Stewart J. Calculus

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people and the number of uninfected people. In an isolated

town of 5000 inhabitants, 160 people have a disease at the

beginning of the week and 1200 have it at the end of the

week. How long does it take for of the population to

become infected?

20. The Brentano-Stevens Law in psychology models the way

that a subject reacts to a stimulus. It states that if represents

the reaction to an amount of stimulus, then the relative rates

of increase are proportional:

where is a positive constant. Find as a function of .

21. The transport of a substance across a capillary wall in lung

physiology has been modeled by the differential equation

where is the hormone concentration in the bloodstream, is

time, is the maximum transport rate, is the volume of the

capillary, and is a positive constant that measures the afﬁn-

ity between the hormones and the enzymes that assist the

process. Solve this differential equation to ﬁnd a relationship

between and .

22. Populations of birds and insects are modeled by the equations

(a) Which of the variables, or , represents the bird pop-

ulation and which represents the insect population?

Explain.

(b) Find the equilibrium solutions and explain their

signiﬁcance.

(d) The direction ﬁeld for the differential equation in part (c)

is shown. Use it to sketch the phase trajectory corre-

20000 40000

100

200

300

60000

400

dy!dx

! !0.2y " 0.000008xy

! 0.4x ! 0.002xy

! !

k " h

SRk

80%

9–11 Solve the initial-value problem.

9. ,

10. ,

11. ,

;

12. Solve the initial-value problem , , and

graph the solution.

13–14 Find the orthogonal trajectories of the family of curves.

13. 14.

15. (a) Write the solution of the initial-value problem

and use it to ﬁnd the population when .

(b) When does the population reach 1200?

16. (a) The population of the world was 5.28 billion in 1990 and

6.07 billion in 2000. Find an exponential model for these

data and use the model to predict the world population in

the year 2020.

(b) According to the model in part (a), when will the world

population exceed 10 billion?

population. Assume a carrying capacity of 100 billion.

Then use the logistic model to predict the population in

2020. Compare with your prediction from the exponential

model.

(d) According to the logistic model, when will the world pop-

ulation exceed 10 billion? Compare with your prediction

in part (b).

17. The von Bertalanffy growth model is used to predict the

length of a ﬁsh over a period of time. If is the largest

length for a species, then the hypothesis is that the rate of

growth in length is proportional to , the length yet to

be achieved.

(a) Formulate and solve a differential equation to ﬁnd an

expression for .

(b) For the North Sea haddock it has been determined that

, cm, and the constant of propor-

tionality is . What does the expression for become

with these data?

18. A tank contains 100 L of pure water. Brine that contains

0.1 kg of salt per liter enters the tank at a rate of 10 L!min.

The solution is kept thoroughly mixed and drains from the

tank at the same rate. How much salt is in the tank after

6 minutes?

19. One model for the spread of an epidemic is that the rate of

spread is jointly proportional to the number of infected

L$t%0.2

L$0% ! 10L

! 53 cm

L$t%

! L

L$t%

t ! 20

P$0% ! 100

! 0.1P

1 !

2000

y ! e

y ! ke

y$0% ! 1y$ ! 3x

y$1% ! 2xy$ ! y ! x ln x

y$0% ! 0$1 " cos x%y$ ! $1 " e

%sin x

r$0% ! 5

" 2tr ! r

652

|| ||

CHAPTER 10 DIFFERENTIAL EQUATIONS

(d) Sketch graphs of the bird and insect populations as func-

tions of time.

24. Barbara weighs 60 kg and is on a diet of 1600 calories per

day, of which 850 are used automatically by basal metabolism.

She spends about 15 cal!kg!day times her weight doing exer-

cise. If 1 kg of fat contains 10,000 cal and we assume that

the storage of calories in the form of fat is efﬁcient,

formulate a differential equation and solve it to ﬁnd her

weight as a function of time. Does her weight ultimately

approach an equilibrium weight?

25. When a ﬂexible cable of uniform density is suspended

between two ﬁxed points and hangs of its own weight, the

shape of the cable must satisfy a differential equa-

tion of the form

where is a positive constant. Consider the cable shown in

the ﬁgure.

(a) Let in the differential equation. Solve the

resulting ﬁrst-order differential equation (in ), and then

integrate to ﬁnd .

(b) Determine the length of the cable.

