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A48 |||| APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL

G THE LOGARITHM DEFINED AS AN INTEGRAL

y	1
y

	y= t
0		area=ln x
		area=ln x


	1	x
	1	x

FIGURE 1

area=_ln x

y=1t

0	x 1

FIGURE 2

y=1t

B C

0 1 2

FIGURE 3

Our treatment of exponential and logarithmic functions until now has relied on our intuition, which is based on numerical and visual evidence. (See Sections 1.5, 1.6, and 3.1.) Here we use the Fundamental Theorem of Calculus to give an alternative treatment that provides a surer footing for these functions.

Instead of starting with ax and deﬁning loga x as its inverse, this time we start by deﬁning ln x as an integral and then deﬁne the exponential function as its inverse. You should bear in mind that we do not use any of our previous deﬁnitions and results concerning exponential and logarithmic functions.

THE NATURAL LOGARITHM

We ﬁrst deﬁne ln x as an integral.

1 DEFINITION The natural logarithmic function is the function deﬁned by

ln x yx 1 dt x 0

1 t

The existence of this function depends on the fact that the integral of a continuous function always exists. If x 1, then ln x can be interpreted geometrically as the area under the hyperbola y 1 t from t 1 to t x. (See Figure 1.) For x 1, we have

				1	1
	ln 1 y1						dt 0
	ln 1 y1				t		dt 0
t	x	1				1		1
	x	1				1		1
For 0 x 1,	ln x y1		dt			yx			dt 0
For 0 x 1,	ln x y1	t	dt			yx		t	dt 0

and so ln x is the negative of the area shown in Figure 2.

V EXAMPLE 1

(a)By comparing areas, show that 12 ln 2 34 .

(b)Use the Midpoint Rule with n 10 to estimate the value of ln 2.

SOLUTION

t(a) We can interpret ln 2 as the area under the curve y 1 t from 1 to 2. From Figure 3 we see that this area is larger than the area of rectangle BCDE and smaller than the area of trapezoid ABCD. Thus we have

			21 1 ln 2 1 21 (1 21 )
				21 ln 2 43
(b) If we use the Midpoint Rule with f t 1 t, n 10, and t 0.1, we get
2	1
ln 2 y1		dt 0.1 f 1.05 f 1.15 f 1.95
ln 2 y1	t	dt 0.1 f 1.05 f 1.15 f 1.95
t			1		1	1
0.1			1		1	1	0.693	M
0.1		1.05				1.95	0.693	M
		1.05		1.15		1.95

APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL |||| A49

Notice that the integral that deﬁnes ln x is exactly the type of integral discussed in Part 1 of the Fundamental Theorem of Calculus (see Section 5.3). In fact, using that theorem, we have

x 1

and so

ln x

We now use this differentiation rule to prove the following properties of the logarithm function.

3	LAWS OF LOGARITHMS	If x and y are positive numbers and r is a rational
number, then
1.	ln xy ln x ln y	2. ln		x	ln x ln y	3. ln xr r ln x
			y

PROOF

1. Let f x ln ax , where a is a positive constant. Then, using Equation 2 and the Chain Rule, we have

f x 1 d ax 1 a 1 ax dx ax x

Therefore f x and ln x have the same derivative and so they must differ by a constant:

ln ax ln x C

Putting x 1 in this equation, we get ln a ln 1 C 0 C C. Thus

ln ax ln x ln a

If we now replace the constant a by any number y, we have

ln xy ln x ln y

2. Using Law 1 with x 1 y, we have

ln y ln

ln 1

and so

ln y

Using Law 1 again, we have

ln x ln

ln x

ln y

The proof of Law 3 is left as an exercise.

A50 |||| APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL

y=ln x

FIGURE 4

y
1
0	1	e	x
	y=ln x

FIGURE 5

In order to graph y ln x, we ﬁrst determine its limits:

(a)

lim ln x

(b) lim

ln x

x l

x l0

PROOF

(a) Using Law 3 with x 2 and r n (where n is any positive integer), we have

ln 2n n ln 2. Now ln 2 0, so this shows that ln 2n l as n l . But ln x is an

increasing function since its derivative 1 x 0. Therefore ln x l as x l .

(b) If we let t 1 x, then t l as x l 0 . Thus, using (a), we have

lim ln x lim ln

lim ln t

x l0

t l

If y ln x, x 0, then

and

d2 y

dx2

which shows that ln x is increasing and concave downward on 0, . Putting this information together with (4), we draw the graph of y ln x in Figure 4.

Since ln 1 0 and ln x is an increasing continuous function that takes on arbitrarily large values, the Intermediate Value Theorem shows that there is a number where ln x takes on the value 1. (See Figure 5.) This important number is denoted by e.

5 DEFINITION

e is the number such that ln e 1.

We will show (in Theorem 19) that this deﬁnition is consistent with our previous deﬁnition of e.

f 1 x y &? f y x

f 1 f x x f f 1 x x

THE NATURAL EXPONENTIAL FUNCTION

Since ln is an increasing function, it is one-to-one and therefore has an inverse function, which we denote by exp. Thus, according to the deﬁnition of an inverse function,

6	exp x y	&?	ln y x

and the cancellation equations are

7	exp ln x x	and	ln exp x x

In particular, we have
	exp 0 1	since	ln 1 0
	exp 1 e	since	ln e 1

We obtain the graph of y exp x by reﬂecting the graph of y ln x about the line y x.

y		y=exp x
		y=exp x
		y=x
1		y=ln x
0	1	x

FIGURE 6

APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL |||| A51

(See Figure 6.) The domain of exp is the range of ln, that is, , ; the range of exp is the domain of ln, that is, 0, .

If r is any rational number, then the third law of logarithms gives

ln er r ln e r

Therefore, by (6),

exp r er

Thus exp x ex whenever x is a rational number. This leads us to deﬁne ex, even for irrational values of x, by the equation

ex exp x

y=´

1
1

0	1	x
	1

FIGURE 7

The natural exponential function

In other words, for the reasons given, we deﬁne ex to be the inverse of the function ln x. In this notation (6) becomes

8	ex y &?	ln y x

and the cancellation equations (7) become

9	eln x x	x 0


10	ln ex x	for all x

The natural exponential function f x ex is one of the most frequently occurring functions in calculus and its applications, so it is important to be familiar with its graph (Figure 7) and its properties (which follow from the fact that it is the inverse of the natural logarithmic function).

PROPERTIES OF THE EXPONENTIAL FUNCTION The exponential function f x ex is an increasing continuous function with domain and range 0, . Thus ex 0 for all x. Also

lim ex 0	lim ex
x l	x l

So the x-axis is a horizontal asymptote of f x ex.

We now verify that f has the other properties expected of an exponential function.

11 LAWS OF EXPONENTS	If x and y are real numbers and r is rational, then
1. ex y exey	2. ex y	ex	3.	ex r erx
1. ex y exey	2. ex y	ey	3.	ex r erx
		ey

A52 \|\|\|\| APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL
PROOF OF LAW 1 Using the ﬁrst law of logarithms and Equation 10, we have
	ln e xe y ln e x ln e y x y ln e x y
Since ln is a one-to-one function, it follows that e xe y e x y.
Laws 2 and 3 are proved similarly (see Exercises 6 and 7). As we will soon see,
Law 3 actually holds when r is any real number.				M
We now prove the differentiation formula for e x.

12		d	e x e x
12			e x e x
		dx

PROOF The function y e x is differentiable because it is the inverse function of y ln x, which we know is differentiable with nonzero derivative. To ﬁnd its derivative, we use the inverse function method. Let y e x. Then ln y x and, differentiating this latter equation implicitly with respect to x, we get

1	dy	1
y
	dx
	dy	y e x	M

	dx

GENERAL EXPONENTIAL FUNCTIONS

If a 0 and r is any rational number, then by (9) and (11),

a r e ln a r e r ln a

Therefore, even for irrational numbers x, we deﬁne

13	a x e x ln a

Thus, for instance,

2s3 e s3 ln 2 e1.20 3.32

The function f x a x is called the exponential function with base a. Notice that a x is positive for all x because e x is positive for all x.

Deﬁnition 13 allows us to extend one of the laws of logarithms. We already know that ln a r r ln a when r is rational. But if we now let r be any real number we have, from Deﬁnition 13,

ln a r ln e r ln a r ln a

Thus

ln a r r ln a

for any real number r

APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL |||| A53

The general laws of exponents follow from Deﬁnition 13 together with the laws of exponents for e x.

15 LAWS OF EXPONENTS If x and y are real numbers and a, b 0, then

1. a x y a xa y 2. a x y a x a y 3. a x y a xy 4. ab x a xb x

PROOF

1. Using Deﬁnition 13 and the laws of exponents for e x, we have

a x y e x y ln a e x ln a y ln a

e x ln ae y ln a a xa y

3.Using Equation 14 we obtain

a x y e y ln ax e yx ln a e xy ln a a xy

The remaining proofs are left as exercises.

The differentiation formula for exponential functions is also a consequence of Deﬁni-

tion 13:

a x a x ln a

PROOF

a x

e x ln a e x ln a

x ln a a x ln a

lim a®=0, lim a®=`

x `

FIGURE 8 y=a®,

a>1

If a 1, then ln a 0,

so d dx a x

a x ln a 0, which shows that

y a x is

increasing (see Figure 8). If 0 a 1, then ln a 0 and so y a x is decreasing (see

Figure 9).

GENERAL LOGARITHMIC FUNCTIONS

If a 0 and a 1, then f x a x is a one-to-one function. Its inverse function is called

the logarithmic function with base a and is denoted by loga. Thus

loga x y

&? a y x

lim a®=`, lim a®=0

x `

In particular, we see that

FIGURE 9

y=a®, 0<a<1

loge x ln x

A54

|||| APPENDIX G THE LOGARITHM DEFINED AS AN INTEGRAL

The laws of logarithms are similar to those for the natural logarithm and can be deduced

from the laws of exponents (see Exercise 10).

To differentiate y loga x, we write the equation as a y x. From Equation 14 we have

y ln a ln x, so

loga x y

ln x

ln a

Since ln a is a constant, we can differentiate as follows:

loga x

ln x

ln a

x ln a

ln a

loga x

x ln a

THE NUMBER e EXPRESSED AS A LIMIT

In this section we deﬁned e as the number such that ln e 1. The next theorem shows that

this is the same as the number e deﬁned in Section 3.1 (see Equation 3.6.5).

e lim 1 x 1 x

x l0

PROOF Let f x ln x. Then f x 1 x, so f 1 1. But, by the deﬁnition of

derivative,

f 1 lim

f 1 h f 1

lim

f 1 x f 1

h l0

x l0

lim

ln 1 x ln 1

lim

ln 1 x lim ln 1 x 1 x

x l0

x l0 x

x l0

Because f 1 1, we have

lim ln 1 x 1 x 1

x l0

Then, by Theorem 2.5.8 and the continuity of the exponential function, we have

e e1 elimx l0 ln 1 x 1 x lim eln 1 x 1 x

lim 1 x 1 x

x l0

EXERCISES

1. (a) By comparing areas, show that

(b) Use part (a) to show that ln 2 0.66.

31 ln 1.5

By comparing areas, show that

(b) Use the Midpoint Rule with n 10 to estimate ln 1.5.

ln n 1

2. Refer to Example 1.

2 3

(a) Find the equation of the tangent line to the curve y 1 t

(a) By comparing areas, show that ln 2 1 ln 3.

that is parallel to the secant line AD.

(b) Deduce that 2 e 3.

5.Prove the third law of logarithms. [Hint: Start by showing that both sides of the equation have the same derivative.]

6.	Prove the second law of exponents for e x [see (11)].
7.	Prove the third law of exponents for e x [see (11)].
8.	Prove the second law of exponents [see (15)].

APPENDIX H COMPLEX NUMBERS |||| A55

9.Prove the fourth law of exponents [see (15)].

10.Deduce the following laws of logarithms from (15):

(a)loga xy loga x loga y

(b)loga x y loga x loga y

(c)loga x y y loga x

Im
		2+3i
_4+2i
i
0	1	Re
_i
_2-2i		3-2i

FIGURE 1

Complex numbers as points in the Argand plane

COMPLEX NUMBERS

A complex number can be represented by an expression of the form a bi, where a and b are real numbers and i is a symbol with the property that i 2 1. The complex number a bi can also be represented by the ordered pair a, b and plotted as a point in a plane (called the Argand plane) as in Figure 1. Thus the complex number i 0 1 i is identiﬁed with the point 0, 1 .

The real part of the complex number a bi is the real number a and the imaginary part is the real number b. Thus the real part of 4 3i is 4 and the imaginary part is 3. Two complex numbers a bi and c di are equal if a c and b d; that is, their real parts are equal and their imaginary parts are equal. In the Argand plane the horizontal axis is called the real axis and the vertical axis is called the imaginary axis.

The sum and difference of two complex numbers are deﬁned by adding or subtracting their real parts and their imaginary parts:

a bi c di a c b d i

For instance,

1 i 4 7i 1 4 1 7 i 5 6i

The product of complex numbers is deﬁned so that the usual commutative and distributive laws hold:

a bi c di a c di bi c di

ac adi bci bdi 2

Since i 2 1, this becomes

a bi c di ac bd ad bc i

EXAMPLE 1

1 3i 2 5i 1 2 5i 3i 2 5i

2 5i 6i 15 1 13 11i

Division of complex numbers is much like rationalizing the denominator of a rational expression. For the complex number z a bi, we deﬁne its complex conjugate to be z a bi. To ﬁnd the quotient of two complex numbers we multiply numerator and denominator by the complex conjugate of the denominator.

	1 3i
EXAMPLE 2 Express the number			in the form a bi.

2		5i

A56 |||| APPENDIX H COMPLEX NUMBERS

z=a+bi

–z=a-bi

FIGURE 2

bi		@	z=a+bi
bi		„„
		b
		„@+
		„
		„a
		œ	b
\|=
	\|z

0			a	Re

FIGURE 3

SOLUTION We multiply numerator and denominator by the complex conjugate of 2 5i, namely 2 5i, and we take advantage of the result of Example 1:

1 3i

2 5i

13 11i

2 5i

22 52

The geometric interpretation of the complex conjugate is shown in Figure 2: z is the reﬂection of z in the real axis. We list some of the properties of the complex conjugate in the following box. The proofs follow from the deﬁnition and are requested in Exercise 18.

PROPERTIES OF CONJUGATES

z w

z n

The modulus, or absolute value, z of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then

z sa 2 b 2

Notice that

zz a bi a bi a 2 abi abi b 2i 2 a 2 b 2

and so		z 2
	zz

This explains why the division procedure in Example 2 works in general:


z	z	w	z	w
w			w 2
	ww

Since i 2 1, we can think of i as a square root of 1. But notice that we also havei 2 i 2 1 and so i is also a square root of 1. We say that i is the principal square root of 1 and write s 1 i. In general, if c is any positive number, we write

s c sc i

With this convention, the usual derivation and formula for the roots of the quadratic equation ax 2 bx c 0 are valid even when b 2 4ac 0:

x b sb 2 4ac

EXAMPLE 3 Find the roots of the equation x 2 x 1 0.

SOLUTION Using the quadratic formula, we have


x	1 s12	4 1	1 s 3	1 s3 i
					M
	2		2	2	M
	2		2	2

Im		a+bi
Im

	r	b
		b
	¨

0	a	Re

FIGURE 4

1+i

œ„2

0	_		π	Re
		6

œ„3-i

FIGURE 5

APPENDIX H COMPLEX NUMBERS |||| A57

We observe that the solutions of the equation in Example 3 are complex conjugates of each other. In general, the solutions of any quadratic equation ax 2 bx c 0 with real coefﬁcients a, b, and c are always complex conjugates. (If z is real, z z, so z is its own conjugate.)

We have seen that if we allow complex numbers as solutions, then every quadratic equation has a solution. More generally, it is true that every polynomial equation

an x n an 1 x n 1 a1 x a0 0

of degree at least one has a solution among the complex numbers. This fact is known as the Fundamental Theorem of Algebra and was proved by Gauss.

POLAR FORM

We know that any complex number z a bi can be considered as a point a, b and that any such point can be represented by polar coordinates r, with r 0. In fact,

		a r cos	b r sin
as in Figure 4. Therefore we have
		z a bi r cos r sin i
Thus we can write any complex number z in the form

		z r cos i sin
					b

where	r z sa 2 b 2		and	tan

					a

The angle is called the argument of z and we write arg z . Note that arg z is not unique; any two arguments of z differ by an integer multiple of 2 .

EXAMPLE 4 Write the following numbers in polar form.

(a) z 1 i

(b)

w s

SOLUTION

(a) We have r z s

and tan 1, so we can take 4.

12 12

Therefore the polar form is

cos

i sin

(b) Here we have r w s

2 and tan

1 s

. Since w lies in the

3 1

fourth quadrant, we take 6 and

w 2

cos

i sin

The numbers z and w are shown in Figure 5.

A58 |||| APPENDIX H COMPLEX NUMBERS

z™ z¡

¨™

¨¡

¨¡+¨™

z¡z™

FIGURE 6

_¨

FIGURE 7

Im	z=1+i
	z=1+i
œ„2	zw
œ„2	2œ„
	π
0	12
0	Re
	2
	w=œ3„-i

FIGURE 8

The polar form of complex numbers gives insight into multiplication and division. Let

		z1 r1 cos 1 i sin 1	z2 r2 cos 2 i sin 2
be two complex numbers written in polar form. Then
	z1z2 r1r2 cos 1 i sin 1 cos 2 i sin 2
	r1r2 cos 1 cos 2 sin 1 sin 2		i sin 1 cos 2 cos	1 sin 2
Therefore, using the addition formulas for cosine and sine, we have

1		z1z2 r1r2 cos 1 2	i sin 1 2

This formula says that to multiply two complex numbers we multiply the moduli and add the arguments. (See Figure 6.)

A similar argument using the subtraction formulas for sine and cosine shows that to divide two complex numbers we divide the moduli and subtract the arguments.

z1	r1	cos 1 2 i sin 1 2 z2 0
z2	r2	cos 1 2 i sin 1 2 z2 0
z2	r2

In particular, taking z1 1 and z2 z (and therefore 1 0 and 2 ), we have the following, which is illustrated in Figure 7.

If z r cos i sin , then	1	1	cos i sin .
	z
		r

EXAMPLE 5 Find the product of the complex numbers 1 i and s3 i in polar form.

SOLUTION From Example 4 we have
					s				4					4

			1 i			2	cos			i sin
		s							6							6
and
and			3 i 2 cos									i sin
So, by Equation 1,							4										4
	s				s		4						6				4	6

1 i (		3 i) 2				2		cos							i sin

				2		2		cos			i sin

					s				12						12
					s				12						12
This is illustrated in Figure 8.																			M

APPENDIX H COMPLEX NUMBERS |||| A59

Repeated use of Formula 1 shows how to compute powers of a complex number. If

then

and

z r cos i sin

z 2	r 2 cos 2 i sin 2
z 3	zz 2 r 3 cos 3 i sin 3

In general, we obtain the following result, which is named after the French mathematician Abraham De Moivre (1667–1754).

2 DE MOIVRE’S THEOREM then

If z r cos

i sin

and n is a positive integer,

z n r cos

i sin

r n cos n

i sin n

This says that to take the nth power of a complex number we take the nth power of the modulus and multiply the argument by n.

EXAMPLE 6 Find (12 12 i)10.

SOLUTION Since 12 12 i 12 1 i , it follows from

form
	1		1	i	s
						2	cos
2
		2		2

Example 4(a) that 12 12 i has the polar

4	4
	i sin

So by De Moivre’s Theorem,


2	2											4				4
2	2						2					4				4
1		1	i	10		s2			10		cos		10	i sin		10
			i								cos			i sin
					25			cos		5		i sin			5		1	i
					210			cos			2	i sin			2		32	i

De Moivre’s Theorem can also be used to ﬁnd the nth roots of complex numbers. An nth root of the complex number z is a complex number w such that

wn z

Writing these two numbers in trigonometric form as

w s cos i sin and z r cos

and using De Moivre’s Theorem, we get

s n cos n i sin n r cos i sin

The equality of these two complex numbers shows that

i sin

and

cos n

s n r

cos

and

sr 1 n sin n

sin

A60 |||| APPENDIX H COMPLEX NUMBERS

From the fact that sine and cosine have period 2

it follows that

n 2k	or	2k
		n

Thus

w r 1 n

cos

i sin

Since this expression gives a different value of w for k 0, 1, 2, . . . , n 1, we have the

following.

ROOTS OF A COMPLEX NUMBER Let z r cos

i sin and let n be a posi-

tive integer. Then z has the n distinct nth roots

wk r 1 n

cos

i sin

where k 0, 1, 2, . . . , n 1.

Notice that each of the nth roots of z has modulus wk r 1 n. Thus all the nth roots

of z lie on the circle of radius r 1 n in the complex plane. Also, since the argument of each

successive nth root exceeds the argument of the previous root by 2 n, we see that the

nth roots of z are equally spaced on this circle.

EXAMPLE 7 Find the six sixth roots of z 8 and graph these roots in the complex

plane.

SOLUTION

In trigonometric form, z 8 cos

i sin

. Applying Equation 3 with n 6,

we get

wk 81 6

cos

i sin

We get the six sixth roots of 8 by taking k 0, 1, 2, 3, 4, 5 in this formula:

w0 81 6

cos

i sin

w1 81 6

cos

i sin

2 i

w¡

w2 81 6

cos

i sin

œ„2i

w™

w¸

w3 81 6

cos

i sin

_œ„2

œ„2 Re

w4 81 6

cos

i sin

∞

_œ„2i

w¢

w5 81 6

cos

i sin

FIGURE 9

The six sixth roots of z=_8

All these points lie on the circle of radius s2 as shown in Figure 9.

N We could write the result of Example 8(a) as

e i 1 0

This equation relates the ﬁve most famous numbers in all of mathematics: 0, 1, e, i, and .

APPENDIX H COMPLEX NUMBERS |||| A61

COMPLEX EXPONENTIALS

We also need to give a meaning to the expression e z when z x iy is a complex number. The theory of inﬁnite series as developed in Chapter 11 can be extended to the case where the terms are complex numbers. Using the Taylor series for e x (11.10.11) as our guide, we deﬁne

		z	n		z	2		z	3
4	e z			1 z
		n!
	n 0			2!			3!

and it turns out that this complex exponential function has the same properties as the real exponential function. In particular, it is true that

5	e z1 z2 e z1e z2

If we put z iy, where y is a real number, in Equation 4, and use the facts that

i 2 1,

i 3 i 2i i, i 4 1,

i 5 i, . . .

we get e iy 1 iy

iy 2

iy 3

iy 4

iy 5

1 iy

y 2

y 3

y 4

y 5

y 2

y 4

y 6

y 3

y 5

cos y i sin y

Here we have used the Taylor series for cos y and sin y (Equations 11.10.16 and 11.10.15). The result is a famous formula called Euler’s formula:

e iy cos y i sin y

Combining Euler’s formula with Equation 5, we get

e x iy e xe iy e x cos y i sin y

EXAMPLE 8 Evaluate:

(a)

e i

(b) e 1 i 2

SOLUTION

(a) From Euler’s equation (6) we have

e i cos

i sin 1 i 0 1

(b) Using Equation 7 we get

e 1 i 2

e 1

cos

i sin

0 i 1

Finally, we note that Euler’s equation provides us with an easier method of proving De Moivre’s Theorem:

r cos i sin n re i n r ne in r n cos n i sin n

A62 |||| APPENDIX H COMPLEX NUMBERS

H EXERCISES

1–14 Evaluate the expression and write your answer in the form a bi.

1.	5 6i 3 2i		2.	(4 21 i) (9 25 i)
3.	2 5i 4 i		4.	1 2i 8 3i

5.	12 7i		6.	2i (21 i)
7.	1	4i	8.	3 2i

	3	2i		1 4i

9.		1	10.	3
	1	i		4 3i

11.	i 3		12.	i 100
13.	s		14.	s		s
		25			3		12

15–17 Find the complex conjugate and the modulus of the number.

15. 12 5i 16. 1 2 s2 i

17.4i

18.Prove the following properties of complex numbers.

(a)

z w

(b)

(c)z n z n, where n is a positive integer [Hint: Write z a bi, w c di.]

19–24 Find all solutions of the equation.

19.	4x 2 9 0	20.	x 4 1
21.	x 2 2x 5 0	22.	2x 2 2x 1 0
23.	z2 z 2 0	24.	z2 21 z 41 0

25–28 Write the number in polar form with argument between 0 and 2 .


25.	3 3i	26.	1 s3 i
27.	3 4i	28.	8i

29–32 Find polar forms for zw, z w, and 1 z by ﬁrst putting z and w into polar form.


29.	z s3 i, w 1 s3 i
30.	z 4 s	3		4i,	w 8i
31.	z 2 s			2i,	w 1 i
			3
32.	z 4(s			i ),	w 3 3i
			3

33–36 Find the indicated power using De Moivre’s Theorem.

33.	1 i 20			34.	(1	s	3	i )5
35.	(2 s		2i )5	36.	1	i 8
35.	(2 s	3	2i )5	36.	1	i 8

37– 40 Find the indicated roots. Sketch the roots in the complex plane.

