | y \neq x\}$

Ahmad Reda
Ahmad Reda
Numerade Educator
02:13

Problem 62

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why, or give an example to show why it is false.
Every point on the level curve $f(x, y)=c$ corresponds to a point on the graph of $f$ that is $c$ units above the $x y$ -plane if $c > 0$ and $|c|$ units below the $x y$ -plane if $c < 0$

Ahmad Reda
Ahmad Reda
Numerade Educator
00:51

Problem 63

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why, or give an example to show why it is false.
$f$ is a function of $x$ and $y$ if and only if for any two points $P_{1}\left(x_{1}, y_{1}\right)$ and $P_{2}\left(x_{2}, y_{2}\right)$ in the domain of $f, f\left(x_{1}, y_{1}\right)=f\left(x_{2}, y_{2}\right)$ implies that $P_{1}\left(x_{1}, y_{1}\right)=P_{2}\left(x_{2}, y_{2}\right)$

Lucas Finney
Lucas Finney
Numerade Educator
02:05

Problem 64

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why, or give an example to show why it is false.
If $f$ is a function of $x$ and $y,$ and $k$ is a real number in the range of $f,$ then there exists at most one point $P(a, b)$ in the domain of $f$ such that $f(a, b)=k$

Ahmad Reda
Ahmad Reda
Numerade Educator