(0, a)

(b, h)

(_b, h)

z ! dy!dx

! k

1 "

y ! f $x%

100%

sponding to initial populations of 100 birds and 40,000

insects. Then use the phase trajectory to describe how

both populations change.

(e) Use part (d) to make rough sketches of the bird and insect

populations as functions of time. How are these graphs

related to each other?

23. Suppose the model of Exercise 22 is replaced by the

equations

(a) According to these equations, what happens to the insect

population in the absence of birds?

(b) Find the equilibrium solutions and explain their

signiﬁcance.

100 birds and 40,000 insects. Describe what eventually

happens to the bird and insect populations.

15000

100

4500025000 35000

120

140

160

180

200

220

240

260

! !0.2y " 0.000008xy

! 0.4x$1 ! 0.000005x% ! 0.002xy

CHAPTER 10 REVIEW

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653

P R O B L E M S P L U S

654

Find all functions such that is continuous and

A student forgot the Product Rule for differentiation and made the mistake of thinking

that . However, he was lucky and got the correct answer. The function that he

used was and the domain of his problem was the interval . What was the

function ?

Let be a function with the property that , , and for

all real numbers and . Show that for all and deduce that .

Find all functions that satisfy the equation

Find the curve such that , , , and the area under the graph

of from to is proportional to the power of .

A subtangent is a portion of the -axis that lies directly beneath the segment of a tangent line

from the point of contact to the -axis. Find the curves that pass through the point and

whose subtangents all have length .

A peach pie is removed from the oven at 5:00 PM. At that time it is piping hot, .

At 5:10 PM its temperature is ; at 5:20 PM it is . What is the temperature of the

room?

Snow began to fall during the morning of February 2 and continued steadily into the after-

noon. At noon a snowplow began removing snow from a road at a constant rate. The plow

traveled 6 km from noon to 1

but only 3 km from 1

to 2

. When did the snow begin

to fall? [Hints: To get started, let be the time measured in hours after noon; let be the

distance traveled by the plow at time ; then the speed of the plow is . Let be the num-

ber of hours before noon that it began to snow. Find an expression for the height of the snow

at time . Then use the given information that the rate of removal (in ) is constant.]

A dog sees a rabbit running in a straight line across an open ﬁeld and gives chase. In a rectan-

gular coordinate system (as shown in the ﬁgure), assume:

(i) The rabbit is at the origin and the dog is at the point at the instant the dog ﬁrst

sees the rabbit.

(ii) The rabbit runs up the -axis and the dog always runs straight for the rabbit.

(iii) The dog runs at the same speed as the rabbit.

(a) Show that the dog’s path is the graph of the function , where satisﬁes the differ-

ential equation

(b) Determine the solution of the equation in part (a) that satisﬁes the initial conditions

when . [Hint: Let in the differential equation and solve the

resulting ﬁrst-order equation to ﬁnd ; then integrate to ﬁnd .]

yzz

z ! dy!dxx ! Ly ! y$ ! 0

1 "

yy ! f $x%

$L, 0%

!hRt

bdx!dtt

x$t%t

65 %C80 %C

100 %C

$c, 1%x

f $x%$n " 1%stx0f

f $1% ! 1f $0% ! 0f $x% & 0y ! f $x%

f $x% dx

f $x%

! !1

f $x% ! e

xf $$x% ! f $x%ba

f $a " b% ! f $a% f $b%f $$0% ! 1f $0% ! 1f

(

, #

)

f $x% ! e

f$ ft%$ ! f $t$

for all real x[ f $x%]

! 100 "

'[ f $t%]

" [ f $$t%]

(

f $f

F I G U R E F O R P R O B L E M 9

(L, 0)

(x, y)

Openmirrors.com

P R O B L E M S P L U S

655

10. (a) Suppose that the dog in Problem 9 runs twice as fast as the rabbit. Find a differential

equation for the path of the dog. Then solve it to ﬁnd the point where the dog catches the

rabbit.

(b) Suppose the dog runs half as fast as the rabbit. How close does the dog get to the rabbit?

What are their positions when they are closest?

11. A planning engineer for a new alum plant must present some estimates to his company regard-

ing the capacity of a silo designed to contain bauxite ore until it is processed into alum. The

ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The

silo is a cylinder 100 ft high with a radius of 200 ft. The conveyor carries and

the ore maintains a conical shape whose radius is 1.5 times its height.