37.	The eighth roots of 1		38.	The ﬁfth roots of 32
39.	The cube roots of i		40.	The cube roots of 1 i

41– 46 Write the number in the form a bi.
41.	e i		42.
		2		e 2 i
43.	e i		44.	e i
		3
45.	e 2 i		46.
				e i
47.	Use De Moivre’s Theorem with n 3 to express cos 3 and
	sin 3 in terms of cos and sin .

48.Use Euler’s formula to prove the following formulas for cos x and sin x :

cos x	eix e ix	sin x	eix e ix
	2		2i

49.If u x f x it x is a complex-valued function of a real variable x and the real and imaginary parts f x and t x are

differentiable functions of x, then the derivative of u is deﬁned to be u x f x it x . Use this together with Equation 7

to prove that if F x e r x, then F x re r x when r a bi is a complex number.

50.(a) If u is a complex-valued function of a real variable, its indeﬁnite integral xu x dx is an antiderivative of u. Evaluate

ye 1 i x dx

(b)By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals

ye x cos x dx

and

ye x sin x dx

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A63

H ANSWERS TO ODD-NUMBERED EXERCISES

CHAPTER 1

EXERCISES 1.1 N PAGE 20
1.	(a) 2 (b)	2.8	(c)	3, 1 (d) 2.5, 0.3
(e)	3, 3 , 2, 3		(f )	1, 3
3.	85, 115	5.	No
7.	Yes, 3, 2 , 3, 2 1, 3
9.	Diet, exercise, or illness

11.T

13.T

midnight

noon

15.amount

price

17.	Height
	of grass

Wed. Wed. Wed. Wed. Wed. t

19. (a) N

600

500

400

300

200

100


0	1990			1992			1994			1996			1998			2000			t

(b)In millions: 92; 485

21. 12, 16, 3a2 a 2, 3a2 a 2, 3a2 5a 4, 6a2 2a 4, 12a2 2a 2, 3a4 a2 2,

9a4 6a3 13a2 4a 4, 3a2 6ah 3h2 a h 2

23.	3 h	25. 1 ax
27.	{x x 31 } ( , 31) (31 , )
29.	0,	31. , 0 5,

33. ,	35. ,
y	y
5

		0	x		0		6	t
					_9
37.	5,			39.	, 0 0,
y					y
y
					4
					2
					0		x
0			5	x
41. ,				43. ,
	y				y
	y
	(0,2)	(0,1)			1
		(0,1)			1
	_2 0	1	x		1	0	x
					1	0	x

45.	f x 25 x 112 , 1 x 5			47. f x 1 s	x
49.	f x x 3 if 0 x 3
	2 x 6 if 3 x 5
51.	A L 10L L2, 0 L 10
53.	A x s	3	x2 4, x 0 55.	S x x2 8 x , x 0
57.	V x 4x3 64x2 240x, 0 x 6
59.	(a) R (%)			(b) $400, $1900
	15
	15
	10
	10

	0	10,000	20,000	I (in dollars)
(c)	T (in dollars)
	2500
	1000
	0	10,000	20,000 30,000	I (in dollars)

61.	f is odd, t is even
63.	(a) 5, 3			(b) 5, 3
65.	Odd	67. Neither			69.	Even
EXERCISES 1.2 N PAGE 34
1.	(a) Root		(b)	Algebraic		(c)	Polynomial (degree 9)
(d)	Rational		(e)	Trigonometric			(f) Logarithmic
3.	(a) h	(b)	f	(c) t

A64

||||

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

(a)

y 2x b,

y b=3 b=0

(c)

y 0.00009979x 13.951 [See graph in (b).]

(f) No

where b is the y-intercept.

b=_1

(d)

About 11.5 per 100 population

(e)

About 6%

23.

(a)

20 (ft)

y=2x+b

1896

2000 (year

(b)

y mx 1 2m,

m=1

Linear model is appropriate

where m is the slope.

m=_1

(b)

y 0.08912x 158.24

(c)

20 ft

(d)

See graph at right.

20 (ft)

(c)

y 2x 3

(2,1)

m=0

y-1=m(x-2)

1896

2000 (year)

Their graphs have slope 1.

25.

y 0.0012937x3 7.06142x2 12,823x 7,743,770;

c=_1

1914 million

c=_2

x c=2

EXERCISES 1.3 N PAGE 43

(a)

y f x 3

(b)

y f x 3

(c)

y f x 3

c=1

(d)

y f x 3

(e) y f x

(f) y f x

c=0

(g)

y 3f x

(h)

y 31 f x

9. f x 3x x 1 x 2

(a)

(b)

(d)

(e)

(a)

(b)

11.

(a)

8.34, change in mg for every 1 year change

(b) 8.34 mg

13.

(a)

(b) 5 , change in F for every

(100, 212)

1 C change; 32, Fahrenheit

F=59 C+32

temperature corresponding

to 0 C

(c)

(d)

(_40,_40)

15.

(a)

T 61 N 3076

(b)

61 , change in F for every chirp per

y s x2 5x 4 1

minute change

11.

17.

(a)

P 0.434d 15

(b)

196 ft

19.

(a)

Cosine

(b)

Linear

y=_x#

21.	(a) 15	Linear model is
		appropriate	0	x	_1	0	x
					y=(x+1)@
	0	61,000	13.
(b)	y 0.000105x 14.521	15	13.	y
		(b)		3
		(b)
		(c)			π
				0	x
					y=1+2cos x
		0	61,000

15.	y	17.	y
	y=sin(x/2)		y=œ„„„„x+3

12π

0	x
	_3		0	x

19.		y		21.		y	2	2
19.		y		21.		y

					x=_1			y=	x+1

_4			0	x						x
_4			0	x		0				x
			_8
1
	y=_(≈+8x)
2

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

|||| A65

45.

t t cos t, f t s

47.

h x x2, t x 3x, f x 1 x

49.

h x s

, t x sec x, f x x4

51.

(a)

4 (b) 3

(c)

(d)

Does not exist; f 6 6 is not

in the domain of t.

(e)

(f)

53.

(a)

r t 60t

(b)

A r t 3600 t2; the area of the

circle as a function of time

55.

(a)

s s

(b)

d 30t

d2 36

(c)

s s

900t2 36;

the distance between the lighthouse and the

ship as a function of the time elapsed since noon

57.

(a)

(b)

120

23.

y=|sin x|

365

25.

L t 12 2 sin

t 80

27.

(a) The portion of the graph of y f x to the right of the

y-axis is reﬂected about the y-axis.

(b)

(c)

y=sin |x|

y=œ„„|x|

29. f t x x3 5x2 1, ,

f t x x3 x2 1, ,

ft x 3x5 6x4 x3 2x2, ,

f t x x3 2x2 3x2 1 , {x x 1 s3 }

31. (a) f t x 4x2 4x, ,

(b)t f x 2x2 1, ,

(c)f f x x4 2x2, ,

(d)t t x 4x 3, ,

33.	(a) f t x 1 3 cos x, ,
(b)	t f x cos 1 3x , ,
(c)	f f x 9x 2, ,
(d)	t t x cos cos x , ,
35.	(a) f t x 2x2 6x 5 x 2 x 1 ,
x x 2, 1
(b)	t f x x2 x 1 x 1 2, {x x 1, 0
(c)	f f x x4 3x2 1 x x2 1 , {x x 0
(d)	t t x 2x 3 3x 5 , {x x 2, 35 }
37.	f t h x 2 x 1
39.	f t h x s
				x6 4x3 1
41.	t x x2 1, f x x10
43.	t x s		, f x x 1 x
		x
	3

(c)	V
	240
	0	5	t

V t 120H t

V t 240H t 5

59.	Yes; m1m2			(b) t x x2 x 1
61.	(a)	f(x x2 6		(b) t x x2 x 1
63.	(a)	Even; even	(b)	Odd; even
65.	Yes

EXERCISES 1.4 N PAGE 51
1.	(c)	3.	150
		_10			30
			_50
5.				7.
		4			3500
				_20	20
4			4
		1			_3500
9.				11.
		1.1		1.5
				0	100
_0.01		0	0.01	_1.5
		0		_1.5

A66 ||||

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

13.

y=20® y=5® y=´

All approach 0 as x l ,

y=2®

all pass through 0, 1 , and

all are increasing. The larger

the base, the faster the rate of

_2π

2π

_ π

increase.

_11

The functions with base

y=”

’

y=”

’

y=10® y=3®

15.

greater than 1 are increasing

and those with base less than

1 are decreasing. The latter

are reﬂections of the former

about the y-axis.

17.

19.

9.05

21.

0, 0.88

23. t

25.

0.85 x 0.85

y=_2–®

27.

(a)

(b)

y=4®-3

œx„

$œx„

Œ„x

y=_3

x %œ„x

^œ„x

11.

y=1

”0, 21 ’

(c)

(d) Graphs of even roots are

Œx„

œx„

similar to sx, graphs of odd

y=1-1 e–®

$œ„x

roots are similar to sx. As n

%œ„x

increases, the graph of

13.

(a)

y ex

(b)

y ex 2

(c)

y ex

y sx becomes steeper near

0 and ﬂatter for x 1.

(d)

y e

(e)

y e

15.

(a) ,

(b) , 0 0,

29.

1 _1.5

-1 -2 -3

17.

f x 3 2x

23.

At x 35.8

25.

(a)

3200

(b)

100 2t 3

(c)

10,159

_2.5

2.5

(d)

60,000

t 26.9 h

If c 1.5, the graph has three humps: two minimum points and a maximum point. These humps get ﬂatter as c increases until at c 1.5 two of the humps disappear and there is only one minimum point. This single hump then moves to the right and approaches the origin as c increases.

31. The hump gets larger and moves to the right.

33. If c 0, the loop is to the right of the origin; if c 0, the loop is to the left. The closer c is to 0, the larger the loop.

EXERCISES 1.5 N PAGE 58
1.	(a) f x ax, a 0	(b)	(c) 0,
(d)	See Figures 4(c), 4(b), and 4(a), respectively.

		0					40
		0
27.	y abt, where a 3.154832569 10 12 and
b 1.017764706; 5498 million; 7417 million
EXERCISES 1.6 N PAGE 70
1.	(a)		See Deﬁnition 1.
(b)	It must pass the Horizontal Line Test.
3.	No			5. Yes		7.	No	9.	No	11.		Yes
13.	No			15. 2		17.	0
19.	F 59 C 32; the Fahrenheit temperature as a function of the
Celsius temperature; 273.15,
21.	f	1		1	2	10	x 0		23. f	1		3
		1		1	2	10				1		3
			x	3 x		3 ,					x sln x

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A67

25. y ex 3

27. f 1 x s4 x 1

f–!

69.	π
	2 y=sin–!x

29.		y=sin x
		y=sin x	The second graph is
y			The second graph is
f–!	π	π	the reﬂection of the
	π	π	ﬁrst graph about the
	_ 2	2	ﬁrst graph about the
	f		line y x.
0	x
		_ π2

31.	(a) It’s deﬁned as the inverse of the exponential function with
base a, that is, loga x y &? ay x.
(b)	0,	(c)		(d) See Figure 11.
33.	(a) 3	(b)	3	35. (a) 3	(b) 2	37. ln 1215

39.	(1 x2)sx
39.	ln
	sin x
41.	3		y=log1.5 x
	3
			y=ln x
			y=log10 x
	0		4
			y=log50 x
	5
43.	About 1,084,588 mi
45.	(a)	y	(b)
	y=log10 (x+5)
	_5 _4	0	x

All graphs approachas x l 0 , all pass through 1, 0 , and all are increasing. The larger the base, the slower the rate of increase.

y=-ln x

0 1


47.	(a)	se (b) ln 5
49.	(a)	5 log2 3 or 5 ln 3 ln 2 (b) 21 (1 s					)
				1 4e
51.	(a)	x ln 10	(b) x 1 e
53.	(a)	( , 21 ln 3]	(b) f 1 x 21 ln 3 x2 , [0, s			)
					3

55.5

								The graph passes the
								Horizontal Line Test.
		_2				4
		_2				4

		_1
f	1	(x) (s		6)(s					s				s		),
f		(x) (s	4	6)(s	D 27x		2	20	s	D 27x	2	20	s	2	),
		3		3			2	3			2	3

where D 3 s3 s27x4 40x2 16; two of the expressions are complex.

57.	(a)	f 1 n 3 ln 2 ln n 100 ; the time elapsed when there
are n bacteria			(b)	After about 26.9 hours
59.	(a)	3	(b)		61.	(a)	4		(b)	4
63.	(a)	10	(b) 3		67.	x
63.	(a)	10	(b) 3		67.	x		1 x2
						s

71.	(a) 32, 0		(b)	2, 2
73.	(a) t 1 x f 1 x c				(b)	h 1 x 1 c f 1 x
CHAPTER 1 REVIEW N PAGE 73
True-False Quiz
1.	False	3. False		5.	True	7.	False	9. True
11.	False	13.	False
Exercises
1.	(a) 2.7	(b)	2.3, 5.6		(c) 6, 6		(d)	4, 4
(e)	4, 4	(f) No; it fails the Horizontal Line Test.
(g)	Odd; its graph is symmetric about the origin.
3.	2a h 2		5. ( , 31 ) (31 , ), , 0 0,
7.	6, ,
9.	(a) Shift the graph 8 units upward.

(b)Shift the graph 8 units to the left.

(c)Stretch the graph vertically by a factor of 2, then shift it 1 unit upward.

(d)Shift the graph 2 units to the right and 2 units downward.

(e)Reﬂect the graph about the x-axis.

(f)Reﬂect the graph about the line y x (assuming f is one-to-one).

11.	y			13.		y
	y
		y=_sin 2x

	0	π	x	y=	1	(1+´)
				y=	2	(1+´)

1 y=12

0 x

15.y

x=_2 y=x+21

		1
		2

		0	x
17.	(a)	Neither	(b) Odd (c) Even (d) Neither
19.	(a)	f t x ln x2 9 , , 3 3,

(b)t f x ln x 2 9, 0,

(c)f f x ln ln x, 1,

(d)t t x x2 9 2 9, ,

21. y 0.2493x 423.4818; about 77.6 years

A68 \|\|\|\|			APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
23.	1	25.	(a) 9 (b) 2 (c) 1 s		(d) 53
				3
27.	(a)	1000			4.4 years

	0				10
	0
			9P
(b)	t ln		1000 P		; the time required for the population
(b)	t ln				; the time required for the population
to reach a given number P.
(c)	ln 81 4.4 years
PRINCIPLES OF PROBLEM SOLVING N						PAGE 81
1.	a 4 s			h, where a is the length of the altitude and
1.	a 4 s	h2 16		h, where a is the length of the altitude and
h is the length of the hypotenuse
3.	37 , 9
5.		y			7.		y
5.		y			7.		y

							0	x

1 x

9.y

01 x

11.5 13. x [ 1, 1 s3 ) (1 s3, 3]

15. 40 mi h

19. fn x x2 n 1

CHAPTER 2

(b) lim x l4 f x means that the values of f x can be made arbitrarily large negative by taking x sufﬁciently close to 4 through values larger than 4.

5.	(a)	2		(b) 3				(c)				Does not exist					(d) 4
(e)	Does not exist
7.	(a)	1			(b)	2						(c)		Does not exist					(d) 2	(e) 0
(f)	Does not exist									(g)			1		(h)	3
9.	(a)				(b)					(c)				(d)				(e)
(f)	x 7, x 3, x 0, x 6
11.	(a)	1		(b)		0			(c) Does not exist
13.				y										15.				y
13.				y										15.				y




											x
											x							0	1	x
17.	32	19.			21	21.					41			23.	53	25.		0	1	x





27.			29.								31.					33.	;
35.	(a) 2.71828						(b)									6

_4		4
_4		4

	_2

37. (a) 0.998000, 0.638259, 0.358484, 0.158680, 0.038851, 0.008928, 0.001465; 0

(b) 0.000572, 0.000614, 0.000907, 0.000978, 0.000993,0.001000; 0.001

39. No matter how many times we zoom in toward the origin, the graph appears to consist of almost-vertical lines. This indicates more and more frequent oscillations as x l 0.

41. x 0.90,	2.24; x sin 1 4 , sin 1 4
EXERCISES 2.3 N	PAGE 106

EXERCISES 2.1 N PAGE 87
1.	(a) 44.4, 38.8, 27.8, 22.2, 16.6
(b)	33.3		(c)	33 31
3.	(a)	(i)	0.333333		(ii)	0.263158		(iii) 0.251256
(iv) 0.250125				(v)	0.2	(vi)	0.238095				(vii) 0.248756
(viii)		0.249875		(b) 41		(c)	y 41 x 41
5.	(a)	(i)	32 ft s		(ii) 25.6 ft s			(iii)		24.8 ft s
(iv) 24.16 ft s				(b) 24 ft s
7.	(a)	(i)	4.65 m s		(ii)	5.6 m s		(iii)		7.55 m s
(iv) 7 m s			(b)	6.3 m s
9.	(a)	0, 1.7321, 1.0847, 2.7433, 4.3301, 2.8173, 0,
2.1651, 2.6061, 5, 3.4202; no								(c)	31.4
EXERCISES 2.2 N PAGE 96
1.	Yes
3.	(a)	lim x l 3 f x means that the values of f x can be

made arbitrarily large (as large as we please) by taking x sufﬁciently close to 3 (but not equal to 3).

1.	(a) 6	(b) 8	(c) 2 (d) 6
(e)	Does not exist		(f) 0

3.	59	5.	390	7.	81	56	9.	0	11.		5	121
13.	Does not exist			15.		56		17.	8		19.	121	21.	6
23.	61	25.	161	27.		1281		29.	21		31.	(a), (b)		32
35.	7	39.	6	41.	4		43.		Does not exist
45.	(a)		y				(b)		(i) 1
			1						(ii)	1
			1						(iii)	Does not exist
									(iii)	Does not exist
			0			x			(iv)	1
47.	(a)	(i) 2	(ii)	2		(b)	No		(c)				y
													2
													0
														1	x
														1

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A69

49.	(a)	(i) 2	(ii)		Does not exist				(iii)	3		29.	x 0						3
(b)	(i)	n 1	(ii)	n	(c) a is not an integer.
55.	8	61. 15; 1
EXERCISES 2.4			N PAGE 117												_4							4
1. 74 (or any smaller positive number)															_4							4
1. 74 (or any smaller positive number)
3.	1.44 (or any smaller positive number)																		_1
5.	0.0906 (or any smaller positive number)											31.	37	33.	1
7. 0.11, 0.012 (or smaller positive numbers)												37.	0, left							39.	0, right; 1, left
9.	(a)	0.031	(b)	0.010								37.	0, left							39.	0, right; 1, left
9.	(a)	0.031	(b)	0.010									y										y
11.	(a) s1000 cm				(b) Within approximately 0.0445 cm								y										y
11.	(a) s1000 cm				(b) Within approximately 0.0445 cm									(0,2)										(1,e)
(c)	Radius; area; s1000					; 1000; 5; 0.0445								(0,2)								(0,2)
																						(0,2)
13.	(a)	0.025	(b)	0.0025									(0,1)									(0,1)		(1,1)
35. (a)		0.093	(b)		B2 3 12 6B1 3 1, where								(0,1)									(0,1)
B 216 108 12 s336							324 81			2			0	(2,0)			x						0		x
																							0		x
41.	Within 0.1
EXERCISES 2.5			N PAGE 128									41.	32	43.	(a)	t x x3 x2 x 1							(b)	t x x2 x
EXERCISES 2.5			N PAGE 128									51.	(b) 0.86, 0.87					53.	(b)	70.347
1.	lim x l4 f x f 4											59.	None		61.	Yes
3.	(a)	4 (removable), 2 ( jump), 2 ( jump), 4 (inﬁnite)
(b)	4, neither; 2, left; 2, right; 4, right											EXERCISES 2.6 N				PAGE 140
5.		y										1.	(a) As x becomes large, f x approaches 5.
												1.	(a) As x becomes large, f x approaches 5.
												(b)	As x becomes large negative, f x approaches 3.
		1										3.	(a)	(b)			(c)			(d)	1	(e)	2
		1										(f)	x 1, x 2, y 1, y 2
		0	2			x						(f)	x 1, x 2, y 1, y 2
		0	2			x						5.								7.
												5.								7.
														y								y	x=2
7.	(a)					(b)	Discontinuous at t 1, 2, 3, 4																x=2
7.	(a)					(b)	Discontinuous at t 1, 2, 3, 4
		Cost												1
		Cost
	(in dollars)													0
																1		x				0			x
		1										9.		y
		1
		0 1			Time
				(in hours)
9.	6																y=3
9.	6													0				x
15.	f 2 is not deﬁned.					17.	lim f x does not exist.							0				x
15.	f 2 is not deﬁned.					17.	lim f x does not exist.
							x l0
	y	x=2								y					x=4
		x=2													x=4
										y=≈		11.	0	13.	23		15.	0	17. 21			19.	12	21.	2
	0			x					y=´	1		23.	3	25.	1		27.	1	b		29.		31.
	0			x											1			1
															6			2 a
										0	x	33.	21	35. 0			37. (a), (b) 21					39.	y 2; x 2
												41.	y 2; x 2, x 1						43. x 5				45.	y 3
															2 x
19.	lim f x f 0					21.	x	x 3, 2				47.	f x x2 x 3
19.	lim f x f 0					21.	x	x 3, 2				49.	,							51.	,
	x l0											49.	,							51.	,
		y											y								y
		y																			y
_π		0	1		x								0		1		x
_π		0	1		x
23.	[21 , )		25. ,				27.	, 1 1,													3
23.	[21 , )		25. ,				27.	, 1 1,													0	1	x

A70 \|\|\|\|			APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
53.	(a)	0	(b)	An inﬁnite number of times
						1
				-25			25
						_0.5
55.	(a)	0	(b)		57. 5
59.	(a)	v*	(b)	1.2			0.47 s
				0			1
61.	N 15		63.		N 6, N 22		65. (a) x 100
EXERCISES 2.7 N				PAGE 150
1.	(a)	f x f 3			(b) lim	f x f 3
1.	(a)	x	3		(b) lim	x 3
		x	3		x l3	x 3
3.	(a) 2		(b) y 2x 1			(c)	6
						_1	0	5
							0
5. y x 5				7. y 21 x 21
9.	(a) 8a 6a2			(b) y 2x 3, y 8x 19
(c)			10
	_2					4
			_3
11.	(a)	Right: 0 t 1 and 4 t 6; left: 2 t 3;
standing still: 1 t 2 and 3 t 4
(b)	v	(m/s)
	1
	0	1			t
		1			(seconds)
					(seconds)
13.	24 ft s			15.	2 a3 m s ; 2 m s; 41 m s; 272			m s
17.	t 0 , 0, t 4 , t 2 , t 2

19.		y				21.	7; y 7x 12
		1
		0
		1			x
23.	(a)	53 ; y 53 x 165			(b)		4
						_1	6
							_2
	2 8a				5		1
25.	2 8a		27.	a 3 2		29. 2 a 2 3 2
31.	f x x10, a 1 or f x 1 x 10 , a 0
33.	f x 2x, a 5
35.	f x cos x, a			or f x cos x , a 0
37.	1 m s; 1 m s
39.		Temperature				Greater (in magnitude)
	72	(in °F)
	72
	38
	0	1	2	Time
			(in hours)


41.	(a)	(i) 11 percent year		(ii) 13 percent year
(iii)	16 percent year
(b)	14.5 percent year		(c)	15 percent year
43.	(a) (i) $20.25 unit		(ii)	$20.05 unit	(b) $20 unit
45.	(a)	The rate at which the cost is changing per ounce of gold

produced; dollars per ounce

(b)When the 800th ounce of gold is produced, the cost of production is $17 oz.

