Bear H.S. Understanding Calculus

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Answers

21.13 0

21.14

21.15 0

21.16

21.17 0

21.18

21.19 0

21.20

21.21

~Ixln

0 if

Ixl

~Ixln

if [x] > 1

21.22

31xl

0 if

Ixl

< 1; n

31xl

[r]

21.23

(lOg;PX

0 if

Ixl

(log;JIXl

Ixl

> 1

21.24

lxl

0 if [x] < 1·

2nl;ln

Ixl

> !

---,;r-

n 2

21.25 (n

2+1)lxl

0 if

Ixl

< 3· (n

2+1)lxl

Ixl

> 3

+n ' 3

+n -

21.26

n!lxl

0 if x = 0;

n!lxl

if x

=1=

n!lxl

0 . .

n!lxl

I I

21.27

(n+2)!

Ixl

(n+2)!

1 X > 1

21.28

(n+~!!lxln

0 if

Ixl

< 1;

(n+~!lxln

[x]

CHAPTER

22.1 1

22.2

22.3 I

22.4 diverges

22.5 1

22.6 4

22.7 diverges

22.8 diverges

22.9 diverges

22.10 2

22.11

diverges

22.12 I

22.13 £

22.14

22.15 diverges

22.16

l;~

22.17 converges

22.18 converges

22.19 converges

22.20 converges

22.21 n

22.22

287

288

22.23 rr

22.24

22.25 (iii)

2e-

= 2,

3e-

dx = 3

·2

4e-

dx = 4 . 3

·2,

...

,10

xne-

= n!

CHAPTER

23.1 500

23.2

!. 100 . 101 = 5050

23.3 210

23.4 100

23.5 N

23.6

23.7 72.38672 =

Hf~>[;n

23.8 diverges;

n~1

23.9 converges; proper alternating series

23.10 converges; proper alternating series

23.11 converges; proper alternating series

23.12 diverges;

-I

1_ > !

ogn

23.13 diverges; logn > !

n n

23.14 converges; geometric series

23.15 converges; integral test

23.16 diverges;

23.17 diverges; integral test

23.18 diverges; integral test

23.19 converges;

sin!

n n n

23.20 diverges;

n~n

23.21 diverges;

. 2

(2)n

23.22 converges, 3n+4n < 3

23.23 a

< a if 0 < a < 1

23.24 sin

a < a if 0 < a < 1

CHAPTER

24.1 1

24.2

24.3 4

24.4

24.5 5

24.6 I

24.7

24.8

Answers

Answers 289

24.9 00

24.10 1

24.11

24.12 1

24.13 10

24.14 1

24.15

(2,4)

24.16

(~,~)

24.17

(-00,00)

24.18

(-6,4)

24.19 e

= 1 + 2x +

t(2x)2

t(2x)3

...

;

(-00,00)

24.20

= 1 - x +

tx2

tx3

+ ... ;

(-00,00)

24.21

log(l-x)

-x

~x2

~x3

~X4

...

;

(-1,1)

2 2

3 2

4 2

S ( I 1)

24.22 log(1 + x) = x - x + T

S-X

..•

;

-2>

24.23 01X)2 = 1 - 2x + 3x

- 4x

+ ;

(-1,1)

2424

-1--I-x3+x6-x9+XI2_

(-11)

l+x

- , ,

24.25

sin2x

= 2x -

t(2x)3

~(2x)s

...

;

(-00,00)

24.26

tan-I

~(~)3

~(~)s

~(~)7

+ ... ;

(-2,2)

24.27 cosx

= 1 - t

~x8

~x12

...

;

(-00,00)

24.28

O:x)2

= x - 2x

+ 3x

- 4x

+ ... ;

(-1,1)

sin x x

( )

24.29 x = 1 - 3f + Sf - 7f + ;

-00,

24.30

l:x

2=x-X

3+x

S-x

;

(-1,1)

24.31 log(1

= x

~x4

~x6

~x8

+ ... ;

(-1,1)

CHAPTER

25.1

PI(x) =

-1

+ 3x; P

2(x)

-1

+ 3x - 2x

;

3(x)

-1

+ 3x - 2x

+ 5x

P4(X) = P

(x )

25.2 P

(x )

I-x

-x

25.3

(x )

= x

-1x3

25.4 P

(x )

= x

answer to (b) is the same

25.5

P5(X)

= 1 +

!(x

- 1) +

ft)(x

- 1)2 +

t~(x

- 1)3

+1-

(-S·3)

(x _ 1)4 +

(7·5·3) (x _

l)s

4! -2

S! 2

25.6

P4(~)

= 1.39558

R4(~)

~(~)s

= .00007

25.7 Jo

P3(X)

dx =

= .3095

R4(X

~XIO;

4(X

l1~S!

= .0008.

25.8

P4(J"X)dx =

ps(J"X) dx = .7639;

IR4(J"X) I= IRs(J"X) I

~lxI3;

1 3 d - 0003.

Jo 6fx x - . ,

cos

dx = .7639 ± .0003.

290

25.9 SIO = 1.4627

25.10

P5(X

dx = 1.4625

1.4625 + .0001 < Jo

dx < 1.4625 + .0003

CHAPTER

Answers

26.1 cos

.9688. Two terms are enough.

26.2 sin

.0998333 ±

10-

26.3 .7786 <

e-~

< .7788

26 4

( )

1 1 1 2 2 1 2·5 3 1 2·5·8 4

• 4 X - + 3

- 2] .

+ 3!33

4Iyr-x

265

(1-x)4

-l-!x

_l.l.x

_l.~x3

_l.!:1:1

.•.