(a) If, at a certain time , the pile is 60 ft high, how long will it take for the pile to reach the

top of the silo?

(b) Management wants to know how much room will be left in the ﬂoor area of the silo when

the pile is 60 ft high. How fast is the ﬂoor area of the pile growing at that height?

the pile reaches 90 ft. Suppose, also, that the pile continues to maintain its shape. How

long will it take for the pile to reach the top of the silo under these conditions?

12. Find the curve that passes through the point and has the property that if the tangent line

is drawn at any point on the curve, then the part of the tangent line that lies in the ﬁrst

quadrant is bisected at .

13. Recall that the normal line to a curve at a point on the curve is the line that passes through

and is perpendicular to the tangent line at . Find the curve that passes through the point

and has the property that if the normal line is drawn at any point on the curve, then

the -intercept of the normal line is always 6.

14. Find all curves with the property that if the normal line is drawn at any point on the curve,

then the part of the normal line between and the -axis is bisected by the -axis.yxP

$3, 2%

20,000

60,000

656

So far we have described plane curves by giving as a function of or

as a function of or by giving a relation between and that deﬁnes

implicitly as a function of . In this chapter we discuss two new methods

for describing curves.

Some curves, such as the cycloid, are best handled when both and are given in

terms of a third variable called a parameter . Other curves, such as

the cardioid, have their most convenient description when we use a new coordinate

system, called the polar coordinate system.

!x ! f "t#, y ! t"t#$t

! f "x, y# ! 0$x

yyx!x ! t"y#$y

x!y ! f "x#$xy

Parametric equations and polar coordinates enable us to

describe a great variety of new curves—some practical,

some beautiful, some fanciful, some strange.

PARAMETRIC EQUATIONS

AND POLAR COORDINATES

CURVES DEFINED BY PARAMETRIC EQUATIONS

Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to

describe C by an equation of the form because C fails the Vertical Line Test. But

the x- and y-coordinates of the particle are functions of time and so we can write

and . Such a pair of equations is often a convenient way of describing a curve and

gives rise to the following deﬁnition.

Suppose that and are both given as functions of a third variable (called a param-

eter) by the equations

(called parametric equations). Each value of determines a point , which we can

plot in a coordinate plane. As varies, the point varies and traces out

a curve , which we call a parametric curve. The parameter t does not necessarily repre-

sent time and, in fact, we could use a letter other than t for the parameter. But in many

applications of parametric curves, t does denote time and therefore we can interpret

as the position of a particle at time t.

EXAMPLE 1 Sketch and identify the curve deﬁned by the parametric equations

SOLUTION Each value of gives a point on the curve, as shown in the table. For instance, if

, then , and so the corresponding point is . In Figure 2 we plot the

points determined by several values of the parameter and we join them to produce

a curve.

A particle whose position is given by the parametric equations moves along the curve

in the direction of the arrows as increases. Notice that the consecutive points marked on

the curve appear at equal time intervals but not at equal distances. That is because the

particle slows down and then speeds up as increases.

It appears from Figure 2 that the curve traced out by the particle may be a parabola.

This can be conﬁrmed by eliminating the parameter as follows. We obtain

from the second equation and substitute into the ﬁrst equation. This gives

and so the curve represented by the given parametric equations is the parabola

. Mx ! y

! 4y " 3

x ! t

! 2t ! "y ! 1#

! 2"y ! 1# ! y

! 4y " 3

t ! y ! 1t

F I G U R E 2

t=0

t=1

t=2

t=3

t=4

t=_1

t=_2

(0,1)

"x, y#

"0, 1#y ! 1x ! 0t ! 0

y ! t " 1x ! t

! 2t

"x, y# ! " f "t#, t"t##

"x, y# ! " f "t#, t"t##t

"x, y#t

y ! t"t#x ! f "t#

tyx

y ! t"t#

x ! f "t#

y ! f "x#

11.1

t x y

!2 8 !1

3 0

0 0 1

1 !1 2

2 0 3

3 3 4

4 8 5

N This equation in and describes

where

the

particle has been, but it doesn’t tell us

when

the

particle was at a particular point. The parametric

equations have an advantage––they tell us

when

the particle was at a point. They also indi-

cate the

direction

of the motion.