(c)Decrease in the short term; increase in the long term

47. The rate at which the temperature is changing at 10:00 AM; 4 F h

49. (a) The rate at which the oxygen solubility changes with respect to the water temperature; mg L C

(b) S 16 0.25; as the temperature increases past 16 C, the oxygen solubility is decreasing at a rate of 0.25 mg L C. 51. Does not exist

EXERCISES 2.8 N PAGE 162

1. (a) 1.5	y
(b) 1		fª
		fª
(c) 0
(d) 4	0	x
(d) 4

(e)0

(f)1

(g) 1.5

3.	(a) II		(b) IV	(c) I	(d) III
5.		y	fª		7.	y
5.		y			7.	y
						fª

	0			x

9.	y	11.	y
		fª		fª
				fª
	0	x
			0	x

13.			y	y=Mª(t)								1963 to 1971
13.			y									1963 to 1971
	0.1
	0.1
	0.05

	_0.03										t
	_0.03

	1950			1960		1970	1980			1990
15.					y		f, fª					f x ex
15.					y							f x ex

				1

				0		1				x
													(c) f x 2x
17.	(a) 0, 1, 2, 4						(b)			1, 2, 4			(c) f x 2x
19.	f x 21 , ,									21. f t 5 18t, ,
23.	f x 3x 2 3, ,
25.	t x 1 s								, [ 21 , ), ( 21 , )
25.	t x 1 s					1 2x			, [ 21 , ), ( 21 , )
27.	G t					4		, , 1 1, , , 1 1,

					t 1 2
29.			f x 4x3, ,								31. (a) f x 4x3 2
33.	(a) The rate at which the unemployment rate is changing, in
percent unemployed per year
(b)
			t			U t					t		U t

		1993				0.80					1998		0.35
		1994				0.65					1999		0.25
		1995				0.35					2000		0.25
		1996				0.35					2001		0.90
		1997				0.45					2002		1.10

35.	4 corner ; 0 discontinuity
37.	1 vertical tangent ; 4 corner

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A71

39. 2

		Differentiable at 1;
	_2	not differentiable at 0
		1
		1

	_1
41.	a f, b f , c f
43.	a acceleration, b velocity, c position

45.6

	fª		f		f x 4 2x,
			f	10 f x 2
	6			10 f x 2
	f·
	10
47.	3	f·			f x 4x 3x2 ,
	f	f·			f x 4 6x,
	f				f x 4 6x,
	4			6	f x 6	,
					f x 6	,
	fª				f 4 x 0
				fªªª
	7

49.	(a) 31 a 2 3
51.	f x	1	if x 6
		1	if x 6
or f x		x 6
or f x		x 6

y		fª
		fª
1
0	6	x
_1

53.	(a)			y			(b)	All x
							(c)	f x 2 x
				0
						x
57.	63
57.	63
CHAPTER 2 REVIEW N PAGE 166
True-False Quiz
1.	False			3. True		5. False		7.	True			9. True
11.	False			13.	True		15. True		17. False			19.	False
Exercises
1.	(a)	(i) 3		(ii)	0	(iii)	Does not exist			(iv)	2
(v)		(vi)			(vii) 4		(viii) 1
(b)	y 4, y 1					(c)	x 0, x 2			(d)	3, 0, 2, 4
3.	1	5.	23	7.		3	9.	11.	74	13.		21
15.				17. 2		19.	2	21.	x 0, y 0				23. 1
29.	(a)	(i)	3	(ii)	0	(iii)	Does not exist			(iv)	0	(v) 0	(vi) 0

A72 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

(b)	At 0 and 3		(c)		y
(b)	At 0 and 3		(c)		y
				3

				0		3	x
31.		35. (a)	8		(b) y 8x 17
37.	(a) (i)	3 m s	(ii) 2.75 m s			(iii) 2.625 m s
(iv) 2.525 m s			(b)	2.5 m s
39.	(a) 10	(b)	y 10 x 16

–4 4

–12

41. (a) The rate at which the cost changes with respect to the interest rate; dollars (percent per year)

(b)As the interest rate increases past 10%, the cost is increasing at a rate of $1200 (percent per year).

(c)Always positive

43.y

fª

45.	(a)	f x 25 3 5x 1 2	(b) ( , 53 ], ( , 53 )
(c)		6
		f
	_3	1
		fª
		_6
47.	4 (discontinuity), 1 (corner), 2 (discontinuity),
5 (vertical tangent)
49.	The rate at which the total value of US currency in circulation
is changing in billions of dollars per year; $22.2 billion year
51.	0

PROBLEMS PLUS N PAGE 170

1.	32	3. 4		5. 1	7. a 21 21 s	5
9.	43	11.	(b) Yes	(c)	Yes; no
13.	(a)	0	(b) 1	(c)	f x x2 1

CHAPTER 3

EXERCISES 3.1 N PAGE 180

1.	(a) See Deﬁnition of the Number e (page 179).
(b)	0.99, 1.03; 2.7 e 2.8
3.	f x 0	5. f t 32	7. f x 3x2 4

9.	f t t3						11. y 52 x 7 5							13. V r 4 r2
														s
15.							17.			G x	1 (2				x) 2ex
	A s	60 s6
19.	F x			5		x4	21.			y 2ax b
				32
23.	y 23 s				(2 s				) 3 (2xs				)
			x					x				x
25.	y 0					27. H x 3x2 3 3x 2 3x 4
29.	u 51 t 4 5 10t3 2									31.	z 10A y11 Bey
33.	y 41 x 43
35.	Tangent: y 2x 2; normal: y 21 x 2
37.	y 3x 1						39. ex 5					41. 45x14 15x2
43.	(a)										(c) 4x3 9x2 12x 7
	50														100

				3			5
3				5
		10				40
45.	f x 4x3 9x2 16, f x 12x2 18x
47.	f x 2 154 x 1 4, f x 1615 x 5 4
49.	(a)	v t 3t2 3, a t 6t			(b) 12 m s2
(c)	a 1 6 m s2			51. 2, 21 , 1, 6
55.	y 12x 15, y 12x 17				57.	y 31 x 31
59.	2, 4		63.	P x x2 x 3
65.	y 163 x3 49 x 3
67.	No	y			y
				ƒ		ƒ
			(1,1)		0	1	x
			(1,1)			1
			0	x

69.

(a) Not differentiable at 3 or 3

if x 3

f x 2x if x 3

(b)

3 0

71.

y 2x2 x

73. a 21 , b 2 75. m 4, b 4

77.

1000

79.

3; 1

EXERCISES 3.2 N PAGE 187

y 5x4 3x2 2x

f x ex x3 3x2 2x 2

y x 2 e x x3

7. t x 5 2x 1 2

V x 14x6 4x3 6

11.

F y 5 14 y2 9 y4

2t t4 4t2 7

13.

x 2 3 x 2

15. y

1 x 2 2

t4 3t2 1 2

17.	y r2 2 er					19.	y 2v 1 v
		4 t1 2								s
21.	f t (2 st)2					23. f x ACex B Cex 2
25.	f x 2cx x2 c 2
27.	x4 4x3 ex; x4 8x3 12x2 ex
	2x2 2x			2
29.	1 2x 2 ; 1 2x 3
31.	y 21 x 21			33.		y 2x; y 21 x
35.	(a)	y 21 x 1			(b)					1.5
									(_1,0.5)
						4				4
										0.5
37.	(a)	ex x 3 x4				39.			xex, x 1 ex
41.	41	43.	(a)	16		(b)			209	(c) 20
45.	7	47.	(a)	0	(b) 32
49.	(a)	y xt x t x							(b) y t x xt x t x 2
(c)	y xt x t x x2
51.	Two, ( 2			s3, (1 s3 ) 2)
53.	$1.627 billion year						55. (c) 3e3x
57.	f x x2 2x ex, f x x2 4x 2 ex,
f x x2 6x 6 ex, f 4 x x2 8x 12 ex,
f 5 x x2 10x 20 ex; f (n) x x2 2nx n n 1 ex
EXERCISES 3.3 N PAGE 195
1.	f x 6x 2 sin x					3.			f x cos x 21 csc2x
5.	t t 3t2 cos t t3 sin t
7.	h csc cot					e		cot csc2
7.	h csc cot					e		cot csc2
		2 tan x x sec2x								sec tan
9.	y	2 tan x 2							11.	f 1 sec 2
13.	y x cos x 2 sin x x3
15.	f x ex csc x x cot x x 1
21.	y 2s3x 32 s3					2			23.	y x 1
25.	(a)	y 2 x		(b)		3π
						2
									” π,π’
									2
						0				π


27.	(a)	sec x tan x 1
29.	cos sin ; 2 cos sin
31.	(a)		f x 1 tan x sec x				(b)	f x cos x sin x
33.	2n 1				31 , n an integer
35.	(a)	v t 8 cos t, a t 8 sin t
(b)	4 s
(b)	4 s	3, 4, 4s3; to the left
37.	5 ft rad			39. 3		41. 3	43.	sin 1
45.	21		47. s
45.	21		47. s		2

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A73

49.

(a)

sec2x 1 cos2x

(b)

sec x tan x sin x cos2x

(c)

cos x sin x cot x 1 csc x

51.

EXERCISES 3.4 N PAGE 203

4 cos 4 x

3. 20x 1 x2 9

5. esx (2 s

)

7. F x 10 x x4 3x2 2 4 2x2 3

9. F x

2 3x2

11.

t t

12t3

4 1 2x x3 3 4

t4 1 4

13.

y 3x2 sin a3 x3

15. y e kx kx 1

17.

t x 4 1 4x 4 3 x x2 7 17 9x 21x2

19.

y 8 2x 5 3 8x2 5 4 4x2 30x 5

21.

12 x x2 1 2

23.

y cos x x sin x ex cos x

x2 1 4

25.

F z 1 z 1 1 2 z 1 3 2

27.

y r2 1 3 2

29.

y 2 cos tan 2x sec2 2x

31.

33.

y 4 sec 2x tan x

y 2sin x ln 2 cos x

4e2x

1 e2x

35.

sin

1 e2x 2

1 e2x

37.

y 2 cos cot sin csc2 sin

39.

f t sec2 et et

etan t sec2t

41.

f t 4 sin esin2 t cos esin2t esin2 t

sin t cos t

43.

t x 2r2 p ln a 2rarx n p 1 arx

45.

cos tan

x sec2

x sins

sin tan

2ssin tan

h x x

47.

x2 1, h x 1 x2 1 3 2

49.

e x

x ; e x 2

2 sin

x 2 cos x

cos x sin

51.

y 20x 1

53.

y x

55.

(a)

y 21 x 1

(b)

(0, 1)

_1.5

57.

(a)

f x 2 2x2

2 x2

59.

2 2n , 3 , 3 2 2n , 1 , n an integer

61.

63.

(a) 30

(b) 36

65.

(a)

(b)

Does not exist

(c)

67.

(a)

F x ex f ex

(b) G x e f x f x

69.

120

71.

75. 250 cos 2x

77.

v t 25

cos 10 t cm s

79.

(a)

cos

2 t

(b) 0.16

5.4

81.

v t 2e 1.5t 2 cos 2 t 1.5 sin 2 t

√

A74 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

83. dv dt is the rate of change of velocity with respect to time; dv ds is the rate of change of velocity with respect to displacement

85.	(a) y abt where a 100.01244 and b 0.000045146
(b) 670.63 A
87.	(b) The factored form
91.	(b) n cos n 1x sin n 1 x
EXERCISES 3.5 N PAGE 213
1.	(a) y y 2 6x x
(b)	y 4 x 2 3x, y 4 x2 3
3.	(a) y y2 x2 (b) y x x 1 , y 1 x 1 2
5.	y x2 y2

7. y

2x y

9. y

3y2 5x4 4x3 y

2y x

x4 3y2 6xy

11.

2xy2 sin y

13. y tan x tan y

2x2 y x cos y

15.

y y ex y

17.

4xys

y2 xex y

x 2x2sxy

19.

ey sin x y cos xy

21.

ey cos x x cos xy

23.

2x4y x3 6xy2

25.

y x 2

4x3y2 3x2 y 2y3

27.

y x 21

29.

x 1340

31.

(a) y 29 x 25

(b)

(1,2)

33.

81 y3

35. 2x y5

Eight; x 0.42, 1.58

37.

(a)

(b)

y x 1, y 31 x 2

39.

( 45 s

41. x0 x a2 y0y b2 1

3, 45 )

45.

47.

1 x

x2 x

49.

G x 1

x arccos x

51.

h t 0

s1 x2

y 2e2 x

x arcsin x

53.

1 e4 x

55. 1

s1 x 2

59.

61.

63.

( s

65. 1, 1 , 1, 1

67. (b) 23 69. 2

3, 0)

EXERCISES 3.6 N PAGE 220

The differentiation formula is simplest.

f x

cos ln x

5. f x

3x 1 ln 2

f x

sin x

cos x ln 5x

5xs ln x 4

2x2 1

11.

F t

13. t x

2t 1

3t 1

x x2 1

15.

f u

1 ln 2

17.

10x 1

u 1 ln 2u 2

5x2 x 2

19.

21.

log10 x

1 x

ln 10

23.

y x 2x ln 2x ; y 3 2 ln 2x

25.y s1 x 2 ; y 1 x 2 3 2

f x

2x 1 x 1 ln x 1

27.

;

x 1 1 ln x 1 2

1, 1 e 1 e,

f x

2 x 1

, 0 2,

29.

;

x x 2

31.

33.

y 3x 2

35. cos x 1 x

24x3

37.

y 2x 1 5 x4 3 6

2x 1

x4 3

39.

sin2x tan4x

2 cot x

4 sec2x

x2 1 2

tan x

x2 1

41.

y xx 1 ln x

43.

y x

sin x

cos x ln x

45.

y cos x x x tan x ln cos x

47.

y tan x 1 x

sec2 x

ln tan x

x tan x

1 n 1 n 1 !

49.

51. f n x

x2 y2

x 1 n

EXERCISES 3.7 N PAGE 230

(a) 3t2 24t 36

(b)

9 ft s

(d)

0 t

2, t 6

(e)

96 ft

(f)

t 8,

(g)

6t 24; 6 m s2

t 6,

s 32

s 0

t 2,

t 0,

s 32

s 0

(h)

(i) Speeding up when

4 or t 6;

slowing down when

√

0 t

2 or 4

t 6

sin

(a)

(b)

s2 ft s

(d)

(e) 4 ft

(f)

t =10,

s=0

t 8, s 1

t 4,

s _1

t 0, s 1

(g)

2 cos t 4 ;

ft s2

(h)

√

(i)

Speeding up when 0

2, 4

slowing down when 2

4, 6

(a) Speeding up when 0

1 or 2

slowing down when 1

(b) Speeding up when 1

2 or 3

slowing down when 0

1 or 2

(a) t 4 s

(b)

t 1.5 s; the velocity has an absolute minimum.

(a) 5.02 m s

(b) s

m s

11.

(a)

30 mm2 mm; the rate at which the area is increasing

with respect to side length as x reaches 15 mm

(b)

A 2x x

13.

(a)

(i) 5

(ii) 4.5

(iii)

4.1

(b)

15.

(a)

8 ft2 ft

(b)

16 ft2 ft

(c)

24 ft2 ft

The rate increases as the radius increases.

17.

(a) 6 kg m

(b) 12 kg m

At the right end; at the left end

19.

(a)

4.75 A

(b) 5 A; t 32 s

21.

(a)

dV dP C P2

(b)

At the beginning

23.

400 3t ln 3;

6850 bacteria h

25. (a) 16 million year; 78.5 million year

(b)P t at3 bt2 ct d, where a 0.00129371, b 7.061422, c 12,822.979, d 7,743,770

(c)P t 3at2 2bt c

(d)14.48 million year; 75.29 million year (smaller)

(e)81.62 million year

27. (a) 0.926 cm s; 0.694 cm s; 0

(b)0; 92.6 cm s cm; 185.2 cm s cm

(c)At the center; at the edge

29. (a) C x 12 0.2 x 0.0015x2

(b)$32 yard; the cost of producing the 201st yard

(c)$32.20

31. (a) xp x p x x2; the average productivity increases as new workers are added.

33. 0.2436 K min

35.	(a) 0 and 0	(b) C 0
(c)	0, 0 , 500, 50 ; it is possible for the species to coexist.

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A75

EXERCISES 3.8 N PAGE 239

About 235

(a)

100 4.2 t

(b)

7409

(c)

10,632 bacteria h

(d)

ln 100 ln 4.2 3.2 h

(a)

1508 million, 1871 million

(b) 2161 million

(c)

3972 million; wars in the ﬁrst half of century, increased life

expectancy in second half

(a)

Ce 0.0005t

(b)

2000 ln 0.9 211 s

(a)

100 2 t 30 mg

(b)

9.92 mg

11.

2500 years

13. (a)

137 F

(b)

116 min

67.74 min

15.

(a)

13.3 C

(b)

17.

(a)

64.5 kPa

(b)

39.9 kPa

19.

(a)

(i) $3828.84

(ii)

$3840.25

(iii) $3850.08

(iv) $3851.61 (v) $3852.01

(vi)

$3852.08

(b)

dA dt 0.05A, A 0 3000

EXERCISES 3.9 N PAGE 245

dV dt 3x2 dx dt

3. 48 cm2 s

3 25 m min

9. 1346

11.

(a)

The plane’s altitude is 1 mi and its speed is 500 mi h.

(b)

The rate at which the distance from the plane to the station is

increasing when the plane is 2 mi from the station

(c)

(d)

y2 x2 1

(e) 250 s

mi h

13. (a) The height of the pole (15 ft), the height of the man (6 ft), and the speed of the man (5 ft s)

(b) The rate at which the tip of the man’s shadow is moving when he is 40 ft from the pole

(c)	(d)	15	x y	(e) 253 ft s
			y
	6

15.65 mi h 17. 837 s8674 8.99 ft s

19.	1.6 cm min										21. 72013		55.4 km h
23.	10,000 800,000 9 2.89 105 cm3 min
25.	103 cm min									27.	6 5 0.38 ft min								29.		0.3 m2 s
31.	80 cm3 min									33. 810107 0.132 s									35. 0.396 m min
37.	(a) 360 ft s									(b)	0.096 rad s			39.				109	km min
41.	1650 s								296 km h				43. 47 s				6.78 m s
41.	1650 s							31	296 km h				43. 47 s		15		6.78 m s
EXERCISES 3.10 N PAGE 252
1. L x 10x 6											3. L x x							2
5. s							1 21 x;
5. s		1 x					1 21 x;							3
	s				0.95,								y=1-	1	x
			0.9											1
			0.9											2
	s					0.995
	s			0.99		0.995										(0,1)
													y=œ„„„„1-x
											_3										3
											_3							(1,0)			3
																		(1,0)

														_1
7.	1.204 x									0.706		9. 0.045						x	0.055		t
11.	(a) dy 2x x cos 2x sin 2x dx															(b)			dy		t
																						dt
																				1 t2		dt

A76 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

13.	(a)			dy									2					du							(b) dy							6r2			dr
13.	(a)			dy			u 1 2											du							(b) dy						1 r3 3				dr
15.	(a)		dy					1			ex 10 dx										(b)				0.01; 0.0101
15.	(a)		dy					10			ex 10 dx										(b)				0.01; 0.0101
17.	(a)			dy sec2x dx																(b) 0.2
19.	y 0.64, dy 0.8																						y			y=2x-≈
19.	y 0.64, dy 0.8																						y
																						1



																												dy		Îy

																								0			1				dx=Îx
21.	y 0.1, dy 0.125
	y																									dx=Îx
																										dx=Îx
					2																											Îy		dy
																																Îy		dy


					y=
	2				y=	x


	1
	1

		0		1																						x
23.	32.08							25. 4.02														27.			1		90 0.965

33.	(a) 270 cm3, 0.01, 1%																							(b) 36 cm2, 0.006, 0.6%
35.	(a) 84 27 cm2;																		1			0.012
35.	(a) 84 27 cm2;																		84			0.012
(b) 1764 2 179 cm3;																				1			0.018
(b) 1764 2 179 cm3;																				56			0.018
37.	(a)			2 rh r										(b)				r 2h
43.	(a)			4.8, 5.2									(b) Too large
EXERCISES 3.11 N PAGE 259
1.	(a)		0		(b)						1		3.					(a) 43								(b)	21 e2 e 2 3.62686
5.	(a)		1		(b)						0
21.	sech x 53 , sinh x 34 , csch x 43 , tanh x 54 , coth x 45
23.	(a)			1	(b)								1				(c)									(d)					(e) 0				(f ) 1
(g)				(h)											(i)		0
31.	f x x cosh x																		33. h x tanh x
35.	y 3ecosh 3x												sinh 3x										37. f t 2et sech2 et tanh et
39.	y				sech2 x															41. G x									2 sinh x
39.	y				1 tanh2 x															41. G x								1 cosh x 2
43.	y							1														45.				y sinh 1 x 3


					2 sx				1				x
					2 sx				1				x
47.	y				1


					xsx			2			1
					xsx			2			1
51.	(a)			0.3572									(b) 70.34°
53.	(b)			y 2 sinh 3x 4 cosh 3x
55.	(ln 1 s											), s				)
55.	(ln 1 s									2		), s			2	)
CHAPTER 3 REVIEW N PAGE 261
True-False Quiz
1.	True				3.					True							5.						False				7. False						9.		True
11.	True

Exercises

1. 6x x4 3x2 5 2 2x2 3

2 2x2 1

2sx

3sx7

7. 2 cos 2 esin 2

sx2 1

cos s

sin s

t2 1

11.

1 t2 2

13.

e1 x 1 2x

15.

1 y4 2xy

4xy

3 x2 3

17.

2 sec 2

tan 2

19. 1 c2 ecx sin x

1 tan 2 2

21.

3x ln x ln 3 1 ln x

23.

x 1 2

25.

2x y cos xy

27.

x cos xy 1

1 2x ln 5

29.

cot x sin x cos x

31.

tan 1 4x

1 16x2

33.

5 sec 5x

35.

6x csc2 3x2 5

3x2

37.

cos(tan

1 x3 )(sec2

1 x3 )

2 s1

39.

2 cos tan sin sec2 sin

41.

x 2 4 3x2 55x 52

43. 2x2 cosh x2 sinh x2

2 sx 1 x

3 8

cosh x

45.	3 tanh 3x				47.	ssinh2x 1
						ssinh2x 1
49.	3 sin(estan 3x)estan 3x sec2 3x							51. 274			53.	5x4 y11
49.					2stan 3x			51. 274			53.	5x4 y11
					2stan 3x
57.	y 2s3x 1 s3 3						59. y 2x 1
61.	y x 2; y x 2
63.	(a)	10 3x				(b) y 47 x 41 , y x 8
63.	(a)	2 s5 x				(b) y 47 x 41 , y x 8
(c)					10
						(4,4)
	_10				(1,2)
	_10					10
					ƒ
					_10
65.	( 4,		s	2 ), (5 4, 2 )			69.	(a)	2	(b) 44
			s			s					t ex ex
71.	2xt x x2t x					73. 2t x t x				75.	t ex ex
77.	t x t x				79.	f x t x 2 t x f x 2
77.	t x t x				79.	f x t x 2
81.	f t sin 4x t sin 4x cos 4x 4								83.	3, 0
85.	y 32 x2 143 x
87.	v t Ae ct c cos t sin t										,
a t Ae ct c2 2 cos t							2c sin t
89.	(a)	v t 3t2 12; a t 6t						(b)	t 2; 0 t				2
(c)	23		(d)		20					(e) t 2; 0			t	2
					y	a
						a
						v
					0			t 3
						position
					15

91.		4 kg m							93. (a) 200 3.24 t									(b) 22,040
(c)		25,910 bacteria h												(d)		ln 50 ln 3.24 3.33 h
95.		(a) C0 e kt								(b)		100 h				97.		34 cm2 min
99.		13 ft s							101.			400 ft h										1.01
103.												3								3
103.		(a) L x 1 x; s1														3x 1 x; s1.03
(b)	0.23							x		0.40							1	109. 41
105.		12 23					16.7 cm2									107.	1					111. 81 x2
105.		12 23					16.7 cm2									107.	32					111. 81 x2
PROBLEMS PLUS N PAGE 266
1. ( 21 s										9. (0, 45 )
1. ( 21 s					3, 41 )					9. (0, 45 )
11.	(a) 4										rad s			(b) 40(cos									) cm
11.	(a) 4					s		3	s	11	rad s			(b) 40(cos						s	8 cos2		) cm
(c) 480 sin (1 cos																			) cm s
(c) 480 sin (1 cos															8 cos2				) cm s
														s
15.		xT 3, , yT 2, , xN (0, 35 ), yN ( 25 , 0)
17.		(b) (i) 53 (or 127 )												(ii)		63 (or 117 )
19.		R approaches the midpoint of the radius AO.
21.		sin a							23. 2s				e	27. 1, 2 , 1, 0
29.		s		58					31.			11.204 cm3 min
29.		s	29	58					31.			11.204 cm3 min

CHAPTER 4

EXERCISES 4.1 N PAGE 277

Abbreviations: abs., absolute; loc., local; max., maximum; min., minimum

1. Absolute minimum: smallest function value on the entire domain of the function; local minimum at c: smallest function value when x is near c

3. Abs. max. at s, abs. min. at r, loc. max. at c, loc. min. at b and r 5. Abs. max. f 4 5, loc. max. f 4 5 and f 6 4,

loc. min. f 2 2 and f 5 3

7.	y						9.	y
	3							3
	2							2
	1							1
	0	1 2 3 4 5				x		0	1 2 3 4 5			x
11. (a)							(b)
		y						y
		1						1
		0	1	2	3		x	0	1	2	3	x
		_1						_1
(c)		y
		2
		1
		0	1	2	3		x
		_1

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A77

13.	(a)																											(b)
										y																										y
							_1


									0							2						x										0						x


15.	Abs. max. f 1 5																								17.				None
19.	Abs. min. f 0 0
21.	Abs. max. f 3 9, abs. and loc. min. f 0 0
23.	Abs. max. f 2 ln 2
25.	Abs. max. f 0 1																								27.				Abs. max. f 3 2
29.	52						31.					4, 2									33.						0, 21 ( 1 s										)	35. 0, 2
29.	52						31.					4, 2									33.						0, 21 ( 1 s									5	)	35. 0, 2
37.	0, 94						39.					0, 78, 4									41.						n				n an integer							43. 0, 32
45.	10						47.					f 0 5, f 2 7
49.	f 1 8, f 2 19																														f 1 21 , f 0 0
51.		f 3 66, f 1 2																									53.				f 1 21 , f 0 0
55.		f (						) 2				, f 1													s
55.		f (					2	) 2				, f 1															3
				s
57.	f 6 23s												3, f 2 0
59.	f 2 2											e		, f 1 1															8	e
											s																	s
61.	f 1 ln 3, f ( 21 ) ln 43
								a										aabb
63.	f
63.	f				a b											a b a b
65.	(a)					2.19, 1.81											(b)			6		s53 2,											6		s53 2
65.	(a)					2.19, 1.81											(b)			25		s53 2,											25		s53 2
67.	(a)					0.32, 0.00											(b)		3			s										69. 3.9665 C
																			3							3, 0
																			16							3, 0
71.	Cheapest, t 0.855 (June 1994);
most expensive, t 4.618 (March 1998)
73.	(a)					r 32 r0										(b) v								4		kr03
73.	(a)					r 32 r0										(b) v								27		kr03
(c)							√
			4				√
			4	kr#¸
			27	kr#¸

							0										2	r¸				r¸ r
							0									3		r¸				r¸ r
EXERCISES 4.2 N PAGE 285
1.	2						3. 49								5. f is not differentiable on ( 1, 1)
7.	0.8, 3.2, 4.4, 6.1																															2 s
9. (a), (b)																												(c)				2 s		2
10																													10
0																					10								0									10
0																													0
11.	0								13. 21 ln[61 (1 e 6)]																							15. f is not continous at 3
23.	16						25.					No				31.							No

A78 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

EXERCISES 4.3 N PAGE 295

Abbreviations: inc., increasing; dec., decreasing; CD, concave downward; CU, concave upward; HA, horizontal asymptote; VA, vertical asymptote; IP, inﬂection point(s)

1.	(a)	1, 3 , 4, 6		(b) 0, 1 , 3, 4 (c) 0, 2
(d)	2, 4 , 4, 6		(e)	2, 3
3.	(a)	I/D Test	(b)	Concavity Test
(c)	Find points at which the concavity changes.
5.	(a)	Inc. on 1, 5 ; dec. on 0, 1 and 5, 6
(b)	Loc. max. at x 5, loc. min. at x 1
7.	x 1, 7
9.	(a)	Inc. on , 3 , 2, ; dec. on 3, 2

(b)Loc. max. f 3 81; loc. min. f 2 44

(c)CU on 12, ; CD on ( , 12); IP ( 12, 372 )

11. (a) Inc. on 1, 0 , 1, ; dec. on , 1 , 0, 1

(b)Loc. max. f 0 3; loc. min. f 1 2

(c)CU on ( , s3 3), (s3 3, );

CD on ( s3 3, s3 3); IP ( s3 3, 229 )


13.	(a) Inc. on 0, 4 , 5 4, 2 ; dec. on 4, 5 4
(b)	Loc. max. f 4 s		2	; loc. min. f 5 4 s2
(c) CU on 3 4, 7 4 ; CD on 0, 3 4 , 7 4, 2 ;
IP 3 4, 0 , 7 4, 0
15.	(a) Inc. on ( 31	ln 2, ); dec. on ( , 31 ln 2)
(b)	Loc. min. f ( 31	ln 2) 2 2 3 21 3 (c) CU on ,
17.	(a) Inc. on 0, e2 ; dec. on e2,

(b)Loc. max. f e2 2 e

(c)CU on e8 3, ; CD on 0, e8 3 ; IP (e8 3, 83 e 4 3 )

19. Loc. max. f 1 7, loc. min. f 1 1 21. Loc. max. f (34 ) 54

23. (a) f has a local maximum at 2.

