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Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 17

Multivariable Calculus

Educators


Problem 1

Let $f(x, y)=2 x-3 y+5 .$ Find the following.
a. $f(2,-1) \quad$ b. $f(-4,1) \quad$ c. $f(-2,-3) \quad$ d. $f(0,8)$

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Problem 2

Let $g(x, y)=x^{2}-2 x y+y^{3} .$ Find the following.
a. $g(-2,4) \quad$ b.g $(-1,-2) \qquad$ c.g $(-2,3) \quad$ d. $g(5,1)$

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Problem 3

Let $h(x, y)=\sqrt{x^{2}+2 y^{2}} .$ Find the following.
a. $h(5,3) \quad$ b. $h(2,4) \quad$ c. $h(-1,-3) \quad$ d. $h(-3,-1)$

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Problem 4

Let $f(x, y)=\frac{\sqrt{9 x+5 y}}{\log x} .$ Find the following.
a. $f(10,2)$
b. $f(100,1)$
c. $f(1000,0)$
d. $f\left(\frac{1}{10}, 5\right)$

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Problem 5

Graph the first-octant portion of each plane.
$x+y+z=9$

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Problem 6

Graph the first-octant portion of each plane.
$x+y+z=15$

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Problem 7

Graph the first-octant portion of each plane.
$2 x+3 y+4 z=12$

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Problem 8

Graph the first-octant portion of each plane.
$4 x+2 y+3 z=24$

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Problem 9

Graph the first-octant portion of each plane.
$x+y=4$

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Problem 10

Graph the first-octant portion of each plane.
$y+z=5$

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Problem 11

Graph the first-octant portion of each plane.
$x=5$

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Problem 12

Graph the first-octant portion of each plane.
$z=4$

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Problem 13

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$.
$3 x+2 y+z=24$

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Problem 14

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$.
$3 x+y+2 z=8$

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Problem 15

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$.
$y^{2}-x=-z$

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Problem 16

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$.
$2 y-\frac{x^{2}}{3}=z$

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Problem 17

Discuss how a function of three variables in the form $w=f(x, y, z)$ might be graphed.

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Problem 18

Suppose the graph of a plane $a x+b y+c z=d$ has a portion in the first octant. What can be said about $a, b, c,$ and $d ?$

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Problem 19

In the chapter on Nonlinear Functions, the vertical line test was presented, which tells whether a graph is the graph of a function. Does this test apply to functions of two variables? Explain.

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Problem 20

A graph that was not shown in this section is the hyperboloid of one sheet, described by the equation $x^{2}+y^{2}-z^{2}=1$ Describe it as completely as you can.

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Problem 21

With its graph in a–f on the next page
$z=x^{2}+y^{2}$

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Problem 22

With its graph in a–f on the next page
$z^{2}-y^{2}-x^{2}=1$

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Problem 23

With its graph in a–f on the next page
$x^{2}-y^{2}=z$

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Problem 24

With its graph in a–f on the next page
$z=y^{2}-x^{2}$

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Problem 25

With its graph in a–f on the next page
$\frac{x^{2}}{16}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1$

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Problem 26

With its graph in a–f on the next page
$z=5\left(x^{2}+y^{2}\right)^{-1 / 2}$

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Problem 27

Let $f(x, y)=4 x^{2}-2 y^{2},$ and find the following.
a. $\frac{f(x+h, y)-f(x, y)}{h}$
b. $\frac{f(x, y+h)-f(x, y)}{h}$
c. $$\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}$$
d. $$\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$$

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Problem 28

Let $f(x, y)=5 x^{3}+3 y^{2},$ and find the following.
a. $\frac{f(x+h, y)-f(x, y)}{h}$
b. $\frac{f(x, y+h)-f(x, y)}{h}$
c. $$\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}$$
d. $$\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$$

