SECTION 8.3
15. A cube with 20cmlong sides is sitting on the bottom of an
aquarium in which the water is one meter deep. Estimate the
hydrostatic force on (a) the top of the cube and (b) one of the
sides of the cube.
16. A dam is inclined at an angle of
726
CHAPTER 11
INFINITE SEQUENCES AND SERIES
Therefore the remainder Rn for the given series satisfies
Rn ! Tn !
1
2n 2
With n ! 100 we have
R100 !
1
! 0.00005
2!100"2
Using a programmable calculator or a computer, we find that
"
#
n!1
100
1
1
$
$ 0.68645
8. Consider a flat metal plate to be placed vertically under water with its top 2 m below the
surface of the water. Determine a shape for the plate so that if the plate is divided into any
number of horizontal strips of equal height, the hydrostatic force
SECTION 10.3
POLAR COORDINATES
655
SOLUTION The points are plotted in Figure 3. In part (d) the point "!3, 3"!4# is located
three units from the pole in the fourth quadrant because the angle 3"!4 is in the second
quadrant and r ! !3 is negative.
5
4
3
O
(
CHAPTER 8
REVIEW
575
Review
8
Concept Check
1. (a) How is the length of a curve defined?
6. Given a demand function p!x", explain what is meant by the
(b) Write an expression for the length of a smooth curve given
by y ! f !x", a ! x ! b.
(c) What if x is
(d) A hemispherical bowl has radius 5 inches and water is running into the bowl at the rate
of 0.2 in3!s.
(i) How fast is the water level in the bowl rising at the instant the water is 3 inches deep?
(ii) At a certain instant, the water is 4 inches deep.
SECTION 10.3
712 Sketch the region in the plane consisting of points whose
polar coordinates satisfy the given conditions.
7. r ! 1
8. 0 " r # 2,
9. r ! 0,
$ " % " 3$!2
41. r 2 ! 9 sin 2%
42. r 2 ! cos 4%
43. r ! 2 & sin 3%
44. r 2% ! 1
45. r ! 1 & 2 cos
592
CHAPTER 9
9.2
DIFFERENTIAL EQUATIONS
Exercises
5. y! ! x $ y # 1
1. A direction field for the differential equation y! ! x cos " y is
shown.
(a) Sketch the graphs of the solutions that satisfy the given
initial conditions.
(i) y!0" ! 0
(ii) y!0" ! 0.5
522
CHAPTER 7
TECHNIQUES OF INTEGRATION
EXAMPLE 3 Evaluate
y
!
"!
1
dx.
1 $ x2
SOLUTION Its convenient to choose a ! 0 in Definition 1(c):
y
!
"!
1
dx !
1 $ x2
y
1
dx $
1 $ x2
0
"!
y
!
0
1
dx
1 $ x2
We must now evaluate the integrals on the right side sep
654
CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
where 0 ! t ! 1. Notice that when t ! 0 we have !x, y" ! !x 0 , y0 " and when t ! 1 we have
!x, y" ! !x 3 , y 3", so the curve starts at P0 and ends at P3.
1. Graph the Bzier curve with control poi
SECTION 9.6
PREDATORPREY SYSTEMS
623
portional to itself, that is,
dW
! !rW
dt
where r is a positive constant
With both species present, however, we assume that the principal cause of death among the
prey is being eaten by a predator, and the birth and s
SECTION 11.10
TAYLOR AND MACLAURIN SERIES
755
To find the radius of convergence we let a n ! x n(n!. Then
! ! !
!
$ $
a n!1
x n!1
n!
x
!
!
!
l 0"1
an
"n ! 1#! x n
n!1
so, by the Ratio Test, the series converges for all x and the radius of convergence
is R
626
CHAPTER 9
DIFFERENTIAL EQUATIONS
TEC In Module 9.6 you can change the
coefficients in the LotkaVolterra equations and
observe the resulting changes in the phase
trajectory and graphs of the rabbit and wolf
populations.
To make the graphs easier to co
SECTION 10.4
7.
8.
AREAS AND LENGTHS IN POLAR COORDINATES
669
35. Find the area inside the larger loop and outside the smaller
loop of the limaon r ! 12 # cos ".
36. Find the area between a large loop and the enclosed small
loop of the curve r ! 1 # 2 cos
504
CHAPTER 7
TECHNIQUES OF INTEGRATION
Its clear that both systems must have expanded !x 2 " 5"8 by the Binomial Theorem and
then integrated each term.