(b) f has a horizontal tangent at 6.

25.y

1 2 3 4

27. y y

_2 0	x	_2 0	2	x
	x=2

31. (a) Inc. on (0, 2), (4, 6), 8, ; dec. on (2, 4), (6, 8)

(b)Loc. max. at x 2, 6; loc. min. at x 4, 8

(c)CU on (3, 6), 6, ; CD on (0, 3)

(d)3 (e) See graph at right.

33. (a) Inc. on , 1 , 2, ; dec. on 1, 2

(b) Loc. max. f 1 7; loc. min. f 2 20

(c)CU on (12 , ); CD on ( , 12 ); IP (12 , 132 )

(d)See graph at right.

35. (a) Inc. on , 1 , 0, 1 ; dec. on 1, 0 , 1,

(b)Loc. max. f 1 3, f 1 3; loc. min. f 0 2

(c)CU on ( 1 s3, 1 s3 );

CD on ( , 1 s3 ), 1 s3, ); IP ( 1 s3, 239 )

(d) See graph at right.

37. (a) Inc. on , 2 , 0, ; dec. on 2, 0

(b)Loc. max. h 2 7; loc. min. h 0 1

(c)CU on 1, ;

CD on , 1 ; IP 1, 3

(d) See graph at right.

39. (a) Inc. on 2, ; dec. on 3, 2

(b)Loc. min. A 2 2

(c)CU on 3,

(d)See graph at right.

41. (a) Inc. on 1, ; dec. on , 1

(b)Loc. min. C 1 3

(c)CU on , 0 , 2, ; CD on 0, 2 ;

IPs 0, 0 , (2, 6 s3 2 )

(d) See graph at right.

29.			43.	(a) Inc. on , 2 ;
29.
	y		dec. on 0,
			(b)	Loc. min. f 1
			(c) CU on 3, 5 3 ;
			CD on 0, 3 , 5 3, 2 ;
	0	x	CD on 0, 3 , 5 3, 2 ;
			IP ( 3, 45 ), (5 3, 45 )
			(d)	See graph at right.
			(d)	See graph at right.

0			2	4		6			8			x
			2	4		6			8
			(_1, 7)	y
			(_1, 7)	y

				0		1	, _13’					x
					”	1	, _13’
						2		2
								(2, _20)
”_	1	,	23’	y			”	1	,	23	’
	œ„3		9					œ„3		9
(_1, 3)				1		(1, 3)




				0				1					x

(_2,7)				y
(_2,7)				7

(_1, 3)

_1 x (0, _1)

_2 x

{2,6Œ„2}

	_4		0					x
			(_1,_3)
y
1		”π3,		5	’	”5π3 ,	5	’


							4
				4
				4

0				π		2π ¨
_1
_1

(π, _1)

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A79

45.	(a) HA y 1, VA x	1, x 1		y
				y

(b)	Inc. on , 1 , 1, 0 ;				y=1
dec. on 0, 1 , 1,					y=1
dec. on 0, 1 , 1,				0
(c)	Loc. max. f 0 0			0
(c)	Loc. max. f 0 0				x
(d)	CU on , 1 , 1, ;
CD on 1, 1			x=_1		x=1
(e)	See graph at right.		x=_1		x=1
(e)	See graph at right.
47. (a) HA y 0			y

(b)Dec. on ,

(c)None

(d)CU on ,

(e)	See graph at right.	1
(e)	See graph at right.
		0	x
49.	(a) VA x 0, x e	y
49.	(a) VA x 0, x e	y
(b)	Dec. on 0, e	x=0	x=e
(c) None		1	(1, 0)
(d) CU on (0, 1); CD on 1, e ;
(d) CU on (0, 1); CD on 1, e ;		0		x
IP (1, 0)
(e)	See graph at right.
51.	(a) HA y 1, VA x 1		y
51.	(a) HA y 1, VA x 1		y

(b)Inc. on , 1 , 1,

(c)None

(d)	CU on , 1 , ( 1, 21 );						y=1
CD on ( 21 , ); IP ( 21 , 1 e2)
					x=_1	0	x
(e)	See graph at right.
53.	3,
55.	(a) Loc. and abs. max. f 1 s
				2, no min.
(b)	41 (3 s		)
		17
57.	(b) CU on 0.94, 2.57 , 3.71, 5.35 ;

CD on 0, 0.94 , 2.57, 3.71 , 5.35, 2 ;

IP 0.94, 0.44 , 2.57, 0.63 , 3.71, 0.63 , 5.35, 0.44

59.	CU on , 0.6 , 0.0, ; CD on 0.6, 0.0
61.	(a)	The rate of increase is initially very small, increases to a
maximum at t 8 h, then decreases toward 0.
(b) When t 8				(c) CU on 0, 8 ; CD on 8, 18								(d) 8, 350
63.	K 3 K 2 ; CD
65.	28.57 min, when the rate of increase of drug level in the blood-
stream is greatest; 85.71 min, when rate of decrease is greatest
67.	f x 91 2x3 3x2 12x 7
EXERCISES 4.4 N PAGE 304
1.	(a)	Indeterminate			(b) 0		(c) 0
(d)	, , or does not exist						(e)	Indeterminate
3.	(a)		(b)	Indeterminate				(c)
5.	2	7. 59	9.			11.		21	13.	p q
15.	0	17.		19.			21.	21		23.	1
25.	ln 35		27. 1	29.		21	31.	0		33.	1 2
35.	21 a a 1			37.	1		39.		41. 3		43.	0
35.	21 a a 1			37.	24		39.		41. 3		43.	0
45.	2		47.	21	49. 21		51.			53. 1

55.	e 2		57. e3	59. 1	61. e4
63.	1 s	e	65. e2	67. 41	71. 1	77. 169 a	79. 56
83.	(a) 0

EXERCISES 4.5 N PAGE 314

1. A. B. y-int. 0; x-int. 0

C. About 0, 0	D. None
E. Inc. on ,	F. None

G.CU on 0, ; CD on , 0 ; IP (0, 0)

H.See graph at right.

3. A. B. y-int. 2; x-int. 2, 12 (7 C. None D. None

E. Inc. on (1, 5);

dec. on , 1 , 5,

F.Loc. min. f 1 5; loc. max. f 5 27

G.CU on , 3 ;

CD on 3, ; IP 3, 11

H. See graph at right.

1 x

3 s5 )

(5,27)

(1, _5)

5. A.	B. y-int. 0; x-int. 4, 0
C. None	D. None

E.Inc. on 3, ; dec. on , 3

F.Loc. min. f 3 27

G.CU on , 2 , 0, ;

CD on 2, 0 ; IP (0, 0), 2, 16 H. See graph at right.

7. A.	B. y-int. 1
C. None	D. None

E.Inc. on , 0 , 1, ; dec. on 0, 1

F.Loc. max. f 0 1; loc. min. f 1 2

G.CU on (1 s3 4, ); CD on ( , 1 s3 4 );

IP (1 3				))
	4, 1 9 (2	3	16
s		s
H. See graph at right.
9. A. x x 1		B. y-int. 0; x-int. 0

C. None D. VA x 1, HA y 1

E.Dec. on , 1 , 1,

F.None

G.CU on 1, ; CD on , 1

H.See graph at right.

(_3,_27)

0, 1

1, _2

y 1

A80 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

11. A. x x 3	B. y-int. 91
C. About y-axis D. VA x 3, HA y 0
E. Inc. on , 3 , 3, 0 ;			y
E. Inc. on , 3 , 3, 0 ;			y
dec. on (0, 3), 3,
F. Loc. max. f 0 91
G. CU on , 3 , 3, ;
CD on 3, 3				x
CD on 3, 3				x 3
H. See graph at right.		x 3		x 3

13. A. B. y-int. 0; x-int. 0 C. About (0, 0) D. HA y 0 E. Inc. on 3, 3 ;

dec. on , 3 , 3,

F.Loc. min. f 3 16 ; loc. max. f 3 16 ;

G.CU on ( 3 s3, 0), (3 s3, );

CD on ( , 3 s3 ), (0, 3 s3 ); IP (0, 0), ( 3 s3, s3 12)

H. See graph at right.

”3, 61 ’

”_3, _61 ’

15. A. , 0 0, B. x-int. 1 C. None D. HA y 0; VA x 0 E. Inc. on 0, 2 ;

dec. on , 0 , 2,

F.Loc. max. f 2 14

G.CU on 3, ;

CD on , 0 , 0, 3 ; IP (3, 29 ) H. See graph at right

”2,41 ’ ”3,29’

17. A.	B.	y-int. 0, x-int. 0
C. About y-axis		D. HA y 1

E.Inc. on 0, ; dec. on , 0

F.Loc. min. f 0 0

G.CU on 1, 1 ;

CD on , 1 , 1, ; IP ( 1, 14 ) H. See graph at right

			y		y=1
”_1,	1	’	”1,	1	’
	4			4
			(0, 0)		x

19. A. , 5 B. y-int. 0; x-int. 0, 5				”	10	,	10œ„„15	’
C. None			y	”	3	,	9	’
C. None	D. None
E. Inc. on ( , 103 ); dec. on (103 , 5)
F. Loc. max. f (103 ) 109 s								x
F. Loc. max. f (103 ) 109 s		15						x
G. CD on , 5
H. See graph at right.

21. A. , 2 1,				y
B. x-int. 2, 1
C. None	D. None

E.Inc. on (1, ); dec. on ( , 2)

F.None

G.CD on , 2 , 1,

H. See graph at right.

_2 0 1

23. A. B. y-int. 0; x-int. 0

C.About the origin

D.HA y 1

E. Inc. on , F. None

G.CU on , 0 ; CD on 0, ; IP 0, 0

H.See graph at right.

y
y=1
(0, 0)	x
y=_1

25.	A. {x x 1, x 0} 1, 0 0, 1
B.	x-int. 1 C. About (0, 0)	y

D.VA x 0

E.Dec. on 1, 0 , 0, 1

F. None

1 x

G. CU on ( 1, s2 3 ), (0, s2 3 );

CD on ( s

2 3, 0), (s

2 3, 1);

IP (

2 3, 1 2 )

H. See graph at right.

27. A.

B. y-int. 0; x-int. 0, 3s

C. About the origin

D. None

E. Inc. on , 1 , 1, ; dec. on 1, 1

F. Loc. max. f 1 2;

loc. min. f 1 2

_1, 2

3œ„3, 0

G. CU on 0, ; CD on , 0 ;

0, 0

IP 0, 0

_3œ„3, 0

1, _2

H. See graph at right.

29. A.

B. y-int. 1; x-int. 1

C. About y-axis D. None

E.Inc. on 0, ; dec. on , 0

F.Loc. min. f 0 1

G.CU on 1, 1 ;

CD on , 1 , 1, ;

(_1,0) 0

(1,0)

IP 1, 0

(0,_1)

H. See graph at right.

31. A. B. y-int. 0; x-int. n

(n an integer)

C. About the origin, period 2

D. None

E. Inc. on 2n

2, 2n

”π, 2’

2 ;

dec. on 2n

2, 2n

3 2

F. Loc. max. f 2n

2π

loc. min. f 2n

2 2

_2π

G. CU on 2n 1 , 2n ;

”_π2, _2’

CD on 2n , 2n 1 ; IP n , 0

H. See graph at right.

33. A. 2,

B. y-int. 0; x-int. 0

C. About y-axis

D. VA x 2

E. Inc. on 0, 2 ;

dec. on 2, 0

x π2

F. Loc. min. f 0 0

G. CU on 2,

H. See graph at right.

35. A. 0, 3 C. None D. None E. Inc. on 3, 5 3 , 7 3, 3 ;

dec. on 0, 3 , 5 3, 7 3

F. Loc. min. f 3 6 12 s3, f 7 3 7 6 12 s3;

loc. max. f 5

3 5

6 2 s3

G. CU on 0,

, 2 , 3 ;

CD on , 2 ;

IP , 2 , 2 ,

5π

7π

H. See graph at right.

2π

3π x

A. All reals except 2n 1

37.

(n an integer)

B. y-int. 0; x-int. 2n

C. About the origin, period 2

D. VA x 2n 1

E. Inc. on 2n 1 , 2n 1

F. None

G. CU on 2n , 2n 1 ; CD on 2n 1 , 2n ;

IP 2n , 0

x=_3π

x=_π y

x=π

x=3π

_2π

2π

39.

B. y-int. 1

C. Period 2

D. None

Answers for E– G are for the interval 0, 2 .

E. Inc. on 0,

2 , 3 2, 2 ; dec. on 2, 3 2

F. Loc. max. f 2 e; loc. min. f 3 2 e 1

G. CU on 0,

, , 2 where sin 1 (21 ( 1 s

)),

; CD on ,

; IP when x

_2π

2π

4π x

41.

B. y-int. 21

C. None

y 1

D. HA y 0, y 1

E. Inc. on

F. None

G. CU on , 0 ; CD on 0, ;

IP (0, 21 ) H. See graph at right.

43.

A. 0,

B. None

C. None

D. VA x 0

E.Inc. on 1, ; dec. on 0, 1

F.Loc. min. f 1 1

G.CU on 0,

H.See graph at right.

(1,1)

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A81

45. A. B. y-int. 14 C. None D. HA y 0, y 1

E. Dec. on F. None

G.CU on (ln 12, ); CD on , ln 12); IP (ln 12, 49)

H.See graph at right.

y=1

”ln 12 ,49 ’

47.	A. All x in 2n , 2n 1 (n an integer)
B.	x-int. 2 2n	C. Period 2	D. VA x n

E. Inc. on 2n ,	2 2n ; dec. on 2 2n , 2n 1
F. Loc. max. f 2 2n 0				G. CD on 2n , 2n 1
H.		y
_4π _3π _2π	_π		π 2π	3π 4π

		0		x

49. A. B. y-int. 0; x-int. 0 C. About (0, 0) D. HA y 0

E.Inc. on ( 1 s2, 1 s2 ); dec. on ( , 1 s2 ), (1 s2, )

F.Loc. min. f ( 1 s2 ) 1 s2e; loc. max. f (1 s2 ) 1 s2e

G.CU on ( s3 2, 0), (s3 2, ); CD on ( , s3 2 ), (0, s3 2 );

IP ( s3 2, s3 2e 3 2 ), 0, 0

H.	y	”	1	,	1	’
H.	y	”	1	,	1	’
			œ„2		œ„„2e

		0				x

51. A.	B. y-int. 2
C. None	D. None

E.Inc. on (15 ln 23, ); dec. on ( , 15 ln 23 )

F.Loc. min. f (15 ln 23) (23)3 5 (23) 2 5

G.CU on ,

H.See graph at right.

local minimum (0, 2)

53.		m		55. y
53.		m		55. y

					0	L/2	L x
	(0, m¸)		√=c
	(0, m¸)		√=c

		0		√

57.	y x 1		59. y 2 x 2

A82 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

61. A. ( , 21 ) (21 , )

B.y-int. 1; x-int. 14 (5 s17 )

C.None

D.VA x 12; SA y x 2

E. Dec. on ( , 21 ), (21, )

F. None

x=_

G. CU on (21, ); CD on ( , 21 )

y=_x+2

H. See graph at right

63. A. x x 0

B. None

(2,4)

C. About (0, 0)

D. VA x 0; SA y x

y=x

E. Inc. on , 2 , 2, ;

dec. on 2, 0 , 0, 2

F. Loc. max. f 2 4;

loc. min. f 2 4

G. CU on 0, ; CD on , 0

(_2,_4)

H. See graph at right.

65. A.

B. y-int. 1; x-int. 1

C. None

D. SA y 2x 1

{œ„3,

œ„3+1}

E. Inc. on ,

F. None

G. CU on ( , s

(0,1)

(0, s

);

CD on ( s

3, 0), (s

3, );

IP ( s

{_œ„3,_

œ„3+1}

y=2x+1

3, 1 23 s3 ), 0, 1

H. See graph at right.

67.	y	71.	VA x 0, asymptotic to y x3
				y
	y x π2			ƒ
	y x π2			10
	0	x			y=˛
	0	x
	y x π		2	0	2	x
	2
				10

EXERCISES 4.6 N PAGE 320

1. Inc. on 0.92, 2.5 , 2.58, ; dec. on , 0.92 , 2.5, 2.58 ; loc. max. f 2.5 4; loc. min. f 0.92 5.12, f 2.58 3.998; CU on , 1.46 , 2.54, ;

CD on 1.46, 2.54 ; IP 1.46, 1.40 , 2.54, 3.999

10		4.04
	ƒ
0	4	ƒ
0	4
_6	2.4	2.7
_6		3.96

3. Inc. on 15, 4.40 , 18.93, ; dec. on , 15 , 4.40, 18.93 ;

loc. max. f 4.40 53,800; loc. min. f 15 9,700,000, f 18.93 12,700,000; CU on , 11.34 , 0, 2.92 ,15.08, ; CD on 11.34, 0 , 2.92, 15.08 ;

IP 0, 0 , 11.34, 6,250,000 , 2.92, 31,800 ,15.08, 8,150,000

	10,000,000	60,000
	f	f
_25	f
_25	25
	_10	10
	_13,000,000	_30,000

5. Inc. on , 1.7 , 1.7, 0.24 , 0.24, 1 ; dec. on 1, 2.46 , 2.46, ; loc. max. f 1 13 ; CU on , 1.7 , 0.506, 0.24 , 2.46, ;

CD on 1.7, 0.506 , 0.24, 2.46 ; IP 0.506, 0.192

	3
_5	5
	_3

7. Inc. on 1.49, 1.07 , 2.89, 4 ; dec. on 4, 1.49 ,1.07, 2.89 ; loc. max. f 1.07 8.79; loc. min.

f 1.49 8.75, f 2.89 9.99; CU on 4, 1.28 ,1.28, 4 ; CD on 1.28, 1.28 ; IP 1.28, 8.77 , 1.28, 1.48

	30	10
	ƒ	ƒ
	ƒ
_4	4
	_2.5	0
	_10	6

9. Inc. on ( 8 s61, 8 s61 ); dec. on ( , 8 s61 ),8 s61, 0 , 0, ; CU on ( 12 s138, 12 s138 ),0, ; CD on ( , 12 s138 ), ( 12 s138, 0)

175

_100

0.95_10

11. (a)	1

œ„1e , 21e

_0.25 1.75

_0.25

(b)limx l 0 f x 0

(c)Loc. min. f (1 se) 1 2e ;

CD on 0, e 3 2 ; CU on e 3 2,

13.	Loc. max. f 5.6 0.018, f 0.82 281.5,
f 5.2 0.0145; loc. min. f 3 0
	y		0.02
		1	8	3.5
			8	3.5


			x
			0.04
			0.04
	500		0.03
1			2
	1500		2.5	8
	1500		0
15.	f x		x x 1 2 x3 18x2 44x 16
15.	f x		x 2 3 x 4 5
			x 2 3 x 4 5

f x 2	x 1 x 6 36x 5 6x 4 628x 3 684x 2 672x 64
	x	2 4 x 4 6

CU on 35.3, 5.0 , 1, 0.5 , 0.1, 2 , 2, 4 , 4, ; CD on , 35.3 , 5.0, 1 , 0.5, 0.1 ;

IP 35.3, 0.015 , 5.0, 0.005 , 1, 0 , 0.5, 0.00001 ,0.1, 0.0000066

17. Inc. on 0, 0.43 ; dec. on 0.43, ; loc. max. f 0.43 0.41; CU on 0.94, ; CD on 0, 0.94 ;

IP 0.94, 0.34

0.5

19.Inc. on 4.91, 4.51 , 0, 1.77 , 4.91, 8.06 , 10.79, 14.34 ,17.08, 20 ;

dec. on 4.51, 4.10 , 1.77, 4.10 , 8.06, 10.79 , 14.34, 17.08 ;

loc. max. f 4.51 0.62, f 1.77 2.58, f 8.06 3.60, f 14.34 4.39;

loc. min. f 10.79 2.43, f 17.08 3.49; CU on 9.60, 12.25 ,15.81, 18.65 ;

CD on 4.91, 4.10 , 0, 4.10 , 4.91, 9.60 , 12.25, 15.81 ,18.65, 20 ;

IPs at 9.60, 2.95 , 12.25, 3.27 , 15.81, 3.91 , 18.65, 4.20

	5
	f
_5	20
	0

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A83

21.	Inc. on , 0 , 0, ;		1
CU on , 0.4 , 0, 0.4 ;				ƒ
CD on 0.4, 0 , 0.4, ;				ƒ
CD on 0.4, 0 , 0.4, ;
IP 0.4, 0.8		_3				ƒ	3
						ƒ

			_1
23.	(a) 2
	0		8
	0		8
	_1
(b)	lim x l0 x1 x 0, lim x l x1 x		1
(c)	Loc. max. f e e1 e	(d)	IP at x 0.58, 4.37
25.	Max. f 0.59 1, f 0.68 1, f 1.96 1;
min. f 0.64 0.99996, f 1.46 0.49, f 2.73 0.51;
IP 0.61, 0.99998 , 0.66, 0.99998 , 1.17, 0.72 ,
1.75, 0.77 , 2.28, 0.34
	1.2
				f
	0				π
	0				π
	1.2
	1		1.2
0.55		_2π						2π


		0.73
		0.73	_1.2
0.9997			_1.2

27. For c 0, there is no IP and only one extreme point, the origin. For c 0, there is a maximum point at the origin, two minimum points, and two IPs, which move downward and away from the origin as c l .

4 4 1 _1 _2

_2.1 2.1

_2.3

A84	\|\|\|\|	APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
29. There is no maximum or minimum, regardless of the					37.	(a)	Positive	(b)	c=4
value of c. For c 0, there is a vertical asymptote at x 0,									c=1
value of c. For c 0, there is a vertical asymptote at x 0,									c=0.5
lim x l0	f x , and lim x l f x 1.								c=0.2
c 0 is a transitional value at which f x 1 for x 0.								12	c=0.1
								12

For c 0, lim x l0 f x 0, lim x l f x 1, and there are
two IPs, which move away from the y-axis as c l .
		2		4				_6	6
		c=0.5		c=_2
		c=0.5		c=_1
				c=_1
	c=1		c=2						c=0
_4			4	c=_0.5				_12	c=_1
			_4	c=_0.5	4			_12	c=_1
			_4		4				c=_4
									c=_4
		_1		_1

31. For c 0, the maximum and minimum values are always12 , but the extreme points and IPs move closer to the y-axis as c

increases. c 0 is a transitional value: when c is replaced by c, the curve is reﬂected in the x-axis.