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Problem 29

Let $f(x, y)=x y e^{x^{2}+y^{2}} .$ Use a graphing calculator or spread- sheet to find each of the following and give a geometric interpretation of the results. (Hint: First factor $e^{2}$ from the limit and then evaluate the quotient at smaller and smaller values of $h . )$
a. $$\lim _{h \rightarrow 0} \frac{f(1+h, 1)-f(1,1)}{h}$$
b. $$\lim _{h \rightarrow 0} \frac{f(1,1+h)-f(1,1)}{h}$$

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Problem 30

The following table provides values of the function $f(x, y)$ However, because of potential errors in measurement, the functional values may be slightly inaccurate. Using the statistical package included with a graphing calculator or spreadsheet and critical thinking skills, find the function $f(x, y)=$ $a+b x+c y$ that best estimates the table where $a, b,$ and $c$ are integers. (Hint: Do a linear regression on each column with the value of $y$ fixed and then use these four regression equations to determine the coefficient $c .$ )

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Problem 31

Production of a digital camera is given by
$$P(x, y)=100\left(\frac{3}{5} x^{-2 / 5}+\frac{2}{5} y^{-2 / 5}\right)^{-5}$$
where $x$ is the amount of labor in work-hours and $y$ is the amount of capital. Find the following.
a. What is the production when 32 work-hours and 1 unit of capital are provided?
b. Find the production when 1 work-hour and 32 units of capital are provided.
c. If 32 work-hours and 243 units of capital are used, what is the production output?

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Problem 32

The multiplier function
$$M=\frac{(1+i)^{n}(1-t)+t}{[1+(1-t) i]^{n}}$$
compares the growth of an Individual Retirement Account (IRA) with the growth of the same deposit in a regular savings account. The function $M$ depends on the three variables $n, i$ and $t,$ where $n$ represents the number of years an amount is left at interest, $i$ represents the interest rate in both types of accounts, and $t$ t represents the income tax rate. Values of $M>1$ indicate that the IRRA grows faster than the savings account. Let $M=f(n, i, t)$ and find the following.

Find the multiplier when funds are left for 25 years at 5$\%$ interest and the income tax rate is 33$\% .$ Which account grows faster?

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Problem 33

The multiplier function
$$M=\frac{(1+i)^{n}(1-t)+t}{[1+(1-t) i]^{n}}$$
compares the growth of an Individual Retirement Account (IRA) with the growth of the same deposit in a regular savings account. The function $M$ depends on the three variables $n, i$ and $t,$ where $n$ represents the number of years an amount is left at interest, $i$ represents the interest rate in both types of accounts, and $t$ t represents the income tax rate. Values of $M>1$ indicate that the IRRA grows faster than the savings account. Let $M=f(n, i, t)$ and find the following.

What is the multiplier when money is invested for 40 years at 6$\%$ interest and the income tax rate is 28$\% ?$ Which account grows faster?

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Problem 34

Find the level curve at a production of 500 for the production functions. Graph each level curve in the $xy$-plane.
In their original paper, $\mathrm{Cobb}$ and Douglas estimated the pro- duction function for the United States to be $z=1.01 x^{3 / 4} y^{1 / 4}$ , where $x$ represents the amount of labor and $y$ the amount of capital. Source: American Economic Review.

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Problem 35

Find the level curve at a production of 500 for the production functions. Graph each level curve in the $xy$-plane.
A study of the connection between immigration and the fiscal problems associated with the aging of the baby boom generation considered a production function of the form $z=x^{0.6} y^{0.4}$ where $x$ represents the amount of labor and $y$ the amount of capital. Source: Journal of Political Economy.

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Problem 36

For the function in Exercise $34,$ what is the effect on z of doubling $x ?$ Of doubling $y ?$ Of doubling both?

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Problem 37

If labor $(x)$ costs $\$ 250$ per unit, materials $(y)$ cost $\$ 150$ per unit, and capital $(z)$ costs $\$ 75$ per unit, write a function for total cost.