If we integrate by hand instead, using the substitution u ! x 2 " 5, we get
y x!x
Derive and the TI89
SECTION 11.6
ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Definition A series
2
733
! a n is called conditionally convergent if it is convergent
but not absolutely convergent.
Example 2 shows that the alternating harmonic series is conditionally conv
712
CHAPTER 11
INFINITE SEQUENCES AND SERIES
49. Let x ! 0.99999 . . . .
(a) Do you think that x ! 1 or x ! 1?
(b) Sum a geometric series to find the value of x.
(c) How many decimal representations does the number 1
have?
(d) Which numbers have more than
SECTION 11.5

NOTE The rule that the error (in using sn to approximate s) is smaller than the first
neglected term is, in general, valid only for alternating series that satisfy the conditions of
the Alternating Series Estimation Theorem. The rule does n
748
CHAPTER 11
INFINITE SEQUENCES AND SERIES
EXAMPLE 3 Find a power series representation of x 3!"x ! 2#.
SOLUTION Since this function is just x 3 times the function in Example 2, all we have to
do is to multiply that series by x 3:
Its legitimate to move
SECTION 9.5
LINEAR EQUATIONS
617
Integrating both sides, we would have
I!x"y ! y I!x" Q!x" dx ! C
so the solution would be
4
y!x" !
1
I!x"
#y
I!x" Q!x" dx ! C
$
To find such an I, we expand Equation 3 and cancel terms:
I!x"y" ! I!x" P!x"y ! (I!x"y)" ! I"!
SECTION 8.4
APPLICATIONS TO ECONOMICS AND BIOLOGY
563
1. Suppose the cups have height h, cup A is formed by rotating the curve x ! f ! y" about the
yaxis, and cup B is formed by rotating the same curve about the line x ! k. Find the value
of k such that
SECTION 9.3
36. Find a function f such that f "3# ! 2 and
"t 2 ! 1# f "t# ! $ f "t#% 2 ! 1 ! 0
t"1
[Hint: Use the addition formula for tan"x ! y# on Reference
Page 2.]
37. Solve the initialvalue problem in Exercise 27 in Section 9.2
to find an expression
668
CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
so, using cos 2" ! sin 2" ! 1, we have
! " ! " ! "
dx
d"
2
dy
d"
!
2
!
!
2
dr
d"
!
dr
cos " sin " ! r 2 sin 2"
d"
cos 2" $ 2r
!
! "
! "
dr
d"
2
sin 2" ! 2r
dr
sin " cos " ! r 2 cos 2"
d"
2
dr
d"
!
SECTION 11.11
x ! 0.2
x ! 3.0
T2!x"
T4!x"
T6!x"
T8!x"
T10!x"
1.220000
1.221400
1.221403
1.221403
1.221403
8.500000
16.375000
19.412500
20.009152
20.079665
ex
1.221403
20.085537
APPLICATIONS OF TAYLOR POLYNOMIALS
769
The values in the table give a numerica
650
CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
we have
y
L!
!" # " #
dx
d"
2!
0
y
!
The result of Example 5 says that the length of
one arch of a cycloid is eight times the radius of
the generating circle (see Figure 5). This was first
proved i
484
CHAPTER 7
TECHNIQUES OF INTEGRATION
1
35. Prove the formula A ! 2 r 2% for the area of a sector of
a circle with radius r and central angle %. [Hint: Assume
0 $ % $ &"2 and place the center of the circle at the origin
so it has the equation x 2 " y 2
SECTION 11.2
2
Definition Given a series
SERIES
705
!n!1 a n ! a 1 " a 2 " a 3 " # # # , let sn denote its
nth partial sum:
n
sn !
"a
i
! a1 " a2 " # # # " an
i!1
If the sequence #sn $ is convergent and lim n l ! sn ! s exists as a real number, then
the s
678
CHAPTER 10
PARAMETRIC EQUATIONS AND POLAR COORDINATES
PF1, PF2 and the ellipse as shown in the figure. Prove that
# ! $. This explains how whispering galleries and lithotripsy
work. Sound coming from one focus is reflected and passes
through the other
752
CHAPTER 11
INFINITE SEQUENCES AND SERIES
33. Use the result of Example 7 to compute arctan 0.2 correct to
(b) Use part (a) to find a power series for
five decimal places.
1
f !x" !
!1 ! x"3
34. Show that the function
(c) Use part (b) to find a power s