0.6

0.2

0.5

1 2

5 4

0.6

33. For c 1, the graph has local maximum and minimum values; for c 1 it does not. The function increases for c 1 and decreases for c 1. As c changes, the IPs move vertically but not horizontally.

	10	c=3	c=1
			c=0.5
			c=0
_15	1		5
			c=_0.5
	_10	c=_3	c=_1

35.		3
	_2	1
	_2	0
		0
_3		3
	1	2
	1
		_3

For c 0, lim x l f x 0 and lim x l f x . For c 0, lim x l f x and lim x l f x 0.

As c increases, the maximum and minimum points and the IPs get closer to the origin.

EXERCISES 4.7 N PAGE 328

1.	(a) 11, 12												(b)				11.5, 11.5												3.			10, 10
5.	25 m by 25 m																7.					N 1
9.	(a)							50																										12,500 ft@
9.	(a)


																					250
			100																			12,500 ft@										120					9000 ft@
			100																			12,500 ft@


												125
																																	75
(b)	x
(b)


													y
(c)	A xy										(d) 5x											2y 750										(e)	A x 375x 25 x2
(f)	14,062.5 ft 2
11.	1000 ft by 1500 ft																				13.					4000 cm3							15. $191.28
17.	( 1728 ,					7		)						19.					( 31 , 34 s										)			21.	Square, side s								r
						7																					2													2
						17																					2													2
23.	L 2, s						3		L 4								25.					Base s						3		r, height 3r 2
27.	4 r3 (3 s3 )																29.					r2(1 s										)	31.			24 cm, 36 cm
27.	4 r3 (3 s3 )																29.					r2(1 s									5	)	31.			24 cm, 36 cm
33.	(a)		Use all of the wire for the square
(b)	40	s			(9							4		s				) m for the square
(b)	40			3	(9							4				3		) m for the square
35.																					3											37.	V		3
35.	Height radius sV cm																															37.	V		2 R (9 s3 )
41.	E 2 4r													csc s																		(b) cos 1(1 s							) 55
43.	(a)		23 S2 csc											csc s											3	cot						(b) cos 1(1 s						3	) 55
(c)	6s[h s (2 s2 )]																									4.85 km east of the reﬁnery
45.	Row directly to B																				47.					4.85 km east of the reﬁnery
49.			3									3
49.	10 s3 (1											s3 ) ft from the stronger source
51.	a2 3								b2 3 3 2
53.	(b)		(i)					$342,491; $342 unit; $390 unit																											(ii) 400
(iii) $320 unit																			1
55.	(a)		p x 19																1			x				(b)			$9.50
55.	(a)		p x 19																3000			x				(b)			$9.50
57.	(a)		p x 550																	1		x (b)							$175				(c) $100
57.	(a)		p x 550																	10		x (b)							$175				(c) $100
61. 9.35 m												65. x 6 in.																	67.			6
69.	At a distance 5 2 s																						5	from A								71. 21 L					W 2
73.	(a)		About 5.1 km from B																											(b)		C is close to B; C is close to
D; W L s																	x, where x BC																	(c) 1.07; no such
D; W L s										25					x2		x, where x BC																	(c) 1.07; no such
value				(d)						s	41			4 1.6

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A85

EXERCISES 4.8 N PAGE 338															54					73.	1588	5.87 ft s2				75.	62,500 km h2 4.82 m s2
1.	(a)	x2 2.3, x3 3								(b)	No			3.	54	5.	1.1797			77.	(a)	22.9125 mi				(b)	21.675 mi			(c) 30 min 33 s
7.	1.1785			9.			1.25			11.		1.82056420					13. 1.217562			(d) 55.425 mi
15.	0.876726						17.		0.724492, 1.220744
19.	1.412391, 3.057104									21.		0.641714								CHAPTER 4 REVIEW N						PAGE 347
23.	1.93822883, 1.21997997, 1.13929375, 2.98984102																			True-False Quiz
25.	1.97806681, 0.82646233																			True-False Quiz
25.	1.97806681, 0.82646233																			1.	False		3. False			5. True			7.	False	9.	True
27.	0.21916368, 1.08422462											29.		(b) 31.622777						1.	False		3. False			5. True			7.	False	9.	True
27.	0.21916368, 1.08422462											29.		(b) 31.622777						11.	True		13.	False			15.	True		17. True		19.	True
35.	(a)	1.293227, 0.441731, 0.507854														(b)	2.0212			11.	True		13.	False			15.	True		17. True		19.	True
35.	(a)	1.293227, 0.441731, 0.507854														(b)	2.0212
37.	0.904557, 1.855277										39.		0.410245, 0.347810							Exercises
41.	0.76286%																			Exercises
41.	0.76286%																			1.	Abs. max. f 4 5, abs. and loc. min. f 3 1;
																				1.	Abs. max. f 4 5, abs. and loc. min. f 3 1;
EXERCISES 4.9					N		PAGE 345													loc. min. f 3 1					2, abs. and loc. min. f ( 1) 9
1. F x 21 x2 3x C											3. F x 21 x					41 x3 51 x4				3.	Abs. max. f 2				2, abs. and loc. min. f ( 1) 9
1. F x 21 x2 3x C											3. F x 21 x					41 x3 51 x4			C	5.	Abs. max. f				5					3	2
5. F x			2	3			1	2 x C					7. F x 4x5 4 4x7 4							5.	Abs. max. f				; abs. min. f 0 0; loc. max.
5. F x			3 x	3			2 x	2 x C					7. F x 4x5 4 4x7 4						C	f 3 3				1								1
9. F x			4x	3 2			6		7 6	C										f 3 3				2 s3; loc. min. f 2 3 2 3								2 s3
9. F x			4x			7 x				C										7.		9.	8	11.		0	13.	1
				5 4x8																7.		9.	8	11.		0	13.	2
				5 4x8						C1		if	x 0							15.		y
11.	F x 5 4x8									C2		if	x 0											x 6
13.	F u 31 u3 6u 1 2									C											2 0
15.	G sin							5 cos			C											1				9	12	x
15.	G sin							5 cos			C
17.	F x 5ex 3 sinh x										C
19.	F x 21 x2 ln x 1 x2												C							17.			y
21.	F x x5 31 x6								4		23.		x3	x4		Cx	D									y=2
25.	203 x8 3		Cx				D			27.	et		21 Ct2		Dt		E
29.	x 3x2				8				31.	4x3 2			2x5 2		4												x
33.	2 sin t			tan t					4 2s3																		x
33.	2 sin t			tan t					4 2s3																	y=_2
35.	23 x2 3 21 if x 0; 23 x2 3 25											if x 0
37.	2x	4	1	3			5x	2	22x			59
37.	2x		3 x				5x		22x			3
39.	sin cos								5	4			41.	x2 2x3			9x	9		19.	A.		B. y-int. 2							y
43.	x2 cos x 21 x									45. ln x				ln 2 x ln 2						C. None			D. None			F. None
47.	10		49.		b															E. Dec. on ,						F. None				2
47.	10		49.		b															G. CU on , 0 ;										2
51.	y																			G. CU on , 0 ;													x
51.	y																			CD on 0, ; IP 0, 2
	1		F

																				H. See graph at right.
																				H. See graph at right.
	0		1											x
																				21.	A.		B. y-int. 0; x-int. 0, 1							y
																				C. None			D. None							2
53.		y										55.				y				E. Inc. on (41 , ), dec. on ( , 41 )
53.		y										55.				y							1			27				1
																							1			27
							(2, 2)													F. Loc. min. f (4 ) 256
																		F		F. Loc. min. f (4 ) 256
		2																F					1			;
		2																					1
																				G. CU on ( , 2 ), 1,
																						1		1		1				0	1		2	x
		1		(1, 1)					(3, 1)											CD on (2 , 1); IP (2 , 16 ), 1, 0										0	1		2	x
		1		(1, 1)					(3, 1)					_2π		0		2π x		CD on (2 , 1); IP (2 , 16 ), 1, 0
														_2π		0		2π x		H. See graph at right.
		0		1			2		3	x										23.	A. x x 0, 3									y
		_1																		23.	A. x x 0, 3									y
																				B. None			C. None
												59. s t 61 t3 t2								D. HA y 0; VA x 0, x 3
57.	s t 1 cos t sin t											59. s t 61 t3 t2						3t	1	E. Inc. on 1, 3 ; dec. on , 0 ,
61.	s t 10 sin t 3 cos t											6 t 3								0, 1 , 3,											0			x
63.	(a)	s t 450 4.9t2									(b)		s450 4.9 9.58 s							F. Loc. min. f 1 41													x 3
(c) 9.8s450 4.9 93.9 m s													(d) About 9.09 s							G. CU on 0, 3 , 3, ; CD on , 0
67.	225 ft			69. $742.08								71. 13011 11.8 s								H. See graph at right.

A86 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

25. A. x x 8

37.

Inc. on 0.23, 0 , 1.62, ; dec. on , 0.23 , 0, 1.62 ;

B. y-int. 0, x-int. 0

C. None

x 8

loc. max. f 0 2; loc. min. f 0.23 1.96, f 1.62 19.2;

D. VA x 8; SA y x 8

CU on , 0.12 , 1.24, ;

E. Inc. on , 16 , 0, ;

CD on 0.12, 1.24 ; IP 0.12, 1.98 , 1.24, 12.1

dec. on 16, 8 , 8, 0

16, 32

y x 8

2.5

F. Loc. max. f 16 32;

loc. min. f 0 0

2.1

G. CU on 8, ; CD on , 8

H. See graph at right.

27. A. 2,

_0.5

0.4

_20

1.5

B. y-int. 0; x-int. 2, 0

C. None

D. None

39.

0.82, 0.22

; (

2 3, e 3 2 )

E. Inc. on ( 34 , ), dec. on ( 2, 34 )

F. Loc. min. f ( 34 ) 94 s

G. CU on 2,

”_4

,_4œ„6 ’

H. See graph at right.

29. A.

B. y-int. 2

C. About y-axis, period 2

D. None

E. Inc. on 2n , 2n

1 , n an integer; dec. on 2n 1 , 2n

41.

2.96, 0.18, 3.01; 1.57, 1.57; 2.16, 0.75, 0.46, 2.21

F. Loc. max. f 2n

1 2; loc. min. f 2n 2

43.

For C 1, f is periodic with period 2 and has local

G. CU on 2n 3 , 2n

3 ;

maxima at 2n

2, n an integer. For C 1, f has no graph.

CD on 2n

3 , 2n

5 3 ; IP (2n

3 , 41 )

For 1 C 1, f has vertical asymptotes. For C 1, f is con-

tinuous on . As C increases, f moves upward and its oscillations

become less pronounced.

_2π

2π

49.

(a) 0

(b) CU on

53. 3 s3 r2

_π

55.

4 3 cm from D

57.

L C

59. $11.50

A. {x x 1}

61.

1.297383

63.

1.16718557

31.

65.

f x sin x sin 1x

5 2

5 3

B. None

C. About (0, 0)

67.

f x 5 x

5 x

D. HA y 0

69.

f t t2

3 cos t

E. Dec. on , 1 , 1,

f (x 21 x2 x3

4x4

_π

71.

F. None

73.

s t t2 tan 1 t

G. CU on 1, ; CD on , 1

75.

(b)

0.1ex

cos x

0.9

(c)

H. See graph at right.

33.

B. y-int. 0, x-int. 0

C. None

D. HA y 0

E. Inc. on ( , 21), dec. on (21, )

F. Loc. max. f (21 ) 1 2e

G. CU on 1, ; CD on , 1 ; IP 1, e 2

{1,e–@}

” 2,

2 e–!’

77.

79.

(b)

About 8.5 in. by 2 in.

(c)

20 3 in., 20

2 3 in.

35.

Inc. on ( s3, 0), (0, s3 );

1.5

PROBLEMS PLUS N

PAGE 352

dec. on ( , 3 ), (

3, );

11.

3.5 a

2.5

2, 4 ,

loc. max. f (s3 ) 92 s3,

13.

m 2, m2 4

15.

a e1 e

loc. min. f ( s3 )

19.

(a)

T1 D c1, T2 2h sec c1

D 2h tan

c2,

9 s3;

CU on ( s6, 0), (s6, );

T3 s4h2

D2 c1

CD on ( , 6 ),

(0,

6 );

(c)

c1 3.85 km s, c2 7.66 km s, h 0.42 km

_1.5

23.

IP (s6, 36 s6 ), ( s6, 36 s6 )

3 (s2

1) 112 h

CHAPTER 5
EXERCISES 5.1 N PAGE 364
1.	(a)	40, 52			(b) 43.2, 49.2
y							y
			y=ƒ					y=ƒ
5							5
0			5		10 x		0			5		10 x
3.	(a)	0.7908, underestimate						(b)	1.1835, overestimate
y									y
1			ƒ=cos x						1			ƒ=cos x
			ƒ=cos x									ƒ=cos x
0		π π 3π π x							0	π π 3π π x
		8	4	8	2					8	4	8	2
5.	(a)	8, 6.875							(b)	5, 5.375
	y				y				y				y
	2				2					2			2
	0		1	x	0	1	x		0		1	x	0	1	x
(c)	5.75, 5.9375				y						y
					2						2
					0		1	x			0	1	x

(d)

0.2533, 0.2170, 0.2101, 0.2050; 0.2

(a) Left: 0.8100, 0.7937, 0.7904;

right: 0.7600, 0.7770, 0.7804

11.

34.7 ft, 44.8 ft

13. 63.2 L, 70 L

15. 155 ft

17.

lim s1

15i n 15 n

19.

lim

cos

2n 2n

n l i 1

21.

The region under the graph of y tan x from 0 to 4

64 n

n2 n 1 2 2n2

2n 1

23.

(a) lim

(b)

(c)

n l

i 1

25.

sin b, 1

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A87

EXERCISES 5.2 N PAGE 376

1. 6

The Riemann sum represents the sum of the areas of the two

rectangles above the x-axis minus the sum of the areas of the three rectangles below the x-axis; that is, the net area of the rectangles with respect to the x-axis.

3. 2.322986

The Riemann sum represents the sum of the areas of the three rectangles above the x-axis minus the area of the rectangle below the x-axis.

y	ƒ=3-1 x
3	ƒ=3-1 x
2			2
2
1			8	10	12	14
			8	10	12	14
0	2	4	6			x

y
6	ƒ=´-2
5	ƒ=´-2
5
4
3
2
1
0	1	2	x
_1

5.	(a)		4		(b) 6					(c) 10				7.	475, 85										9.	124.1644
11.	0.3084				13.				0.30843908, 0.30981629, 0.31015563
15.
15.				n							Rn			The values of Rn appear to be
				n							Rn

				5			1.933766							approaching 2.
				5			1.933766
				10			1.983524
				50			1.999342
				100			1.999836

17.	x26 x ln 1						x2 dx					19.		x18 s								dx			21.				42
17.	x26 x ln 1						x2 dx					19.		x18 s			2x			x2		dx			21.				42
														n					2		4i n					4
23.	4			25. 3.75						29.		lim							2		4i n					4
23.	3			25. 3.75						29.		lim				1 2 4i n								5			n
	3											n l												5
														i 1
				n				5 i					2
31.		lim				sin		5 i					2
31.		lim				sin		n			n
	n l i 1							n			n	5											43
33.	(a)			4	(b)		10				(c)	3			(d)				2		35.		43
37.	3			9			39.			2.5		41. 0							43.		3		45.			e		5	e			3
37.	3			4			39.			2.5		41. 0							43.		3		45.			e			e
47.	x5		1	f x dx			49. 122
51.	2m x02 f x dx 2M by Comparison Property 8
																				3
55.	3 x14 sx dx 6											57.								3	tan x dx										s3
55.	3 x14 sx dx 6											57.		12							tan x dx					12					s3
														12				y		4						12
				2											x01 x4 dx
59.	0 y0				xe x dx 2 e							69.			x01 x4 dx						71.			21
EXERCISES 5.3 N PAGE 387
1. One process undoes what the other one does. See the
Fundamental Theorem of Calculus, page 387.
3.	(a)		0, 2, 5, 7, 3											(d)					y
(b)			(0, 3)
(c)			x		3																				g
(c)			x		3



																			1

																			0	1										x

A88 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

(a), (b) x2

y=t@

	0								1						x							t
7. t x 1 x3															1
9.	t y y2 sin y																11.						F x s									1		sec x
												arctan 1 x
																															y stan x
13.		h x																							15.														stan x sec2 x
13.		h x														x	2								15.														stan x sec2 x
																x
17.	y				3 1 3x 3																		19.					3		21. 63
17.	y		1						1 3x									2					19.					4		21. 63
			1						1 3x
23.	95		25.				87						27.					1567							29.				403				31.		1		33.			493
35.	ln 3						37.											39.						e2 1						41. 0
43.	The function f x x 4 is not continuous on the interval
2, 1 , so FTC2 cannot be applied.
45.	The function f sec																									tan is not continuous on the
interval 3, , so FTC2 cannot be applied.
47.	2434		49.						2
51.	3.75				y						y=˛
51.	3.75				y

	1
	1
					0																	x
					0												2					x
53.	t x					2 4x2 1																			3 9x2 1
53.	t x									4x2							1										9x2			1
																				sin s
						7 2									3												x
55.	y			3x		7 2				sin x					3												x			57.			s257				59.			29
																							4
																							4
	(a) 2 s									, s													2 sx
61.	(a) 2 s							n		, s				4n 2, n an integer 0
(b)	0, 1 , ( s
(b)	0, 1 , ( s										4n 1, s4n 3 ), and (s																							4n 1, s4n 1 ),
n an integer 0															(c)			0.74
63.	(a) Loc. max. at 1 and 5;																												y
loc. min. at 3 and 7																													1
																													1
(b)	x 9																														2		4				6 8
	1
(c)	1																												0											x
(c)	(2 , 2), 4, 6 , 8, 9																											_1
(d)	See graph at right.																											_1
(d)	See graph at right.
																												_2
65.	41		73.						f x x3 2, a 9
65.	41		73.						f x x3 2, a 9
75.	(b) Average expenditure over 0, t ; minimize average
expenditure
EXERCISES 5.4 N PAGE 397
5. 31 x3 1 x														C					7. 51 x5 81 x4														81 x2 2x C
9.	2t t2							31 t3 41 t4													C							11.			31 x3 4 s								C
9.	2t t2							31 t3 41 t4													C							11.			31 x3 4 s					x			C
13.	cos x								cosh x									C							15.				21	2			csc					C
17.	tan					C

19.	sin x	41 x2		C				20	10	5	0
											0
											_5
											_5
					_10						10
								_6
21.	18	23.	2		1 e		25.	52	2s5
27.	256	29.	63		31.		55	33.			35. 8
27.	15	29.	4		31.		63	33.			35. 8
37.	1	4	39.		2565	34	41. 6		43.	3.5
45.	0, 1.32; 0.84				47.	34
49.	The increase in the child’s weight (in pounds) between the
ages of 5 and 10
51.	Number of gallons of oil leaked in the ﬁrst 2 hours
53.	Increase in revenue when production is increased from
1000 to 5000 units
55.	Newton-meters (or joules)							57.	(a) 23 m		(b)	416	m
59.	(a) v t 21 t2				4t		5 m s	(b) 416 32 m
61.	46 32 kg		63.	1.4 mi			65.	$58,000
67.	(b) At most 40%; 365

EXERCISES 5.5 N PAGE 406

e x

92 x3

1 3 2

5. 41 cos4

21 cos x2

9. 631

3x 2 21

11.

31 2x

x2 3 2

13.

31 ln 5 3x

15.

(1 cos

17. 32 s3ax

bx3

19.

31 ln x 3

21.

2 sin st

23.

71 sin7

25.

32 1

ex 3 2

27.

21 1

z3 2 3

29.

etan x

31.

1 sin x

33.

32 cot x 3 2

35.

ln 1

cos2 x

37. ln sin x

39.

31 sec3x

41.

ln sin 1 x

43.

tan 1x

21 ln 1

45.

74 x

2 7 4 38 x

2 3 4

47.

81 x2 1 4

49.

41 sin4x

0.35

_0.35

51.

53.

1829

55.

63. 31 (2 s2 1)a3

57.

59.

e se

61.

65.	1516		67. 2	69.	ln e	1	71. s	3	31
73.	6		75. All three areas are equal.					77. 4512 L
79.		5	1 cos	2 t	L	81. 5	87.	2 4
		4		5

CHAPTER 5 REVIEW N PAGE 409
True-False Quiz
1.	True		3. True		5. False		7. True		9. True
11.	False		13.	False	15.	False

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A89

Exercises
1.	(a)	8
y
2			y=ƒ
2
			6
0		2	x

(b)	5.7
y
2		y=ƒ
2
		6
0	2	x

3. 21			4			5. 3				7. f is c, f is b, x0x f t dt is a
9.	37		11.			9		13.		76		15.	214	17. Does not exist
						10
19.	31 sin 1					21.		0		23.	1 x 2 ln x				x C
25.	s	x2		4x			C			27. 1 (2 sin2 t				C
29.	2esx			C				31. 21 ln cos x 2					C
33.	41 ln 1				x4				C	35.	ln 1		sec	C	37.	233
39.	2 s		1	sin x					C	41. 645			43. F x x2 1			x3
45.	t x 4x3cos x8									47.		y (2ex esx ) 2x
49.	4 x13 s								dx 4 s			55. 0.280981
					x2			3			3
57.	Number of barrels of oil consumed from Jan. 1, 2000, through
Jan. 1, 2008
59. 72,400						61.		3		63.	c 1.62
65.	f x e2 x 1								2x 1 e x				71. 32
PROBLEMS PLUS N PAGE 413
1.	2			3. f x 21 x							5. 1		7. e 2		9. 1, 2

11. (a) 12 n 1 n (b) 12 b 2b b 1 12 a 2a a 1

17. 2(s2 1)

CHAPTER 6

EXERCISES 6.1 N PAGE 420

32						3. e			1 e			10		5. 19.5						7.	1	9.	1
1. 3						3. e			1 e			3		5. 19.5						7.	6	9.	ln 2 2
11.	31					13.		72			15. 2 2 ln 2							17.		1259		19.	323
21.	38					23.		21	25.			32		27.					ln 2		29.	6.5
31.	23 s			1					33. 0.6407					35.				0, 0.90; 0.04					37. 8.38
31.	23 s	3		1					33. 0.6407					35.				0, 0.90; 0.04					37. 8.38
39.	12s				6		9		41.			117 31 ft			43. 4232 cm2
45.	(a) Car A								(b) The distance by which A is ahead of B after
1 minute							(c) Car A					(d) t 2.2 min
47.	245 s		3					49.		42 3		51.		6
53.	0 m 1; m ln m 1
EXERCISES 6.2 N PAGE 430
1. 19 12									y				y=2-			1	x			y
														2
									1		x=1				x=2
									1



									0			1 y=0 2							x	0			x

5. 162	y		y
	y=9
		(6,9)
	x=0
		x=2œ„y
	0	x	0	x
7. 4 21	y		y
		(1,1)
	y=x
		y=˛
	0	x	0	x

9. 64 15

		¥=x	(4,2)
			(4,2)
			x=2y
		0	x	0	x
11.	6	y		y
		y=œ„x	(1,1)		y=1
		y=œ„x	(1,1)
		y=x
		0
			x	0	x

13. 2 (34

)

y	y=3	”π3,3’	y
”_π3,3’	y=3	”π3,3’

	y=1+sec x
		y=1		y=1
0		x	0	x

A90 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

15. 16 15	y	x=1	y	5.	1 1 e
		x=1		x=1	y		y
	x=¥	(1,1)			y		y
	x=¥	(1,1)			1	y=e_≈
					1	y=e_≈
	0		x 0	x
					0	1 x	0
		(1,_1)			0	1 x	x
		(1,_1)					x
							x

17.	29 30
				y		x=¥				(1,1)															y
						x=¥				(1,1)
								y=≈

_1				0								x														0			x
_1												x														0			x
																						x=_1
19.	7						21.			10					23.					2			25.		7 15
27.	5	14					29. 13								30				31.						tan3 x 2 dx
27.	5	14					29. 13								30				31.				x0 4 1		tan3 x 2 dx
33.				12 1						sin x 2 dx
33.	x0			12 1						sin x 2 dx
35.	x2		2ss2		2 [52 (s														2)2] dy
35.	x2		2ss2		2 [52 (s						1							y2	2)2] dy
37.	1.288, 0.884; 23.780																		39.			118	2
41.	Solid obtained by rotating the region 0 y cos x,
0 x				2 about the x-axis
43.	Solid obtained by rotating the region above the x-axis bounded
by x y2 and x y4 about the y-axis
45.	1110 cm3									47. (a) 196											(b) 838				49. 31 r2h
51.	h2 (r 31 h)									53.							32 b2h				55. 10 cm3				57.				24
59.	31			61.			8
59.	31			61.			15
63.	(a)	8 R x0r s														dy			(b)			2 2r2R
63.	(a)	8 R x0r s							r2 y2							dy			(b)			2 2r2R
65.	(b)	r2h									67.		5	r3					69.				8 x0r s				s			dy
													5											R2 y2				r2 y2
													12											R2 y2				r2 y2
EXERCISES 6.3 N PAGE 436
1.	Circumference 2 x, height x x 1 2; 15

3. 2

7.	16
y	y
	y=4(x-2)@

7y=≈-4x+7

(1,4)		(3,4)
0	2	x	x	2	x
			x
9. 21 2

11. 768 7		13. 16 3	15.	7 15	17. 8 3
19.	5 14	21. x12 2 x ln x dx
23.	x01 2 x	1 sin x 2 x4 dx

		y ssin y dy	27.	3.68
25. x0 2 4
29.	Solid obtained by rotating the region 0 y x4, 0 x 3

about the y-axis

31. Solid obtained by rotating the region bounded by

(i) x 1 y2, x 0, and y 0, or (ii) x y2, x 1, and y 0 about the line y 3

33.	0.13		35.		1	3	37.		8		39.	2 12 4 ln 4
					32
41.	34		43. 34	r3			45.	31 r2h
EXERCISES 6.4 N PAGE 441
1.	588 J		3. 9 ft-lb				5.	180 J			7.	154 ft-lb
9.	(a)	2425 1.04 J				(b)	10.8 cm				11.	W2 3W1
13.	(a)	625 ft-lb				(b)	18754 ft-lb			15. 650,000 ft-lb
17.	3857 J		19.			2450 J			21.	1.06 106 J
23.	1.04 105 ft-lb						25.		2.0 m			29. Gm1 m2	1	1
													a	b

EXERCISES 6.5 N PAGE 445
1.	38	3. 2845	5.	1	1 e 25	7. 2 5
1.	38	3. 2845	5.	10	1 e 25	7. 2 5
9.	(a) 1	(b)	2, 4		(c)
9.	(a) 1	(b)	2, 4		(c)

11. (a) 4				(b) 1.24, 2.81
(c)

15.	3831			17.			50						28 F 59 F							19. 6 kg m
21.	5 4 0.4 L
CHAPTER 6 REVIEW N PAGE 446
Exercises
1.	38	3.		7				5.				34		4		7. 64 15					9.	1656 5
1.	38	3.		12				5.				34		4		7. 64 15					9.	1656 5
	4						2			3 2												2			1
11.	3	2ah h					2			3 2				13.	3							2	x 4 ) dx
11.	3	2ah h												13.	x 3 2 ( 2 x)(cos								x 4 ) dx
15.	(a)	2 15							(b) 6						(c)	8 15
17.	(a)	0.38				(b) 0.87
19.	Solid obtained by rotating the region 0 y cos x,
0 x 2 about the y-axis
21.	Solid obtained by rotating the region 0 x ,
0 y 2 sin x about the x-axis
23.	36			25.		1253			s				m3		27.	3.2 J
23.	36			25.		1253			s		3		m3		27.	3.2 J
29.	(a)	8000 3 8378 ft-lb														(b) 2.1 ft					31.	f x
PROBLEMS PLUS N PAGE 448
1.	(a)	f t 3t2										(b)		f x s					3.		2732
1.	(a)	f t 3t2										(b)		f x s			2x		3.		2732
5.	(b)	0.2261						(c)					0.6736 m
(d)	(i)	1 105 0.003 in s (ii) 370 3 s 6.5 min
9.	y 329 x2																	s						y1 4
11.	(a)	V			h					f		y 2		dy	(c)		f y			kA C				y1 4
			x0

Advantage: the markings on the container are equally spaced.