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Problem 38

The rate of heat loss (in watts) in harbor seal pups has been approximated by
$$H(m, T, A)=\frac{15.2 m^{0.67}(T-A)}{10.23 \ln m-10.74}$$
where $m$ is the body mass of the pup (in $\mathrm{kg} ),$ and $T$ and $A$ are the body core temperature and ambient water temperature, respectively $\left(\text { in }^{\circ} \mathrm{C}\right) .$ Find the heat loss for the following data. Source: Functional Ecology.
a. Body mass $=21 \mathrm{kg} ;$ body core temperature $=36^{\circ} \mathrm{C} ;$ ambient water temperature $=4^{\circ} \mathrm{C}$
b. Body mass $=29 \mathrm{kg} ;$ body core temperature $=38^{\circ} \mathrm{C} ;$ ambient water temperature $=16^{\circ} \mathrm{C}$

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Problem 39

The surface area of a human (in square meters) has been approximated by
$$A=0.024265 h^{0.3964} m^{0.5378}$$
where $h$ is the height $(\mathrm{in} \mathrm{cm})$ and $m$ is the mass (in $\mathrm{kg} ) .$ Find $A$ for the following data. Source: The Journal of Pediatrics.
a. Height, $178 \mathrm{cm} ;$ mass, 72 $\mathrm{kg}$
b. Height, $140 \mathrm{cm} ;$ mass, 65 $\mathrm{kg}$
c. Height, $160 \mathrm{cm} ;$ mass, 70 $\mathrm{kg}$
d. Using your mass and height, find your own surface area.

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Problem 40

An article entitled “How Dinosaurs Ran” explains that the locomotion of different sized animals can be compared when they have the same Froude number, defined as
$$F=\frac{v^{2}}{g l}$$
where $v$ is the velocity, $g$ is the acceleration of gravity $\left(9.81 \mathrm{m} \text { per } \sec ^{2}\right),$ and $l$ is the leg length (in meters). Source: Scientific American.

a. One result described in the article is that different animals change from a trot to a gallop at the same Froude number, roughly 2.56. Find the velocity at which this change occurs for a ferret, with a leg length of 0.09 m, and a rhinoceros, with a leg length of 1.2 m.
b. Ancient footprints in Texas of a sauropod, a large herbivo- rous dinosaur, are roughly 1 m in diameter, corresponding to a leg length of roughly 4 m. By comparing the stride divided by the leg length with that of various modern creatures, it can be determined that the Froude number for these dinosaurs is roughly 0.025. How fast were the sauropods traveling?

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Problem 41

According to research at the Great Swamp in New York, the percentage of fish that are intolerant to pollution can be estimated by the function
$$P(W, R, A)=48-2.43 W-1.81 R-1.22 A$$
where $W$ is the percentage of wetland, $R$ is the percentage of residential area, and $A$ is the percentage of agricultural area surrounding the swamp. Source: Northeastern Naturalist.

a. Use this function to estimate the percentage of fish that will be intolerant to pollution if 5 percent of the land is classified as wetland, 15 percent is classified as residential, and 0 percent is classified as agricultural. (Note: The land can also be classified as forest land.)
b. What is the maximum percentage of fish that will be intolerant to pollution?
c. Develop two scenarios that will drive the percentage of fish that are intolerant to pollution to zero.
d. Which variable has the greatest influence on P?

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Problem 42

In tropical regions, dengue fever is a significant health problem that affects nearly 100 million people each year. Using data collected from the 2002 dengue epidemic in Colima, Mexico, researchers have estimated that the incidence $I$ (number of new cases in a given year) of dengue can be predicted by the following function.
$$\begin{aligned} I(p, a, m, n, e)=&(25.54+0.04 p-7.92 a+2.62 m\\ &+4.46 n+0.15 e )^{2} \end{aligned}$$
where $p$ is the precipitation $(\mathrm{mm}), a$ is the mean temperature $\left(^{\circ} \mathrm{C}\right), m$ is the maximum temperature $\left(^{\circ} \mathrm{C}\right), n$ is the minimum temperature $\left(^{\circ} \mathrm{C}\right),$ and $e$ is the evaporation $(\mathrm{mm}) .$ Source: Journal of Environmental Health.
a. Estimate the incidence of a dengue fever outbreak for a region with 80 $\mathrm{mm}$ of rainfall, average temperature of $23^{\circ} \mathrm{C}$ , maximum temperature of $34^{\circ} \mathrm{C}$ , minimum temperature of $16^{\circ} \mathrm{C},$ and evaporation of 50 $\mathrm{mm} .$
b. Which variable has a negative influence on the incidence of dengue? Describe this influence and what can be inferred mathematically about the biology of the fever.