(x,y)={ f(t), g(t)}

F I G U R E 1

657

No restriction was placed on the parameter in Example 1, so we assumed that t could

be any real number. But sometimes we restrict t to lie in a ﬁnite interval. For instance, the

parametric curve

shown in Figure 3 is the part of the parabola in Example 1 that starts at the point and

ends at the point . The arrowhead indicates the direction in which the curve is traced

as increases from 0 to 4.

In general, the curve with parametric equations

has initial point and terminal point .

EXAMPLE 2 What curve is represented by the following parametric equations?

SOLUTION If we plot points, it appears that the curve is a circle. We can conﬁrm this

impression by eliminating Observe that

Thus the point moves on the unit circle . Notice that in this example

the parameter can be interpreted as the angle (in radians) shown in Figure 4. As

increases from 0 to , the point moves once around the circle in

the counterclockwise direction starting from the point .

EXAMPLE 3 What curve is represented by the given parametric equations?

SOLUTION Again we have

so the parametric equations again represent the unit circle . But as

increases from 0 to , the point starts at and moves twice

around the circle in the clockwise direction as indicated in Figure 5.

Examples 2 and 3 show that different sets of parametric equations can represent the

same curve. Thus we distinguish between a curve, which is a set of points, and a parametric

curve, in which the points are traced in a particular way.

"0, 1#"x, y# ! "sin 2t, cos 2t#2

" y

! 1

" y

! sin

2t " cos

2t ! 1

0 $ t $ 2

y ! cos 2tx ! sin 2t

F I G U R E 4

3π

t=0

(1,0)

(cost, sint)

t=2π

t=π

"1, 0#

"x, y# ! "cos t, sin t#2

" y

! 1"x, y#

" y

! cos

t " sin

t ! 1

0 $ t $ 2

y ! sin tx ! cos t

" f "b#, t"b##" f "a#, t"a##

a $ t $ by ! t"t#x ! f "t#

"8, 5#

"0, 1#

0 $ t $ 4y ! t " 1x ! t

! 2t

658

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CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES

F I G U R E 3

(8,5)

(0,1)

t=0,π,2π

F I G U R E 5

(0,1)

EXAMPLE 4 Find parametric equations for the circle with center and radius .

SOLUTION If we take the equations of the unit circle in Example 2 and multiply the expres-

sions for and by , we get , . You can verify that these equations

represent a circle with radius and center the origin traced counterclockwise. We now

shift units in the -direction and units in the -direction and obtain parametric equa-

tions of the circle (Figure 6) with center and radius :

EXAMPLE 5 Sketch the curve with parametric equations , .

SOLUTION Observe that and so the point moves on the parabola

. But note also that, since , we have , so the para-

metric equations represent only the part of the parabola for which . Since

is periodic, the point moves back and forth inﬁnitely often

along the parabola from to . (See Figure 7.) M

y=sin2tx=cost y=sin2t

x=cost

F I G U R E 8

"1, 1#"!1, 1#

"x, y# ! "sin t, sin

t#sin t

!1 $ x $ 1

!1 $ x $ 1!1 $ sin t $ 1y ! x

"x, y#y ! "sin t#

! x

y ! sin

tx ! sin t

F I G U R E 6

x=h+rcost, y=k+rsint

(h,k)

0 $ t $ 2

y ! k " r sin tx ! h " r cos t

r"h, k#

ykxh

y ! r sin tx ! r cos tryx

r"h, k#

SECTION 11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS

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659

Module 11.1A gives an animation of the

relationship between motion along a parametric

curve , and motion along the

graphs of and as functions of . Clicking on

TRIG gives you the family of parametric curves

If you choose and click

on animate, you will see how the graphs of

and relate to the circle in

Example 2. If you choose ,

, you will see graphs as in Figure 8. By

clicking on animate or moving the -slider to

the right, you can see from the color coding how

motion along the graphs of and

corresponds to motion along the para-

metric curve, which is called a Lissajous ﬁgure.

y ! sin 2t

x ! cos t

d ! 2

a ! b ! c ! 1

y ! sin tx ! cos t

a ! b ! c ! d ! 1

y ! c sin dtx ! a cos bt

ttf

y ! t"t#x ! f "t#

TE C

F I G U R E 7

(1,1)

(_1,1)

GRAPHING DEVICES

Most graphing calculators and computer graphing programs can be used to graph curves

deﬁned by parametric equations. In fact, it’s instructive to watch a parametric curve being

drawn by a graphing calculator because the points are plotted in order as the correspon-

ding parameter values increase.