13. b 2a 15. B 16A

CHAPTER 7

EXERCISES 7.1 N PAGE 457

1.	31 x3 ln x 91 x3	C 3. 51 x sin 5x	1	cos 5x C
			25
5.	2 r 2 er 2	C

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A91

7.	1		x2 cos x				2		x sin			x		2		cos x	C
								2							3

9. 21 2x 1 ln 2x 1 x C
11.	t arctan 4t 81 ln 1								16t2				C
13.	21 t tan 2t 41 ln sec 2t											C
15.	x ln x 2 2x ln x								2x			C
17.	1		2										C
17.	13 e			2 sin 3 3 cos 3									C
19.	3				21. 1 1 e							23.	21 21 ln 2				25.	41 43 e 2
27.	61 (			6 3 s3 )					29.			sin x ln sin x 1						C
31.	32			2	64	ln 2			62
31.	5	ln 2			25	ln 2			125
33.	2 sx sin sx					2 cos sx						C		35. 21			4
37.	1		2	1 ln 1			x		1		2		1		3	C
37.	2 x			1 ln 1			x		4 x				2 x		4	C
39.	2x			1 ex		C										7
														ƒ		F
							3.5										1.5
															1
41.	31 x2 1				x2 3 2 152 1					x2 5 2				C
													4
									F
							_2									2
											f
													_4
43.	(b)			41 cos x sin3x					83 x 163				sin 2x			C
45.	(b)			32 , 158	51.		x ln x 3 3x ln x 2									6x ln x 6x			C
53.	254	754 e 2				55.		1.0475, 2.8731; 2.1828									57. 4 8
59.	2 e				61.	29 ln 3 139						63.		2 e t t2			2t	2 m
65.	2

EXERCISES 7.2 N PAGE 465


1.	51 cos5x 31 cos3x										C				3.	11
																384
5.	1		sin3 x				2				sin5			x			1		sin7 x				C
	3																7
				5											23						41 sin 2
7.	4			9. 3 8								11.					2 sin								C
13.	16			15.		2		s						45 18 sin2							15 sin4 C
										sin
						45
17.	21	cos2x ln cos x												C	19. ln sin x								2 sin x			C
21.	21	tan2 x		C					23.			tan x x						C
25.	51	tan5 t		32 tan3 t							tan t				C		27.			1178
29.	31	sec3x sec x								C
31.	41	sec4x tan2x								ln sec x							C
33.	61	tan6		41 tan4							C
35.	x sec x ln sec x												tan x				C			37.	s		31
																						3
39.	31	csc3		51 csc5								C			41.		ln csc x cot x								C
43.	61 cos 3x				1		cos 13x								C		45.			81 sin 4				1	sin 6	C
					26																			12
47.	21	sin 2x		C					49.				1	tan5 t2				C
													10

A92 \|\|\|\|											APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
51.	41 x2 41 sin x2 cos x2																							C				53.		61 sin 3x 181						sin 9x				C	37.	87 s2 tan 1															x 2											3x 8									C
														π																					1			ƒ			37.	87 s2 tan 1																s2							4 x2 4x										6		C
								F																														F			39.	ln					s								1 1									C
	_π																							π					_2									F		2							sx								1 1
											f																														41.	2						ln 259											43.			103 x2					1 5 3 43 x2										1 2 3			C
																																									45.	2 sx												3								6							6			1					C
																																									45.	2 sx										3 sx									6 sx				6 ln sx							1					C
													_π																						1						47.	ln					ex								2 2								C
55.	0						57.				1					59. 0								61.			2 4					63.			(2	s		2	5 )		47.	ln	ex													1							C
55.	0						57.				1					59. 0								61.			2 4					63.			(2			2	5 )				ex													1
65.	s 1 cos3 t 3																																						2		49.	ln tan t														1 ln tan t												2				C								1
65.	s 1 cos3 t 3																																								49.	ln tan t														1 ln tan t												2				C
EXERCISES 7.3 N PAGE 472																																									51.	(x 21 ) ln x2 x																						2 2x							s7 tan 1 2xs7										C
1.	sx2 9 9x															C					3.		31 x2 18 sx2											9	C						53.	21 ln 3 0.55
5.	24								s3 8 41											7.		s25 x2 25x												C								1							x 2																			1			2 tan x 2 1
9.	ln(sx2									16					x)				C					11.			41 sin 1 2x							21 xs1 4x2						C	55.	2 ln										x									C				59.			5 ln				tan x 2						2		C
13.	61 sec 1 x 3 sx2 9 2x2																												C												61.	4 ln				2							2							63.			1				11	ln 2
15.	161	a4									17.				sx2 7									C																	61.	4 ln				3							2							63.			1				3	ln 2
15.	161	a4									17.				sx2 7									C																	65.	t ln P																	1	ln 0.9P							900					C, where C 10.23
19.	ln (s1										x2 1) x											s1						x2		C				21.	5009						65.	t ln P																	9	ln 0.9P							900					C, where C 10.23
19.	ln (s1										x2 1) x											s1						x2		C				21.	5009						67.	(a)			24,110														9	1					668					1							9438			1
23.	9	sin				1		x			2 3									1	x	2 s5								4x x				2	C						67.	(a)					4879										5x				2				323			2x				1				80,155			3x 7
23.	2	sin						x			2 3									2	x	2 s5								4x x					C												4879										5x				2				323			2x				1				80,155			3x 7
25.	sx2								x				1 21 ln(sx2													x			1	x		21 )			C							1							22,098x												48,935
27.	21 x							1 sx2								2x 21 ln x												1		sx2				2x				C			260,015												x2						x				5
29.	41 sin 1 x2														41 x2 s1 x4											C																4822							ln 5x											2				334			ln 2x						1				3146			ln 3x 7
33.	61 (s48 sec 1 7)																				37.			0.81, 2; 2.10																	(b)	4879							ln 5x											2				323			ln 2x						1				80,155			ln 3x 7
41.	r	s		R2 r2											r2 2 R2 arcsin r R																	43.			2 2Rr2						11,049							ln x2										x				5					75,772								tan 1 2x					1	C
		s																																							11,049							ln x2										x				5					75,772						19		tan 1 2x					1	C
EXERCISES 7.4													N		PAGE 481																										260,015																								260,015						s		19					s	19
EXERCISES 7.4													N		PAGE 481																																																								s							s
						A										B										A				B					C						The CAS omits the absolute value signs and the constant of
1.	(a)					A										B						(b)				A				B					C						integration.
1.	(a)			x					3					3x					1			(b)				x			x		1			x			1 2				integration.
				x					3					3x					1							x			x		1			x			1 2
3.	(a)			A						B						C				Dx					E																EXERCISES 7.5 N PAGE 488
				x					x2						x3						x2				4																EXERCISES 7.5 N PAGE 488
																																																			1
																																									1.	sin x									1					3	x				C
			A												B							C								D											1.	sin x									3 sin						x				C
(b)			A												B							C								D											3.	sin x									ln csc x cot x																	C
(b)	x					3					x					3		2			x	3							x 3			2									3.	sin x									ln csc x cot x																	C
	x					3					x					3					x	3							x 3												5. 4 ln 9																7. e 4 e									4
5.	(a)		1						x		A							x	B		1				Cx				D												9.	2435 ln 3 24225																			11.		21 ln x2 4x											5			tan 1 x 2				C
									x		1							x			1					x2			1												13.	81 cos8										61 cos6											C (or				41 sin4 31 sin6											81 sin8			C)
	At						B						Ct				D					Et				F															13.	81 cos8										61 cos6											C (or				41 sin4 31 sin6											81 sin8			C)
(b)	At						B						Ct				D					Et				F																x				1 x
(b)	t	2					1						t	2			4				t		2			4		2													15.														2				C
		2												2									2					2													15.																		C
																																											s
7.	x			6 ln					x 6										C																						17.	41 x2 21 x sin x cos x																							41 sin2 x							C
9.	2 ln x								5				ln				x 2									C			11.		21 ln 23										(or	41 x2 41 x sin 2x 81 cos 2 x																											C)
13.	a ln					x b										C					15.			67			ln 32														19.	ee x								C							21.				x			1 arctan sx sx														C
17.	275		ln 2 59 ln 3 (or 59 ln 38)																																						23.	409745										25.					3x					233 ln x 4 35 ln x															2			C
19.				1			ln x								5				1				1						1	ln	x 1						C				27.	x ln 1															ex						C				29.		15				7 ln 72
19.			36				ln x								5				6		x				5				36	ln	x 1						C				31.	sin		1				x s1 x2																C
			36																6		x				5				36												31.	sin						x s1 x2																C
21.	21 x2 2 ln x2																4				2 tan 1 x 2									C											33.	2 sin 1											x					1						x		1		s	3 2x x2									C
23.	2 ln x											1 x							3 ln x							2				C											33.	2 sin 1											x		2			1						x	2	1			3 2x x2									C
23.	2 ln x											1 x							3 ln x							2				C																									2										2
25.	ln			x 1											21 ln x2						9 31 tan 1 x 3													C							35.	0							37.						8 41										39.			ln sec						1 ln sec							C
27.	1						2									(1 s2 ) tan								1		(x s2 )					C										41.	tan										21				2 ln								sec					C					43.			32 1		ex 3 2		C
27.	2	ln x									1					(1 s2 ) tan										(x s2 )					C										45.		1		x				3					1 e				x 3
																				23 tan 1 x									1												45.		3		x									1 e									C
29.	21 ln x2										2x					5				23 tan 1 x								2	1			C									47.	ln x 1 3 x 1 1 23 x 1 2 31 x 1 3																																							C
																														1			1 2x					1			49.	ln			s				4x							1 1								C				51.				ln				s	4x2		1	1	C
31.	1	ln			x						1					1	ln x			2			x							1	tan		1 2x					1	C		49.	ln			s											1					1			C				51.				ln				s		2x			C
31.	3	ln			x						1					6	ln x						x			1 s3					tan				s3				C						s4x											1					1																	2x
	1				8										1	ln x							1		ln x2					4				1								1																					2											2
33.	4	ln			3						35.				16	ln x							32		ln x2					4		8 x2				4			C		53.	m x				2 cosh mx														m2 x sinh mx													m3			cosh mx				C

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A93

55.

2 ln s

2 ln(1

)

47. F x

sin3x cos7x

sin x cos7x

sin x cos5x

160

57.

73 x

c 7 3 43 c x

c 4 3

sin x cos3x

sin x cos x

128

256

59.

sin sin x 31 sin3 sin x

61. 2(x 2s

2)esx C

max. at , min. at 0; IP at 0.7,

2, and 2.5

63.

tan 1 cos2 x

65. 32 x

1 3 2 x3 2

0.04

67.

ln (2

) ln (1

)

69.

ex ln 1

71.

21 arcsin x 2

1 x2

73.

81 ln x 2

ln x2

4 81 tan 1 x 2 C

75.

2 x 2

2 ln

77.

32 tan 1 x3 2

79.

31 x sin3 x

31 cos x 91 cos3x

81.

xex2

EXERCISES 7.6 N PAGE 493

1 x

sin 1(

)

7 2x2

sec x tan x

sec x

tan x

tan2 x

ln cos x

11.

e 2

4x2

13.

21 tan2 1 z ln cos 1 z

15.

21 e2 x

1 arctan ex 21 ex

2y 1

87 sin 1

2y 1

17.

4y 4y2

6 4y 4y2 3 2

19.

91 sin3x 3 ln sin x 1

21.

2s3

ex s3

83 ln sec x

tan x

23.

41 tan x sec3x

83 tan x sec x

25.

21 ln x

2 ln[ln x

]

ln x 2

cos 1 e x

27.

e2 x 1

29.

51 ln x5

31.

x10 2

35.

31 tan x sec2 x

32 tan x

37.

41 x x2

2 s

2 ln(s

x 2

39.

1 2x 5 2 61 1

2x 3 2

41.

ln cos x

21 tan2x

tan4x

43.

(a)

both have domain 1, 0 0, 1

45.

F x 21 ln x2 x

1 21 ln x2

1 ;

max. at 1, min. at 1; IP at 1.7, 0, and 1.7

0.6

EXERCISES 7.7 N PAGE 505

1. (a) L2 6, R2 12, M2 9.6

(b)L2 is an underestimate, R2 and M2 are overestimates.

(c)T2 9 I (d) Ln Tn I Mn Rn

3.	(a)	T4 0.895759 (underestimate)
(b)	M4 0.908907 (overestimate)
T4 I M4
5.	(a)	5.932957,	EM 0.063353
(b)	5.869247, ES 0.000357
7.	(a)	2.413790	(b)	2.411453	(c)	2.412232
9.	(a)	0.146879	(b)	0.147391	(c)	0.147219

11.	(a)	0.451948	(b)	0.451991	(c)	0.451976
13.	(a)	4.513618	(b)	4.748256	(c)	4.675111
15.	(a)	0.495333	(b) 0.543321			(c) 0.526123
17.	(a)	1.064275	(b)	1.067416	(c)	1.074915

19. (a) T8 0.902333, M8 0.905620

(b) ET 0.0078, EM 0.0039

21. (a) T10 1.983524, ET 0.016476;

M10 2.008248, EM 0.008248;

S10 2.000110, ES 0.000110

(b)ET 0.025839, EM 0.012919, ES 0.000170

(c)n 509 for Tn, n 360 for Mn, n 22 for Sn

23.	(a) 2.8		(b)	7.954926518		(c) 0.2894
(d)	7.954926521			(e)	The actual error is much smaller.
(f )	10.9		(g) 7.953789422			(h)	0.0593
(i)	The actual error is smaller.					(j)	n 50
25.
25.		n	Ln		Rn		Tn	Mn
		n	Ln		Rn		Tn	Mn
		5	0.742943		1.286599		1.014771	0.992621
		10	0.867782		1.139610		1.003696	0.998152
		20	0.932967		1.068881		1.000924	0.999538


		n	EL		ER		ET	EM
		5	0.257057		0.286599		0.014771	0.007379
		10	0.132218		0.139610		0.003696	0.001848
		20	0.067033		0.068881		0.000924	0.000462

1.1	Observations are the same as after Example 1.

A94 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

27.
27.		n	Tn	Mn	Sn
		n	Tn	Mn	Sn

	6		6.695473	6.252572	6.403292
	12		6.474023	6.363008	6.400206


		n	ET	EM	ES

		6	0.295473	0.147428	0.003292
		12	0.074023	0.036992	0.000206

Observations are the same as after Example 1.

29.	(a)	19.8	(b)	20.6	(c)	20.53
31.	(a)	23.44	(b)	0.3413		33.	37.73 ft s
35.	10,177 megawatt-hours					37.	828	39.	6.0	41.	59.4
43.	y
	1
	0	0.5	1	1.5	2	x

EXERCISES 7.8 N PAGE 515
Abbreviations: C, convergent; D, divergent
1.	(a)	Inﬁnite interval								(b)	Inﬁnite discontinuity
(c)	Inﬁnite discontinuity									(d) Inﬁnite interval
3.	21 1 2t2 ; 0.495, 0.49995, 0.4999995; 0.5
5.	1		7. D				9. 2e 2				11.	D	13.	0	15.		D
5.	12		7. D				9. 2e 2				11.	D	13.	0	15.		D
17.	D	19.				1	21.			D	23.	9
17.	D	19.				25	21.			D	23.	9
25.	21	27.			D		29.			323	31.	D	33.	754
35.	D		37. 2 e							39.	38 ln 2 98
41.	e										43. 2 3
			y											0.5			2
														2	y=
														2
						x 1								9		≈+9
														9		≈+9
	y ex
	y ex
											_7						7
		0			1			x			_7			0			7
		0			1			x						0
45.	Inﬁnite area						20
							20
											y=sec@x
							0							π
							0							2
47.	(a)
47.	(a)								t
									t
				t					y1 sin2x x 2 dx
					2					0.447453
					5					0.577101
					10					0.621306
			100							0.668479
			1,000							0.672957
			10,000							0.673407

It appears that the integral is convergent.

(c)	1
		ƒ=	1
			≈
		©=sin@x
					≈
	1						10
	0.1
49.	C	51.		D	53.	D	55.	57.	p 1, 1 1 p
59.	p 1, 1 p 1 2						65. s2GM R
67.	(a)	y
		1			y=F(t)
					y=F(t)
		0			700		t
							(in hours)

(b)The rate at which the fraction F t increases as t increases

69.	1000
71.		(a)					F s 1 s, s 0														(b)		F s 1 s 1 , s 1
(c)	F s 1 s2, s 0
77.		C 1; ln 2										79.					No
CHAPTER 7 REVIEW N PAGE 518
True-False Quiz
1.	False							3.				False						5.		False						7.	False
9.	(a)				True						(b) False									11.			False				13. False
Exercises
1.	5 10 ln 32											3.			ln 2					5.				2
1.	5 10 ln 32											3.			ln 2					5.				15
7.	cos ln t C															9.		645 ln 4 12425									2) C
11.						1						13.				3		x	3							3
						1										3		x	3							3
	s3 3														3es				(sx2							2 sx
15.	21					ln x 23 ln x 2 C
17.		x sec x ln sec x tan x C
19.		1	ln 9x2 6x 5 91 tan 1[21 (3x 1)] C
19.		18	ln 9x2 6x 5 91 tan 1[21 (3x 1)] C
21.	ln x 2 s																		C
21.	ln x 2 s												x2 4x						C

							sx2			x		1 1
23.		ln														C									tan 1(x s				) C
25.		23 ln x2 1 3 tan 1x s
25.		23 ln x2 1 3 tan 1x s																					2					2
27.	52					29.		0				31.				6 23
33.							x		sin 1								x	C

		s															2
		s		4 x2													2
35.		4s									C					37.					21 sin 2x 81 cos 4x C
						1 s
						1 s				x
39.		81 e 41										41.		1				43. D
39.		81 e 41										41.		36				43. D
45.		4 ln 4 8										47.				34				49.						4
51.		x 1 ln x2 2x 2 2 arctan x 1 2x C
53.	0
55.		41 2x 1 s
55.		41 2x 1 s										4 x2 4x 3											2x 1 s								C
																						ln
																						ln								4x2 4x 3

57.	21 sin xs							4 sin2x										2 ln(sin x s								4 sin2x				) C
61.	No
63.	(a)			1.925444												(b) 1.920915									(c)		1.922470
65.	(a)			0.01348, n 368																			(b) 0.00674, n 260
67.	8.6 mi
69.	(a)			3.8						(b)				1.7867, 0.000646											(c)			n 30
71.	C			73.						2				75.						3		2
71.	C			73.						2				75.						16		2
PROBLEMS PLUS N PAGE 521
1.	About 1.85 inches from the center																										3.	0
7.	f 2													11.							bba a 1 b a e 1
13.	2 sin 1(2														)
13.	2 sin 1(2													5	)
											s
CHAPTER 8

EXERCISES 8.1											N	PAGE 530
										x02																	4
	4s5												s1 sin2 x dx														4	s9y4 6y2 2 dy
1.	4s5			3.									s1 sin2 x dx												5.		y1	s9y4 6y2 2 dy
7.	2	(82 s							1)										9.		1261240			11.	323
	2						82
	243						82
13.	ln(s					1)								15.					ln 3 21
13.	ln(s				2	1)								15.					ln 3 21
17.	s										s				ln(									1) 1 ln(							s		1)
17.			1 e2										2		ln(								1 e2	1) 1 ln(								2	1)
																				s
19.	s			ln(1 s													)				21. 463				23.		5.115840
19.	s		2	ln(1 s												2	)				21. 463				23.		5.115840
25.	1.569619
27.	(a), (b)																												L1 4,
																													L2 6.43,
																													L4 7.50

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES																																																										\|\|\|\| A95
9.	2s												2 ln( s																																		)			11. 212
9.	2s			1 2									2 ln( s																											1					2		)			11. 212
13.		1				(145 s														10 s														)								15.				a2
		1													145															10
		27													145															10
17.	9.023754																19.								13.527296
21.	41				[4 ln(													4) 4 ln(																							1)										4										]
21.	41				[4 ln(									17				4) 4 ln(																					2		1)								s	17	4						s		2		]
											s																										s												s								s
23.	61				[ln(s											3) 3 s																				]
23.	61				[ln(s							10				3) 3 s																10				]
27.						31	a2											(b)					4556		s								a2
27.		(a)				31	a2											(b)					4556		s				3				a2
																					a2b sin 1(s																									a)
29.		(a)				2						b2									a2b sin 1(s																	a2 b2								a)
29.		(a)				2						b2
																											sa2 b2
																ab2 sin 1(s
(b)	2						a2									ab2 sin 1(s															b2 a2 b)

																							sb2 a2
																							sb2 a2
31.		xab 2 c f x s																																									dx						33. 4 2r2
31.		xab 2 c f x s																						1 f x 2																			dx						33. 4 2r2
EXERCISES 8.3 N PAGE 547
1.	(a)				187.5 lb ft2																		(b)							1875 lb														(c)				562.5 lb
3.	6000 lb																5. 6.7 10 4 N																									7.				9.8 103 N
9.	1.2 10 4 lb																						11. 32					ah														13.				5.27 105 N
15.	(a) 314 N																		(b) 353 N
17.		(a)				5.63 103 lb																						(b) 5.06 104 lb
(c)	4.88 104 lb																						(d)					3.03 105 lb																				10; 1; (							, 2110 )
19.		2.5 10 5 N																					21.					230; 237														23.											1
19.		2.5 10 5 N																					21.					230; 237														23.											21
25.		0, 1.6															27.								1										,		e					1							29. (			9		,		9		)
25.		0, 1.6															27.								e						1				,		4												29. (			20		,		20		)
																									1
									2		4														1
31.				s															,																							33.						(2, 0)
				4 (s						1)												4 (s					1)
				4 (s				2		1)												4 (s				2	1)
35. 60; 160; (38, 1)																									37.								0.781, 1.330																	41. (0,										1		)
35. 60; 160; (38, 1)																									37.								0.781, 1.330																	41. (0,										12		)
45.	31		r2h

(c)	x04 s1 4 3 x 3 4 x 2 3 2 dx	(d) 7.7988
29.	s5 ln(21 (1 s5 )) s2 ln(1 s2 )
31.	6