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Problem 43

Using data collected by the U.S. Forest Service, the annual number of deer-vehicle accidents for any given county in Ohio can be estimated by the function
$$\begin{aligned} A(L, T, U, C)=& 53.02+0.383 L+0.0015 T+0.0028 U \\ &-0.0003 C \end{aligned}$$
where $A$ is the estimated number of accidents, $L$ is the road length (in kilometers), $T$ is the total county land area (in hundred-acres (Ha)), $U$ is the urban land area (in hundred- acres), and $C$ is the number of hundred-acres of crop land. Source: Ohio Journal of Science.
a. Use this formula to estimate the number of deer-vehicle accidents for Mahoning County, where $L=266 \mathrm{km}, T=$ $107,484 \mathrm{Ha}, U=31,697 \mathrm{Ha},$ and $C=24,870$ Ha. The actual value was $396 .$
b. Given the magnitude and nature of the input numbers, which of the variables have the greatest potential to influence the number of deer-vehicle accidents? Explain your answer.

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Problem 44

Using data collected by the U.S. Forest Service, the annual number of deer that are harvested for any given county in Ohio can be estimated by the function
$$N(R, C)=329.32+0.0377 R-0.0171 C$$
where $N$ is the estimated number of harvested deer, $R$ is the rural land area (in hundred-acres), and $C$ is the number of hundred-acres of crop land. Source: Ohio Journal of Science.
a. Use this formula to estimate the number of harvested deer for Tuscarawas County, where $R=141,319$ Ha and $\mathrm{C}=$ $37,960$ Ha. The actual value was 4925 deer harvested.
b. Sketch the graph of this function in the first octant.

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Problem 45

Pregnant sows tethered in stalls often show high levels of repetitive behavior, such as bar biting and chain chewing, indicating chronic stress. Researchers from Great Britain have developed a function that estimates the relationship between repetitive behavior, the behavior of sows in adjacent stalls, and food allowances such that
$$\ln (T)=5.49-3.00 \ln (F)+0.18 \ln (C)$$
where $T$ is the percent of time spent in repetitive behavior, $F$ is the amount of food given to the sow (in kilograms per day), and $C$ is the percent of time that neighboring sows spent bar biting and chain chewing. Source: Applied Animal Behaviour Science.
a. Solve the above expression for $T$
b. Find and interpret $T$ when $F=2$ and $C=40$ .

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Problem 46

Extra postage is charged for parcels sent by U.S. mail that are more than 84 in. in length and girth combined. (Girth is the distance around the parcel perpendicular to its length. See the figure.) Express the combined length and girth as a function of $L, W,$ and $H$

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Problem 47

Refer to the figure for Exercise $46 .$ Assume $L, W,$ and $H$ are in feet. Write a function in terms of $L, W,$ and $H$ that gives the total area of the material required to build the box.

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Problem 48

The holes cut in a roof for vent pipes require elliptical templates. A formula for determining the length of the major axis of the ellipse is given by
$$L=f(H, D)=\sqrt{H^{2}+D^{2}}$$
where $D$ is the (outside) diameter of the pipe and $H$ is the "rise "of the roof per $D$ units of "run'; that is, the slope of the roof is $H / D$ . (See the figure below.) The width of the ellipse (minor axis) equals $D .$ Find the length and width of the ellipse required to produce a hole for a vent pipe with a diameter of 3.75 in. in roofs with the following slopes.
a. 3$/ 4$
b. 2$/ 5$

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