EXAMPLE 6 Use a graphing device to graph the curve .

SOLUTION If we let the parameter be , then we have the equations

Using these parametric equations to graph the curve, we obtain Figure 9. It would be

possible to solve the given equation for y as four functions of x and

graph them individually, but the parametric equations provide a much easier method. M

In general, if we need to graph an equation of the form , we can use the para-

metric equations

Notice also that curves with equations (the ones we are most familiar with—

graphs of functions) can also be regarded as curves with parametric equations

Graphing devices are particularly useful when sketching complicated curves. For

instance, the curves shown in Figures 10, 11, and 12 would be virtually impossible to pro-

duce by hand.

One of the most important uses of parametric curves is in computer-aided design

(CAD). In the Laboratory Project after Section 11.2 we will investigate special parametric

curves, called Bézier curves, that are used extensively in manufacturing, especially in the

automotive industry. These curves are also employed in specifying the shapes of letters and

other symbols in laser printers.

THE CYCLOID

EXAMPLE 7 The curve traced out by a point on the circumference of a circle as the

circle rolls along a straight line is called a cycloid (see Figure 13). If the circle has

radius and rolls along the -axis and if one position of is the origin, ﬁnd parametric

equations for the cycloid.

Pxr

_6.5 6.5

F I G U R E 1 0

x=t+2sin2t

y=t+2cos5t

2.5

_2.5

2.5

F I G U R E 1 1

x=1.5cost-cos30t

y=1.5sint-sin30t

_2.5

F I G U R E 1 2

x=sin(t+cos100t)

y=cos(t+sin100t)

y ! f "t#x ! t

y ! f "x#

y ! tx ! t"t#

x ! t"y#

"x ! y

! 3y

y ! tx ! t

! 3t

t ! y

x ! y

! 3y

660

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CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES

An animation in Module 11.1B

shows how the cycloid is formed as the

circle moves.

TE C

_3 3

F I G U R E 9

SOLUTION We choose as parameter the angle of rotation of the circle when is at

the origin). Suppose the circle has rotated through radians. Because the circle has been

in contact with the line, we see from Figure 14 that the distance it has rolled from the

origin is

Therefore the center of the circle is . Let the coordinates of be . Then

from Figure 14 we see that

Therefore parametric equations of the cycloid are

One arch of the cycloid comes from one rotation of the circle and so is described by

. Although Equations 1 were derived from Figure 14, which illustrates the

case where , it can be seen that these equations are still valid for other

values of (see Exercise 39).

Although it is possible to eliminate the parameter from Equations 1, the resulting

Cartesian equation in and is very complicated and not as convenient to work with as

the parametric equations.

One of the ﬁrst people to study the cycloid was Galileo, who proposed that bridges be

built in the shape of cycloids and who tried to ﬁnd the area under one arch of a cycloid.

Later this curve arose in connection with the brachistochrone problem: Find the curve

along which a particle will slide in the shortest time (under the inﬂuence of gravity) from

a point to a lower point not directly beneath . The Swiss mathematician John

Bernoulli, who posed this problem in 1696, showed that among all possible curves that

join to , as in Figure 15, the particle will take the least time sliding from to if the

curve is part of an inverted arch of a cycloid.

The Dutch physicist Huygens had already shown that the cycloid is also the solution to

the tautochrone problem; that is, no matter where a particle is placed on an inverted

cycloid, it takes the same time to slide to the bottom (see Figure 16). Huygens proposed

that pendulum clocks (which he invented) swing in cycloidal arcs because then the pendu-

lum takes the same time to make a complete oscillation whether it swings through a wide

or a small arc.

FAMILIES OF PARAMETRIC CURVES

EXAMPLE 8 Investigate the family of curves with parametric equations

What do these curves have in common? How does the shape change as increases?a

y ! a tan t " sin tx ! a " cos t

BABA

ABA

0 $

$ 2

! !y ! r "1 ! cos

#x ! r"

! sin

y !

! r ! r cos

! r"1 ! cos

x !

! r

! r sin

! r"

! sin

"x, y#PC"r

, r#

! arc PT ! r

! 0

F I G U R E 1 3

SECTION 11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS

|| ||

661

F I G U R E 1 5

cycloid

F I G U R E 1 4

C(r¨,r)

r¨

P Q

F I G U R E 1 6