33.	s x		2	[ 1 9x 3 2 10 s										10	]	35. 2 s		2	(s	1 x		1)
33.	s x		27	[ 1 9x 3 2 10 s										10	]	35. 2 s		2	(s	1 x		1)
37.	209.1 m							39. 29.36 in.					41. 12.4
EXERCISES 8.2 N PAGE 537
1.	(a) x01 2 x4 s									dx			(b) x01 2 xs								dx
1.	(a) x01 2 x4 s						1 16x6			dx			(b) x01 2 xs				1 16x6				dx

	1												1
3.	(a) y0 2 tan 1x 1															dx
3.	(a) y0 2 tan 1x 1										1 x2 2					dx
	1					1
(b) y0 2 x 1												dx
(b) y0 2 x 1									1 x2 2			dx
5.	1	(145 s						1) 7.					983
	1				145
	27				145

EXERCISES 8.4 N PAGE 553

1.	$38,000				3.	$43,866,933.33			5.	$407.25
7.	$12,000				9.	3727; $37,753			13.		1 k b		1 k	a	1 k
11.	32 (16s2 8) $9.75 million								13.		1 k b		1 k	a	1 k
													2 k		2 k

15.	1.19 10 4 cm3 s										2 k b			a
15.	1.19 10 4 cm3 s
17.	6.60 L min					19.	5.77 L min
EXERCISES 8.5 N PAGE 560
1.	(a)	The probability that a randomly chosen tire will have a
lifetime between 30,000 and 40,000 miles
(b)	The probability that a randomly chosen tire will have a
lifetime of at least 25,000 miles
3.	(a)	f x 0 for all x and x						f x dx 1
(b)	1 83 s		3	0.35
5.	(a)	1			(b)	21
7.	(a)	f x 0 for all x and x						f x dx 1 (b)				5
11.	(a)	e 4 2.5 0.20					(b) 1 e 2 2.5 0.55					(c) If you
aren’t served within 10 minutes, you get a free hamburger.
13.	44%					(b) 5.21%
15.	(a)	0.0668				(b) 5.21%
17.	0.9545

A96 \|\|\|\|	APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
19. (b) 0; a0	(c) 1x1010	EXERCISES 9.2 N PAGE 578

				1.	(a)		y	(b)	y 0,
							3		y 2,
								(iv)	y 2,
								(iv)	y 2
									y 2
				4x10–10	(i)
		0		4x10–10	(ii)	_3		3 x
					(ii)	_3		3 x
(d)	1 41e 8 0.986		(e)	23 a0
							_3
CHAPTER 8 REVIEW N		PAGE 562					(iii)


Exercises
1.	152	3. (a)				1621	(b)	1041		5.	3.292287				7. 1245
9.	458 lb					11.	(58 , 1)		13.		(2, 32 )		15.		2 2
17.	$7166.67
19.	(a)	f x 0 for all x and x								f x dx 1
(b)	0.3455					(c)	5, yes
21.	(a)	1 e 3 8 0.31							(b)	e 5 4 0.29
(c)	8 ln 2 5.55 min
PROBLEMS PLUS N PAGE 564
1.	32 21 s
1.	32 21 s		3
3.	(a)	2 r r d					(b)	3.36 106 mi2
(d)	7.84 107 mi2
5.	(a)	P z P0 t x0z x dx
(b)	P0 0tH r2							0tHeL H xr r ex H 2 s								dx
(b)	P0 0tH r2							0tHeL H xr r ex H 2 s						r2 x2		dx
	Height s							28				3
7.				2	b			28				3	9. 0.14 m
7.						, volume (27			s6 2)			b	9. 0.14 m

11. 2 , 1

CHAPTER 9

EXERCISES 9.1 N PAGE 571

3.	(a)	21 , 1	5. (d)
7.	(a)	It must be either 0 or decreasing
(c)	y 0		(d) y 1 x 2
9.	(a)	0 P 4200 (b) P 4200
(c)	P 0, P 4200

13. (a) At the beginning; stays positive, but decreases

P(0)

3. III

5. IV

7.	y		9.
			(b)
			(a)
_3		3	x
	_3		(c)
	_3
11.	y		13.
	3
_3		3 x
	_3
15.		4
_3			3
		_2
17.	y	c=3
		2
			c=1
_1	0	1	t
			c=_1
		_2
		c=_3

y 3

_3	3 x
	_3
	y
	3
_3	3 x
	_3

2 c 2; 2, 0, 2

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A97

19.

(a)

(i) 1.4

(ii)

1.44

(iii)

1.4641

27.

(a), (c)

(b) y s2 x C

(b)

y=´

Underestimates

1.5

h=0.1

1.4

h=0.2

h=0.4

1.3

1.2

1.1

1.0

0.1

0.2

0.3 0.4

29.

y Cx2

31.

x2 y2 C

(c)

(i)

0.0918

(ii)

0.0518

(iii) 0.0277

≈-¥=C

It appears that the error is also halved (approximately).

21.

1, 3, 6.5, 12.25

23.

1.7616

xy=k

25.

(a)

(i) 3

(ii) 2.3928

(iii)

2.3701

(iv) 2.3681

(c)

(i)

0.6321

(ii) 0.0249

(iii) 0.0022

(iv) 0.0002

It appears that the error is also divided by 10 (approximately).

27.

(a), (d)

(b)

(c)

Yes; Q 3

(e)

2.77 C

33.

Q t 3 3e 4t; 3

35. P t M Me kt; M

37.

(a)

x a (kt 2

a)2

(b)

tan 1

b x

ksa b

a b

39.

(a)

C t C0 r k e kt r k

(b)

r k; the concentration approaches r k regardless of the

EXERCISES 9.3

PAGE 586

value of C0

41.

(a)

15e t 100 kg

(b) 15e 0.2 12.3 kg

1. y Kx

y Ksx2 1

43.

About 4.9%

45.

t k

y ln sec y

31 x3 x C

CesM kt 1

y s 3 tet et C 2 3 1

u Ae2t t 2 2 1

47.

(a)

dA dt ksA M A

(b) A t

M CesM kt 1

11.

y sx2 9

13.

cos x x sin x y2 31 e3y 32

where C sM sA0

and A0 A 0

15.

u st2 tan t 25

17.

y 4a sin x a

sM sA0

y ex2 2

19.

21.

y Kex x 1

EXERCISES 9.4 N PAGE 598

23.

(a)

sin 1y x2 C

(b)

y sin x2 , s 2 x s 2

1. (a) 100; 0.05

(b)

Where P is close to 0 or 100;

on the line P 50; 0 P0 100; P0 100

y=sin ≈

(c)

150

P¸=140

P¸=120

P¸=80

100

P¸=60

_œπ/2„„„

œπ/2„„„

P¸=40

25.

cos y cos x 1

P¸=20

Solutions approach 100; some increase and some decrease, some

have an inﬂection point but others don’t; solutions with P0 20

and P0 40 have inﬂection points at P 50

(d)

P 0, P 100; other solutions move away from P 0 and

toward P 100

2.5

(a) 3.23 107 kg

(b)

1.55 years

A98

|||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

(a)

dP dt

P 1 P 100 , P in billions

EXERCISES 9.5 N

PAGE 606

265

(b)

5.49 billion (c) In billions: 7.81, 27.72

Yes 3. No

(d)

In billions: 5.48, 7.61, 22.41

5. y 3 e

7. y x2 ln x Cx2

9. y 32 s

C x

(a)

dy dt ky 1 y

(b) y

y0 1 y0 e kt

11.

xsin x2 dx C

13. u

2t 2C

(c)

3:36 PM

sin x

2 t 1

11.

PE t 1578.3 1.0933 t 94,000;

15.

y x 1 3ex

17.

v t3et 2

5et 2

32,658.5

PL t

94,000

1 12.75e 0.1706t

19.

y x cos x x

					130,000
						PL
				P
			(in thousands)				PE
					0		45
					90,000
					1960	1980	2000
						t (year)
13.	(a)	P t mk P0 mk ekt					(b) m kP0
(c)	m kP0, m kP0				(d)	Declining
15.	(a)	Fish are caught at a rate of 15 per week.
(b)	See part (d)			(c)	P 250, P 750
(d)	P						0 P0 250: P l 0;
	1200						P0 250: P l 250;
	800						P0 250: P l 750
	800
	400
	0			40	80	120 t
(e)	P t		250 750ket 25			1200
(e)	P t		1	ket 25
			1	ket 25
where k 111 , 91
						0	120

17.	(b)	P
		1400
		1200
		1000
		800
		600
		400
		200
		0	20	40	60	80	100	t

0 P0 200: P l 0; P0 200: P l 200;

P0 200: P l 1000

(c)	P t	m K P0 K P0 m e K m k K t
		K P0 P0 m e K m k K t

19.	(a) P t P0e k r sin rt sin (b) Does not exist

21.y x 1 ex C

25. y Cx4 52x 1 2

c=3		c=3
c=5		c=5
c=7	5	c=7
	5
	c=1
_3		3
c=_5	_5	c=_5
c=_5		c=_5
c=_3		c=_3
c=_1		c=_1

27.	(a) I t 4 4e 5t	(b) 4 4e 1 2 1.57 A
29.	Q t 3 1 e 4t , I t 12e 4t
31.	P t M Ce kt	P(t)
31.	P t M Ce kt	P(t)
		M
		P(0)
		0
		0	t
			t
33.	y 52 100 2t 40,000 100 2t 3 2; 0.2275 kg L
35.	(b) mt c (c) mt c t m c e ct m m2t c2

EXERCISES 9.6 N PAGE 612

1. (a) x predators, y prey; growth is restricted only by predators, which feed only on prey.

(b) x prey, y predators; growth is restricted by carrying capacity and by predators, which feed only on prey.

3. (a) The rabbit population starts at about 300, increases to 2400, then decreases back to 300. The fox population starts at 100, decreases to about 20, increases to about 315, decreases to 100, and the cycle starts again.

(b)	R		F
	2500		300
	2000	R	300
	2000	R	F
	1500		200
	1000		100
	500		100
	500
	0	t¡ t™ t£	t

(b) t 10 ln 572 33.5

(b) L t 53 43e 0.2t

5.Species 2

200 t=2

		150			t=3
		150

		100				t=1
		100				t=1
					t=4
		50			t=0,5
		50


			0	50 100 150 200 250 Species 1
9.	(a)	Population stabilizes at 5000.
(b)	(i)	W 0, R 0: Zero populations

(ii)W 0, R 5000: In the absence of wolves, the rabbit population is always 5000.

(iii)W 64, R 1000: Both populations are stable.

(d)	R	W
	1500	80
	1500	W
		60

1000

R40

500	20
	20

0	t

CHAPTER 9 REVIEW N PAGE 615

True-False Quiz

1. True

3. False

5. True

7. True

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A99

2000

15. (a) P t 1 19e 0.1t ; 560

17. (a) L t L L L 0 e kt

19. 15 days 21. k ln h h R V t C

23. (a) Stabilizes at 200,000

(b) (i) x 0, y 0: Zero populations

(ii)x 200,000, y 0: In the absence of birds, the insect population is always 200,000.

(iii)x 25,000, y 175: Both populations are stable.

(c)

The populations stabilize at 25,000 insects and 175 birds.

(d)

(insects)

(birds)

45,000

250

35,000

birds

200

25,000

insects

150

15,000

100

5,000

25.

(a)

y 1 k cosh kx a 1 k or

y 1 k cosh kx 1 k cosh kb h

(b) 2 k sinh kb

PROBLEMS PLUS N PAGE 618

1. f x 10ex

5. y x1 n

7. 20 C

x2 L2

(b)

f x

21 L ln

11.

(a)

9.8 h

(b)

31,900

100,000 ft2; 6283 ft2 h

(c)

5.1 h

Exercises

13.

x2 y 6 2 25

(a)

(b)

0 c 4;

y 0, y 2, y 4

CHAPTER 10

(iv)

(iii)

EXERCISES 10.1

PAGE 626

t=5

t= π {0,π@}

(ii)

{1+œ„5, 5}

(i)

1 t

t=0

t=4

(1, 0)

(3, 0)

(a)

y 0.3 0.8

t=0 (0, 0)

5 x

(a)

(b)

y 32 x 133

2 3 x

(_2,3)

(1,5)

t=1

t=2

(b)

0.75676

(_5,1)

(c)

y x and y x; there is a local maximum or minimum

t=0

y (21 x2 C)e sin x

y sln x2 2x3 2 C

(_8,_1)

r t 5et t 2

11.

y 21 x ln x 2 2x

13. x C 21 y2

t=_1

A100 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

(a)

(b) x 41 y 5 2 2,

(7,11)

3 y 11

t=_3

(_2,5)

t=0

(14,_3)

”4

,0’ t=2

t=4

(a)

(b)

y 1 x2, x 0

(0,1) t=0

(1,0) t=1

(2,_3) t=4

y 1 x, y 1

11.

(a)

x2 y2 1, x 0

13.

(a)

(b)

(0,1)

(1,1)

(0,_1)

15.

(a)

y 21 ln x 1

17.

(a)

y2 x2 1, y 1

(b)

31.	(b) x 2 5t, y 7 8t, 0 t 1
33.	(a) x 2 cos t, y 1 2 sin t, 0 t 2
(b)	x 2 cos t, y 1 2 sin t, 0 t 6
(c)	x 2 cos t, y 1 2 sin t, 2 t 3 2
37.	The curve y x2 3 is generated in (a). In (b), only the portion

with x 0 is generated, and in (c) we get only the portion with x 0.

41. x a cos , y b sin ; x2 a2 y2 b2 1, ellipse

43.y

45.(a) Two points of intersection

(b)One collision point at 3, 0 when t 3 2

(c)There are still two intersection points, but no collision point.

47. For c 0, there is a cusp; for c 0, there is a loop whose size increases as c increases.

3		1
	_1		1
	_1	1	2
		1	2
	0
0	1.5	0	1.5
_3		_1

49. As n increases, the number of oscillations increases;

19. Moves counterclockwise along the circle

a and b determine the width and height.

x 3 2 y 1 2 4 from 3, 3 to 3, 1
21. Moves 3 times clockwise around the ellipse							EXERCISES 10.2 N PAGE 636
x2 25 y2 4 1, starting and ending at 0, 2							EXERCISES 10.2 N PAGE 636
23.	It is contained in the rectangle described by 1 x 4							2t 1	y x
and 2 y 3.							1.	3.	y x
and 2 y 3.								t cos t sin t
25.			27.					t cos t sin t
25.	y		27.	y	1		5.	y 2 e x 3	7. y 2x 1
				1 t=	1		5.	y 2 e x 3	7. y 2x 1
				1 t=	2		9.	1	20
							9.	y 6 x	20
		(0,1)	t=1
	(_1,0)		x	t=0		1	x
	(_1,0)			t=0		1	x
	t=0	(0,_1)	t=_1
		(0,_1)	t=_1
								_10	10
29.							11.	1 2 t, 3 4t , t 0	_2
29.		3					11.	1 2 t, 3 4t , t 0
		3						3
							13.	e t, e t 1 et , t 0
	3		3				15.	23 tan t, 43 sec3t, 2 t 3 2
	3		3				17.	Horizontal at 6, 16 , vertical at 10, 0
							17.	Horizontal at 6, 16 , vertical at 10, 0
							19.	Horizontal at ( s2, 1) (four points), vertical at 2, 0
		3					21.	0.6, 2 ; (5 6 6 5, e6 1 5 )

23.	7.5	25.	y x, y x
			y
8.5	3		x
8.5	3		0
	1

27.	(a) d sin r d cos									29. (2716, 299 ), 2, 4
31.	ab					33.	3 e 35.			2 r2 d2
37.	x12 s						dt 3.1678
37.	x12 s			1 4t2			dt 3.1678
	x02
39.		s3				2 sin t 2 cos t dt 10.0367 41. 4s2 2
39.		s3				2 sin t 2 cos t dt 10.0367 41. 4s2 2
43.	s				3 ln(3 s				) s			ln(1 s		)
43.	s		10		3 ln(3 s			10	) s		2	ln(1 s	2	)

45.	s2 e					1				8
45.	s2 e					1				8

2.5

47. e3 11 e 8

				1									21
				1
49.	612.3053						51. 6 s
49.	612.3053						51. 6 s		2, s2			t 0, 4
55.		(a)		15								t 0, 4
	15										15
	15										15

		294		15
(b)		294
57.		x01 2 t2 1 ets											dt 103.5999
57.		x01 2 t2 1 ets						e2t t 1 2 t2 2t 2					dt 103.5999
59.		2	(247 s				64)			61. 56 a2		63. 59.101
		2			13
		1215			13
65.		245 (949 s				1)				71. 41
65.		245 (949 s		26		1)				71. 41
EXERCISES 10.3 N PAGE 647
1.	(a)							”2, π3 ’			(b)

O			3π
”1,_	3π’	_	3π
	4		4

2, 7 3 , 2, 4 3

1, 5 4 , 1,

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A101

(c)
				π
				2
				O
				”_1, π2 ’
	1, 3 2 , 1, 5 2
3.	(a)					(b)
			(1,π)		π
			(1,π)						O
				O						_2π
										_2π
										3
							”2,_2π’
							3
(c)			1, 0				( 1, s3 )
(c)
				3π
				4
				O
					”_2,3π4 ’
	(	s	2, 2 )
		s		s		(ii) ( 2s2, 3 4)
5.	(a)		(i)	(2s2, 7 4)		(ii) ( 2s2, 3 4)
(b)	(i)		2, 2 3		(ii)	2, 5 3
7.						9.
					r=2				¨=π6
	r=1							O
								O
				O			¨=_	π
							¨=_	π		r=4
								2		r=4
11.					¨=7π
						3
					r=3
				r=2
				O
					¨=5π
						3
13.	2s3			15.	Circle, center O, radius 2
17.	Circle, center (0, 23 ), radius 23
19. Horizontal line, 1 unit above the x-axis
21.	r 3 sec				23.	r cot	csc		25.	r 2c cos
27.	(a)			6	(b)	x 3
29.						31.				”1, π2’
										”1, π2’

¨=_	π	O
	6

A102 \|\|\|\|		APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES
33.			35.			69.	Center b 2, a 2 , radius sa			2		2
						69.	Center b 2, a 2 , radius sa				b 2
	O					71.		2.6		73.			3.5
				O
							_3.4		1.8			_3		3
												_3		3
37.		¨=π	39.		¨=π
		3			¨=π
	5	2			8			_2.6					_2.5
	5	2						_2.6					_2.5
	6	1				75.		7
	6	1						7
	3	4
							7		7
41.	5π	¨= π	43.
¨=	5π	¨= π
	6	6
		O						7
		O
				O		77.	By counterclockwise rotation through angle 6,							3,
						77.	By counterclockwise rotation through angle 6,							3,
						or about the origin
						79.	(a) A rose with n loops if n is odd and 2n loops if n is even
45.			47.			(b) Number of loops is always 2n
45.			47.	¨=2π3	¨=π3	81.	For 0 a 1, the curve is an oval, which develops a dimple as
						a l1 . When a 1, the curve splits into two parts, one of which
						has a loop.
				(3,π)	(3,0)
						EXERCISES 10.4 N PAGE 653
						1.	5 10,240	3. 12 81 s3				5. 2	7.	414
						9.	49			11.		4
49.	2		51.
	2
													O
	1			(2, 0)	(6, 0)		O
				(2, 0)	(6, 0)		O
	O	1
						13.				15.		3
53.		55.	(a) For c 1, the inner loop										3
53.		55.	(a) For c 1, the inner loop
		begins at		sin 1 1 c and				¨=	π
		ends at		sin 1 1/c ;				¨=	6
		ends at		sin 1 1/c ;

	for c 1, it begins at		3
O	sin 1	3
		1 c and
	ends at 2 sin 1 1 c .


57.	s3		59.		61. 1																		3
63.	Horizontal at (3				4), ( 3
				2,		2, 3 4);
				2,		2, 3 4);						17.	8	19.		20	21.		2 s3		23.	3	2 s3
			s		s							17.	8	19.		20	21.		2 s3		23.	3	2 s3
													1			9			3			1	1
vertical at 3, 0 , 0, 2															4				5		1			1
vertical at 3, 0 , 0, 2															4				5		1			1
			3		3				, 5 3);			25.	4 s3 3			27.		29.		24	4 s3		31.	2	1
			3		3								4 s3 3							24	4 s3			2
65.	Horizontal at (2 , 3), 0, [the pole], and (2											33.	81	41		35.	41 ( 3 s3 )
vertical at (2, 0), (21 , 2 3), (21 , 4 3)											),	37.	(23,	6), (23, 5 6), and the pole
67.	Horizontal at 3, 2 , 1, 3 2 ; vertical at (23 21 s									3,	),	39.	1,	where 12, 5 12, 13 12, 17 12
(23 21 s			) where		sin 1( 21 21 s			)
(23 21 s		3,	) where		sin 1( 21 21 s		3	)				and 1, where 7 12, 11 12, 19 12, 23 12

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A103

41.	(21 s
41.	(21 s		3, 3), (21 s3, 2 3), and the pole
43.	Intersection at								0.89, 2.25; area 3.46														45.
47.	38			2 1 3 2 1										49. 29.0653								51. 9.6884
	16
53.	16			1												55.		(b)		2		(2	s2 )
53.	3			1												55.		(b)		2		(2	s2 )
_0.75																1.25
_0.75																1.25

				_1
EXERCISES 10.5 N PAGE 660
1. 0, 0 , (81 , 0), x 81																			3. 0, 0 , (0,					1		), y						1
1. 0, 0 , (81 , 0), x 81																			3. 0, 0 , (0,					16		), y						16
		y																					y
					”	1	,0’																				y=		1
						1																							16
																													16
				8
																															x
															x											”0, _		1		’
																												1
																												16
x=_		1
x=_		8

5. 2, 3 , 2, 5 , y 1																7. 2, 1 , 5, 1 , x 1
	(_2, 5)							y															y
	(_2, 5)

									y=1							(_5,_1)							0				x
																(_5,_1)

																	(_2,_1)

												x
																									x=1
9.	x y2, focus ( 41 , 0), directrix x 41
11.	3, 0 , 2, 0															13.	0, 4 , (0, 2s										)
11.	3, 0 , 2, 0															13.	0, 4 , (0, 2s								3		)
						y																y
					œ„5																	4
					œ„5

_3				0									3 x
_3				0									3 x				_2					0	2					x
																	_2					0	2					x
				_œ„5
																						_4
																						_4
15.	1, 3 , (1, s										)					17.			x2		y2		1, foci (0, s											)
										5									x2		y2												5
										5							4																5
					y												4					9
					y		(1,3)
							(1,3)

	1			0					3 x
							(1,_3)

19. 12, 0 , 13, 0 ,			21. 0, 2 , (0, 2s	2	),
y	5	x	y x
	12

y	y
y
y=125 x		y=x
	2
0	x
0	0
12	0	x
	_2

23. 4, 2 , 2, 2 ;

(3 s5, 2);

y 2 2 x 3

0	x
(2,_2)	(4,_2)
(3-œ„5,_2)	(3+œ„5,_2)

25.	Parabola, 0, 1 , (0, 43)
27.	Ellipse, ( s
27.	Ellipse, ( s										2, 1), 1, 1
29.	Hyperbola, 0, 1 , 0, 3 ; (0, 1 s																											)			31. x2 8y
29.	Hyperbola, 0, 1 , 0, 3 ; (0, 1 s																										5	)			31. x2 8y
33.		y2 12 x 1														35.					y 3 2 x 2 2
		x2				y2											x2								y 4 2
37.								1							39.														1
37.	25				21			1							39.									16					1
	25				21											12								16
41.		x 1 2									y 4 2					1								43.		x2				y2	1

			12									16														9				16
			12									16														9				16
45.		y 1 2									x 3 2					1								47.		x2				y2	1

			25									39														9				36
			25									39														9				36
49.				x2											y2			1
49.												3,753,196						1
	3,763,600											3,753,196
51.	(a)				121x2										121y2								1			(b) 248 mi
51.	(a)														3,339,375								1			(b) 248 mi
			1,500,625											a	3,339,375
55.	(a)			Ellipse										a						b c
55.	(a)			Ellipse									(b)		Hyperbola (c) No curve
59.	9.69				61.							b2c			ab ln									a		where c2 a2 b2
59.	9.69				61.										ab ln											where c2 a2 b2
EXERCISES 10.6											N			PAGE 668
1.	r				42										3.	r								15
1.	r		4 7 sin												3.	r						4 3 cos
5.	r				8										7. r									4

			1 sin																2 cos
9.	(a) 1					(b)			Parabola								(c)					y 1
(d)						”	1	, π2 ’
(d)							1
							2								y=1
															y=1

										O

A104 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

11. (a) 41	(b)	Ellipse		(c) y 12
(d)			”4, π2 ’
			”4, π2 ’

	(3,π)	O	(3,0)

123π

”5 , 2 ’

y=_12

25.	r	2.26 108
25.	r	1 0.093 cos


27.	35.64 AU		29.	7.0 107 km		31. 3.6 108 km
CHAPTER 10 REVIEW N PAGE 669
True-False Quiz
1.	False	3.	False		5. True	7. False	9. True

Exercises

13. (a) 31	(b) Ellipse	(c) x 29
(d)		x=29
		x=29

3 π

”2, 2’

”49,π’	”89,0’
	O

3 3π

”2, 2 ’

15. (a) 2	(b) Hyperbola	(c) x 83
(d)
”-43 ,0’	” 41 ,π’
	O

	x=_	3
		8
17.	(a) 2, y 21		1
		_2	2
			y=-1
			2
			_3

(b) r	1
	1 2 sin 3 4

	1. x y 2 8y 12											3. y 1 x
				y									y
							(0,6), t=_4
						(5,1),								(1,1), ¨=0
										t=1

										x								x

	5. x t, y st; x t 4, y t 2;
	x tan2 t, y tan t, 0 t											2
	7. (a) ”4, 2π’													(b) (3	s
																2, 3 4),
																2, 3 4),
3
3									2π					( 3s2, 7 4)
														( 3s2, 7 4)

								3

							O
		( 2, 2s					)

					3
9.										π		11.							¨=		π
9.												11.

										”1, 2’											6
			(2,π)																(1,0)


									O
13.										”1, 3π2 ’		15.					”_3,	3π	’
										”1, 3π2 ’




1																		2
1
2			(2,π)								(2,0)
			(2,π)								(2,0)								y=	3

							O
							O													2

_1
_1

	2													”1, π2 ’			O
	2																O

19. The ellipse is nearly circular	e=0.4	e=1.0
19. The ellipse is nearly circular
when e is close to 0 and becomes


more elongated as e l 1 . At
e 1, the curve becomes a		e=0.6
e 1, the curve becomes a		e=0.8
parabola.

17. r

cos

sin

	19.	0.75

		r=sin ¨
		¨
	-0.3	1.2
		-0.75

21.	2		23.		1
25.		1 sin t			,	1 cos t sin t			27.	(11	,	3 )
25.		1 cos t			,				27.	(11	,	3 )
		1 cos t					1 cos t 3			8		4
29.		Vertical tangent at										y
(23 a, 21 s			3	a), 3a, 0 ;
(23 a, 21 s			3	a), 3a, 0 ;
horizontal tangent at
a, 0 , ( 21 a, 23 s								a)
a, 0 , ( 21 a, 23 s							3	a)	( 3a, 0)				(a, 0)

												0	x

31.	18 33. 2, 3							35. 21 1
37.	2(5 s				1)
37.	2(5 s			5	1)
		2 s				s			2 s
39.		2 s	2 1			s	4 2 1	ln	2 s	4 2 1

	2								s 2 1
	2								s 2 1
41.	471,295 1024
43.	All curves have the vertical asymptote x 1. For c 1, the

curve bulges to the right. At c 1, the curve is the line x 1. For 1 c 0, it bulges to the left. At c 0 there is a cusp at (0, 0). For c 0, there is a loop.

45.	1, 0 , 3, 0				47.	( 2425 , 3), 1, 3
		y					y
		2 œ„2
			(1,0)					(_1,3)
								(_1,3)
	3	0	3	x
		2 œ„2					0	x
	x 2	y 2	1	y 2		x 2	1
49.	25	9	1	51. 72 5		8 5	1
	x 2	8y 399 2					4
53.	25	160,801		1	55.	r	3 cos
57.	x a cot		sin cos , y a 1 sin2
PROBLEMS PLUS N PAGE 672
1. ln 2
3. [ 43 s3, 43 s3] 1, 2
5.	(a) At (0, 0) and (23 , 23 )
(b)						3	3
(b)	Horizontal tangents at (0, 0) and (s2, s4 );
				3	3
vertical tangents at (0, 0) and (s4, s2 )
(d)			y		(g) 23
	y x 1			x
	y x 1

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A105

CHAPTER 11

EXERCISES 11.1 N PAGE 684

Abbreviations: C, convergent; D, divergent

1. (a) A sequence is an ordered list of numbers. It can also be deﬁned as a function whose domain is the set of positive integers.

(b)The terms an approach 8 as n becomes large.

(c)The terms an become large as n becomes large.

3.	0.8, 0.96, 0.992, 0.9984, 0.99968													5. 3, 23 , 21 , 81 ,						1
3.	0.8, 0.96, 0.992, 0.9984, 0.99968													5. 3, 23 , 21 , 81 ,						40
7.	3, 5, 9, 17, 33					9.	an 1 2n 1								11.			an 5n 3
13.	an ( 32 )n 1					15. 31 , 52 , 73 , 94 ,				5	,	6	; yes; 21
13.	an ( 32 )n 1					15. 31 , 52 , 73 , 94 ,				11	,	13	; yes; 21
17.	1	19.	5			21.	1	23.	1					25.	0			27.	D
29.	0	31.	0			33.	0	35.	0					37.	1			39.	e 2
41.	ln 2	43. D					45.	D	47. 1						49. 21
51.	D	53.	0
55.	(a)	1060, 1123.60, 1191.02, 1262.48, 1338.23																(b) D
57.	1 r 1
59.	Convergent by the Monotonic Sequence Theorem; 5 L 8
61.	Decreasing; yes						63.	Not monotonic; no
65.	Decreasing; yes						67.	2	69. 21 (3 s								)
65.	Decreasing; yes						67.	2	69. 21 (3 s							5	)
71.	(b)	21 (1 s			)		73.	(a) 0			(b)			9, 11
71.	(b)	21 (1 s		5	)		73.	(a) 0			(b)			9, 11
EXERCISES 11.2 N PAGE 694
1.	(a) A sequence is an ordered list of numbers whereas a series is
the sum of a list of numbers.
(b)	A series is convergent if the sequence of partial sums is a con-
vergent sequence. A series is divergent if it is not convergent.
3.	2.40000, 1.92000,								1

2.01600, 1.99680,2.00064, 1.99987,2.00003, 1.99999,2.00000, 2.00000; convergent, sum 2

5. 1.55741, 0.62763,0.77018, 0.38764,2.99287, 3.28388,2.41243, 9.21214,9.66446, 9.01610; divergent

7. 0.29289, 0.42265, 0.50000, 0.55279, 0.59175, 0.62204, 0.64645, 0.66667, 0.68377, 0.69849; convergent, sum 1

sand

ssnd

sand

ssnd

_10

{sn}

{an}

65. The series is divergent.

A106 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

9.	(a) C		(b) D				11.	9		13.	D		15. 60				17. 71
19.	D	21.			D		23.	D		25.	25	27.		D		29. D
31.	D	33.			e e 1			35. 23			37.		116			39.	e 1
41.	92	43. 1138 333						45.		5063 3300
47.	3 x 3;						x	49.		41 x 41 ;						1
47.	3 x 3;						x	49.		41 x 41 ;				1		4x
					3		x							1		4x
51.	All x;				2			53. 1
51.	All x;		2 cos x					53. 1
			2 cos x
55.	a1 0, an						2		for n 1, sum 1

						n n 1
57.	(a) Sn			D 1 c n					(b)	5	59.	1	(		1)
				D 1 c n								1		3
												2		3
					1 c							2	s
					1 c

63. n n 1

71. sn is bounded and increasing.

73. (a) 0, 19 , 29 , 13 , 23 , 79 , 89 , 1

75.	(a)	21 , 65 , 2423 , 120119 ;	n 1 ! 1		(c) 1

				n 1 !
EXERCISES 11.3 N PAGE 703
1.	C	y
		y=		1
				x1.3

	a™		a£	a¢	a∞ . . .
0	1	2	3	4	x

21.	1.0000, 0.6464,							1
0.8389, 0.7139, 0.8033,									ssnd
0.7353, 0.7893, 0.7451, 0.7821,
0.7505; error 0.0275								0						10
0.7505; error 0.0275								0	sand					10
									sand
								_1
23.	5	25.	4		27.	0.9721		29.		0.0676
31.	An underestimate					33. p is not a negative integer
35.	bn is not decreasing
EXERCISES 11.6 N PAGE 719
Abbreviations: AC, absolutely convergent;
CC, conditionally convergent
1.	(a) D	(b)		C	(c)	May converge or diverge
3.	AC	5.	CC		7.	AC	9.	D		11.	AC	13. AC
15.	AC	17.		CC		19. AC		21.		AC	23.	D
25.	AC	27.		D	29. D		31. (a) and (d)
35.	(a) 960661 0.68854, error 0.00521
(b)	n 11, 0.693109
EXERCISES 11.7 N PAGE 722
1.	C	3. D		5. C		7. D		9.		C	11. C		13. C
15.	C	17.	D		19.	C	21.	C		23.	D	25.	C
27.	C	29.	C		31.	D	33.	C		35.	C	37.	C
EXERCISES 11.8 N PAGE 727
1.	A series of the form n 0 cn x a n, where x is a variable
and a and the cn’s are constants
3.	1, 1, 1			5.	1, 1, 1			7. , ,

3.	D	5. C		7.	C	9.	D	11.	C	13.	D	15. C	9.	2, 2, 2				11. 21 , ( 21 , 21 ]					13.	4, 4, 4
3.	D	5. C		7.	C	9.	D	11.	C	13.	D	15. C	15.	1, 1, 3				17. 31 , [ 133 , 113 )					19. , ,
17.	C	19.	C		21.	D	23.	C	25.	C	27.	p 1	15.	1, 1, 3				17. 31 , [ 133 , 113 )					19. , ,
29.	p 1			31.	1,								21.	b, a b, a b						23.	0, {21 }			25.	41, [ 21, 0]
33.	(a)	1.54977, error 0.1					(b)	1.64522, error 0.005					27.	, ,				29.		(a) Yes		(b) No				31. k k			33. No
(c)	n 1000												35.	(a)	,
35.	0.00145			41.	b 1 e								35.	(a)	,
35.	0.00145			41.	b 1 e								(b), (c)							2	s¸	s™	s¢
													(b), (c)							2	s¸	s™	s¢
EXERCISES 11.4				N PAGE 709																				J¡
																								J¡
1. (a) Nothing				(b) C		3.	C	5.	D	7. C		9. C			_8									8
11.	C	13.	C		15.	C	17.	D	19.	D	21.	C
23.	C	25.	D		27.	C	29.	C	31.	D
33.	1.249, error			0.1		35.	0.76352, error 0.001													_2	s¡	s£	s∞
45.	Yes																			_2	s¡	s£	s∞
45.	Yes													1, 1 , f x 1 2x 1 x 2
													37.	1, 1 , f x 1 2x 1 x 2										41.	2
EXERCISES 11.5				N PAGE 713									EXERCISES 11.9 N PAGE 733
1.	(a) A series whose terms are alternately positive and												EXERCISES 11.9 N PAGE 733
1.	(a) A series whose terms are alternately positive and																										1

negative		(b) 0 bn 1 bn and limn l bn 0,											1.	10	3.	1 nx n, 1, 1							5.	2				x n,	3, 3

																											n 1
where bn an				(c)		Rn bn 1										n 0								n 0	3
																n 0								n 0	3
3.	C	5. C		7.	D	9.	C	11.	C	13.	D						1
3.	C	5. C		7.	D	9.	C	11.	C	13.	D		7.	1 n			1		x 2 n 1,	3, 3		9. 1 2 x n, 1, 1
15.	C	17.	C		19.	D							7.	1 n			n 1		x 2 n 1,	3, 3		9. 1 2 x n, 1, 1
15.	C	17.	C		19.	D								n 0	9										n 1

11.

1 n 1

x n, 1, 1

n 1

n 0

13.

(a)

1 n n 1 x n, R 1

n 0

(b)

1 n n 2 n 1 x n, R 1

2 n 0

(c)

1 nn n 1 x n, R 1

n 2

x n

n 2

15.

ln 5

n , R 5

17.

n 1

x n, R 2

n 1 n5

n 3

1 n

19.

n 1 x 2n 1, R

n 0

s£

0.25

s¡

s∞

s™

s¢

s™

s¢

s∞

s¡

s£

_0.25

2 n 1

s£

s™

21.

, R 1

s¡

n 0

2n 1

t 8n 2

23.

, R 1

n 0

2 n 1

25.

C 1 n 1

, R 1

n 1

27.

0.199989

29.

0.000983

31.

0.09531

33.

(b) 0.920

37.

1, 1 , 1, 1 , 1, 1

EXERCISES 11.10

N PAGE 746


1.	b8 f 8 5 8!						3. n 1 x n, R 1
								n 0

5.	n 1 x n, R 1
	n 0
						2 n 1
7.		1 n						x 2 n 1, R

					2n
	n 0				2n		1 !
		5	n						x	2 n 1
9.		5		x n , R				11.	x		, R
9.				x n , R				11.	2n 1 !		, R
	n 0	n!						n 0	2n 1 !

13. 1 2 x 1 3 x 1 2 4 x 1 3 x 1 4,

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A107

15.

x 3 n, R

n 0

17.

1 n 1

2n, R

2n !

n 0

1 3 5 2n 1

19.

1 n

9 n, R 9

2n 1

n 0

1 3 5 2n 3

1 n 1

25.

x n, R 1

n 2

n 1 n 2

27.

1 n

x n, R 2

n 4

n 0

2 n 1

29.

1 n

x 2 n 1, R

2n 1 !

n 0

31.

x n , R

n 0

33.

1 n

x 4 n 1 , R

2 n

2n !

n 0

1 3 5 2n 1

21 x 1 n

35.

n! 2

3 n 1

x 2 n 1, R

n 1

1 n 1 2

2 n 1

37.

x 2 n, R

n 1

2n !

39.

1 n

x 4 n, R

2n !

n 0

1.5

T¸=T¡=T™=T£

_1.5

1.5

Tˆ=T˜=T¡¸=T¡¡

_1.5

T¢=T∞=Tß=T¶

n 1

41.

x n, R

n 1

n 1 !

T£

T∞

T¡

T™

T¢

Tß

T™

T£

T¢ T∞ Tß

43.

0.81873

1 3 5 2n 1

45.

(a)

x 2 n

n 1

1 3 5 2n 1

(b)

1 2

x 2 n 1

n 1

A108 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

							x 6n 2
47.	C		1 n									, R
47.	C		1 n				6n					, R
		n 0					6n	2 2n !
						1
49.	C		1 n			1			x 2 n, R					51. 0.440


		n 1					2n 2n !
53.	0.40102		55. 31					57.			1		59.	1 23 x 2 2425 x 4
53.	0.40102		55. 31					57.			120		59.	1 23 x 2 2425 x 4
61.	1 61 x 2			7	x 4			63.		e x 4
61.	1 61 x 2			360	x 4			63.		e x 4
65.	1		67.			e 3 1
65.	1	2	67.			e 3 1
	s

EXERCISES 11.11 N PAGE 755

1. (a) T0 x 1 T1 x , T2 x 1 12 x 2 T3 x , T4 x 1 12 x 2 241 x 4 T5 x ,

T6 x 1 12 x 2 241 x 4 7201 x 6

				T¢=T∞		2
									T¸=T¡

								f
			_2π						2π
			_2π						2π

					Tß	_2 T™=T£
(b)
(b)		x	f		T0 T1		T2 T3	T4 T5		T6
		x	f		T0 T1		T2 T3	T4 T5		T6

			0.7071		1		0.6916	0.7074		0.7071

	4
	4
			0		1		0.2337	0.0200		0.0009

	2
	2
			1		1		3.9348	0.1239		1.2114

3. 12 14 x 2 18 x 2 2 161 x 2 3

T£

	2	0			4
	2	6		2
		1 x			3
5. x		1 x
		1.1
			T£	f
			T£
		0			π
					π
					2

T£

_1.1

7. x 16 x 3

1.6

T£

f_1.6

9. x 2x 2 2x 3

1.5

							T£ f			4				_4										3
										4								4						3				4
																					2		8						3
11.	T5 x 1 2 x												2 x													x
														3							4				15				4
															10							4			64					5
							T¢ T∞									x											x
							T¢ T∞
					5
					T£			f
					T™			f
					T™
																						T¢
																					π
																					2			T™
																								T™
					0																					2
					0									π												2
														π												f
														4												f
														4							T∞		T£
																					T∞		T£
					_2
13.	(a)	2 41 x 4								1	x 4 2							(b) 1.5625 10 5
13.	(a)	2 41 x 4								64	x 4 2							(b) 1.5625 10 5
15.	(a)	1 32 x 1 91 x 1 2															4		x 1 3							(b)		0.000097
15.	(a)	1 32 x 1 91 x 1 2															81		x 1 3							(b)		0.000097
17.	(a)	1 21 x 2				(b) 0.0015										19.			(a)			1 x 2					(b)		0.00006
21.	(a)	x 2 61 x 4					(b)		0.042					23.				0.17365								25.		Four
27.	1.037 x 1.037													29. 0.86 x 0.86
31.	21 m, no				37.			(c) They differ by about 8 10 9 km.
CHAPTER 11 REVIEW N PAGE 759
True-False Quiz
1.	False		3. True						5.			False						7.			False
9.	False		11.		True				13.					True				15.				False
17.	True		19.		True
Exercises
1.	21	3.	D		5.			0				7.		e 12		9.				2			11.			C
13.	C		15. D					17.		C				19.		C					21.		C			23.			CC
25.	AC		27.	1				29.				4				31.				e e			35. 0.9721
25.	AC		27.	11				29.				4				31.				e e			35. 0.9721

37. 0.18976224, error 6.4 10 7
41.		4, 6, 2						43.		0.5, [2.5, 3.5)
															2 n																2 n 1
															2 n																2 n 1
	1							1												s3
45.			1 n
										x															x
										x							2n 1 !								x
		2 n 0						2n !				6					2n 1 !											6
																					x		n
47.		1 nx n 2, R 1											49.								x			, R 1
47.		1 nx n 2, R 1											49.											, R 1
		n 0															n 1					n
									x 8 n 4
51.			1 n								, R
51.			1 n								, R
		n 0					2n 1 !
	1					1 5 9 4n 3
53.																						x n, R			16
53.	2									n! 2		6 n 1										x n, R			16
	2			n 1						n! 2
										x n
55.		C ln x
55.		C ln x								n n!
									n 1	n n!
57.		(a)	1 21 x 1 81 x 1 2																1			x 1 3
57.		(a)	1 21 x 1 81 x 1 2																16			x 1 3
(b)	1.5																						(c)		0.000006
			T£
			f
	0																2
59.	61
PROBLEMS PLUS N PAGE 762
1.	15! 5! 10,897,286,400
3.	(b)		0 if x 0, 1 x cot x if x k , k an integer
5.	(a)		sn 3 4n, ln 1 3n, pn 4n 3n 1																						(c)			52 s	3
9.	1, 1 ,				x 3 4x 2 x										11.		ln 21
9.	1, 1 ,							1 x 4							11.		ln 21
13.		(a)	101250		e n 1 5 e n 5											(b)				101250
APPENDIXES
EXERCISES A N PAGE A9
1.	18		3.					5.		5 s						7.				2 x
1.	18		3.					5.		5 s			5			7.				2 x
									x 1			for		x 1
9.	x 1 x 1											for		x 1										11. x 2 1
13.		2,														15.	1,

	_2							0															_1		0
17.		3,														19.		2, 6

							3													2										6
21.		0, 1														23.	[ 1, 21 )

							0	1															_1		1
																											2
25.		, 1 2,														27.	[ 1, 21 ]

							1	2															_1		1
																										2

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A109

29.	,	31.	( s
29.	,	31.	( s		3, s3 )

				_œ„3		0		œ„3
33.	, 1	35.	1, 0 1,

	0 1		_1			0	1

37. , 0 (14 , )

39.10 C 35 41. (a) T 20 10h, 0 h 12

(b) 30 C T 20 C														43. 23				45.		2, 34
47.	3, 3						49. 3, 5									51. , 7 3,
53.	1.3, 1.7							55.				4, 1 1, 4
57.	x a b c ab													59.		x c b a
EXERCISES B N PAGE A15
1. 5	3.			s							5. 2 s					7. 2		9.		29
1. 5	3.			s	74						5. 2 s				37	7. 2		9.		29
17.		y	x=3													19.					y		xy=0
			x=3																				xy=0

	0					3						x								0					x

21.	y 6x 15										23.			2x 3y 19 0
25.	5x y 11										27.			y 3x 2				29.			y 3x 3
31.	y 5			33.						x 2y 11 0								35.		5x 2y 1 0
37.	m 31 ,										39.			m 0,				41.					m 43 ,
	b 0													b 2									b 3
		y														y							y

																0		x					0			x
																	y=_2
	0							x									y=_2
																_2								_3

43.				y									x			45.						y

				0																0					x
47.				y												49.			y
47.				y												49.			y		y		4
																					y		4
																								x	2

	_2			0					2				x
	_2			0					2				x

																		0							x

A110 |||| APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES

51.		y		y=1+x				27.		Parabola				29. Parabola
51.		y						27.		Parabola				29. Parabola
										y						y
				0, 1													2

	0				x
	0				x						(3,4)	0							4 x
				y=1-2x								0							4 x

																	2
																	2

										0			x
53.	0, 4			55. (a)	4, 9	(b) 3.5, 3	57. 1, 2


								31.		Ellipse		33.
59.	y x 3		61.		(b) 4x 3y 24 0				y						y	(3, 9)
																(3, 9)
EXERCISES C N PAGE A23
EXERCISES C N PAGE A23									0	1	3		5 x
1. x 3 2 y 1 2 25						3. x 2 y 2 65
																0		x
5. 2, 5 , 4			7. ( 21 , 0), 21			9. (41 , 41 ), s10 4										0		x

								35.		y x 2 2x
11.	Parabola					13. Ellipse		35.		y x 2 2x
		y				y		37.				39.
	0										y					y
	0


_4		0		4 x
			0		1	x
	_2		0		1	x
	_2					0	1	x
						0	1	x
						1

15. Hyperbola	17. Ellipse	EXERCISES D N	PAGE A32

	y				y		1. 7 6			20
	y=54 x				y	1	1. 7 6		3.	20
							11.	67.5		13.	3
_5	0	5	x	_ 1		1	17.		y
_5	0	5	x	_ 1	0	1	x
					0
				2		2
	y=_	4				_1
	y=_	5 x							0		x
								315°	0		x
								315°

19. Parabola

21. Hyperbola

21.

y=_3x

2 rad

5.	5	7. 720°	9. 75°
cm	15.	32 rad 120

19.y

3π

_ 4

23. Hyperbola		25. Ellipse
	y	y
		(1,2)

0		x
		0

23.

sin 3 4 1

2, cos 3 4 1

2, tan 3 4 1,

csc 3 4

2, sec 3 4

2, cot 3 4 1

25.

sin 9 2 1, cos 9 2 0, csc 9 2 1, cot 9 2 0,

tan 9 2 and sec 9 2 undeﬁned

27.

sin 5 6 21 , cos 5 6 s

2, tan 5 6 1 s

csc 5 6 2, sec 5 6 2

3, cot 5 6

29.

cos 54 , tan 43 , csc 35 , sec

45 , cot 34

APPENDIX H ANSWERS TO ODD-NUMBERED EXERCISES |||| A111

31.		sin s						3, cos 32 , tan s													2, csc 3 s
31.		sin s			5			3, cos 32 , tan s												5	2, csc 3 s					5,
cot	2
cot	2					5
				s
33.		sin 1 s											cos 3 s					tan				31 ,
33.		sin 1 s										10,	cos 3 s				10,	tan				31 ,
csc s									sec s						3
csc s				10,					sec s					10	3
35.		5.73576 cm										37. 24.62147 cm							59.			1	(4 6 s		)
																						1		2
																						15		2
61.		1	(3 8 s							)	63. 2524			65.		3, 5 3
		1					2
		15					2
67.		4, 3 4, 5 4, 7 4												69.		6,			2, 5 6, 3 2
71.	0, , 2										73. 0 x 6 and 5 6 x 2
75.		0 x 4, 3 4 x 5 4, 7 4 x 2

77.y

1		1
	2

0		π	5π	x

79.y

0 π π	3π 2π	5π 3π	x
2	2	2

81.y

	_π		0						π		2π	x
89.	14.34457 cm2
EXERCISES E N PAGE A38
1.	s		s		s		s		s		3. 34 35 36
1.	s	1	s	2	s	3	s	4	s	5	3. 34 35 36
5.	1 31 53 75 97								7. 110 210 310 n10
												10
9.	1 1 1 1 1 n 1										11.	i

i 1

	19			i				n					5			n
13.							15.	2i				17. 2i				19. x i
			i 1
	i 1							i 1						i 0		i 1
21.	80	23. 3276							25.	0			27.		61	29. n n 1
31.	n n 2 6n 17 3								33. n n 2 6n 11 3
35.	n n 3 2n 2 n 10 4
41.	(a)		n 4		(b)		5100 1			(c)			97		(d)	an a0
													300
43.	31	45.			14			49.	2n 1 n 2 n 2
EXERCISES G N PAGE A54
1.	(b)	0.405
EXERCISES H N PAGE A62
1.	8 4i				3.		13 18i			5.			12 7i			7. 1311 1310 i
9.	21 21 i				11.		i		13.	5i			15.		12 5i, 13
17.	4i, 4				19.		23 i		21.	1 2i
23.	21 (s					2)i		25. 3 s				cos 3 4 i sin 3 4
					7						2
27.	5{cos[tan 1(34 )] i sin[tan 1(34)]}
29.	4 cos 2 i sin 2 , cos 6 i sin 6 ,
21 cos 6 i sin 6
31.	4 s			cos 7 12 i sin 7 12 ,
			2

(2 s2 ) cos 13 12 i sin 13 12 , 14 cos 6 i sin 6

33.

1024

35. 512 s3 512i

37.

1, i, (1 s

) 1 i

39. (s

2) 21 i, i

1 Re

41.

i 43.

21 (s

2)i

45. e 2

47.

cos 3 cos3 3 cos sin2 ,

sin 3 3 cos2 sin sin3

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