· - 2 2! 2

2"'

4! 2

(

+x2)4

- 1 +

1 x

+ l.1.3

6 _l.1.3.5

8 +

...

- 2 2! 2! 3! 23 4!

26 6

x 3 3( 3) e

3 2 e

3 3

· e = e + e x - +

(x - ) + 3f(x - ) + ...

e' =

e-1(x

+ 1) +

e;/

(x + 1)2 +

e;/

(x + 1)3 + ...

26.7

= ! .:

~(x

- 2) +

~(x

- 2)2 -

~(x

- 2)3 + ...

26.8 log x = log 2 + !(x - 2)

! .

;ft

(x - 2)2 + 1.

(x - 2)3 - *.

(x - 2)4 + ...

26 9

cosx-I

1 x

---xr-

-2]

+ 4f - 6f + Sf -

...

26.10 .94608 <

Si~x

dx < .94611

CHAPTER

27.1

y=!sinx

2+2

27.2 y = tan

x-I

27.3 y = tan "

+ %

27.4

!log(x

2+1)+5

27.5

1x3

27.6 y = log

(ex

+ 1)

27.7

tan-

x + c

27.8 log Iyl =

sin-

x + c

27.9 y =

ce"

27.10 y =

-2

ce"

27.11 y =

ce'

27.12 y =

-1

+ ce

x 2

27.13 y =

cxe'

27.14 y =

-2

+ c(x - 1)

27.15 T = 20 +

80G)~;

T(lO) = 25

27.16 y =

100(5)~;

y = 1000 at t = 4.29 hrs; y = 2500 at t = 6 hrs

CHAPTER 28

28.1 y =

Cx-

28.2 y =

~x2

Cx-

28.3 y =

Ce-

x 2

Answers

28.4 y = ! +

Ce-

28.5

y =

ce:'

28.6 y =

2xe-

ce::

28.7 y =

-2x

- 1 + Ce

28.8 y

= 2x

28.9 y

= cos 3x + 2 sin 3x + Ce

28.10

y=!cos2x+~sin2x+Ce-x

28.11

y =

cos2x +

sin2x

+ Ce:"

28.12 y =

-x

+cos3x

-2sin3x+Ce

28.13

y = e

+ ce:"

28.15 v = 13.8 ft/sec

28.16 T = 50.6

CHAPTER 29

29.2 y =

Cle-

2t;

-7e-

+8e-

29.3 y

= C

cv:'.

y = 3e

29.4 y

= C

le-

c-«:

y =

-e-

29.5 y

= C

+ cv:'. y = 2e

d2y 2

2 0

([if

- r dt + r y =

29.7 y = Cse' + Cite'

29.8 y = C

le-

+ C2te-5t

29.10 y

= Clet cost +

C2et

sinr: y = 3e

cost + 'le' sint

29.11

y = Clet cos2t +

C2et

sin2t;

y = 3e

cos2t

+4e

sin2t

29.12 y

= C

cost +C

sinr: y = 6e

sint

29.13 y

= C

cos4t +

e2t

sin4t; y

= e

cos4t

29.14 y

= CI cos 2t +

sin 2t; frequency =

29.15 y =

2sin(2t

29.16 (a) l = 3.24ft; (b) e = .158sinrrt, .158rad = 9.01°

CHAPTER 30

30.1 (b) J

te'i

~[ct

- 1]

ctdt

~[c2t2

- 2ct + 2]

3ectdt

= ::[c

- 3c

2t2

+ 6ct - 6]

30.2 y =

e:'

30.3 y =

-tet

30.4 y = (2t

- 6t +7)e

30.5

y = (3t

+ 2t)e

30.6 y

= 2t + 3

30.7 y =

-3

sin t + cos t + C

+ C2e-t

30.8 y

= 3t sin t + CI cos t +

sin t

30.9

y =

-4cos2t

3sin2t

+ Clet + Czte'

30.10 y =

!te

sint

+ Clet cost +

C2et

sint

291

292

CHAPTER

Understanding Calculus

31.1

31.2

31.3

31.4

31.5

31.6

31.7

31.8

31.9

31.10

31.11

31.12

31.13

31.14

(a) -31 -

v10

(b)

- 2J;

vTI

(c)

-41-6j;

V32

(d)

12J;

R =(0, 4)

R=(-I,2)

A=-0sI-~J

-3.

1+-2.-·

v13

vTIJ

-L.+-L.

v10 I

ViOJ

cos 6 I + sin 6J

u=!1+1j;

(5, 8) and

(-1,

(5,4)

(a)D=

~I-~j;

E =-1;

F=-~i+

~J;

(b)

D=B-A;

E=-A;

F=-B;

G=-v'2B+A;

H=A-B

1;::-t.

--+

1--+;::-t.

WnteQ=2PK-

PQandR=2PQ-

IIFJI

2061bs; 6

Course65

;

speed 588

mph.

CHAPTER

32.1 6 =

¥;

1!Allcos

6 = °

32.2

6=~;

IIA1lcos

vTI/V2

32.3 8

1.75 rad;

IlAllcos

6 =

-.4

32.4 8

.073 rad;

IlAllcos

6.32

32.5

A=~i+~j

32.6 A =

~(i

+ 3j)

32.7 A =

~(i

- 2j)

32.8

1.11

rad

63.4

32.9 6

71.6

1.25

rad

32.10

=~I+~j;

A2=-f7I+~j

32.11 56.3

;

- Sj =

~)u

(~)v

32.12 Write

IIA

BII

= (A + B) · (A

+ B) etc.

CHAPTER

33.1 R(t) = t(2i + J+ 2k); cos a

=~,

cos

=!,

COS"

33.2

R(t)=(2+4t)I+(4+4t)j+(I-t)k;

cosa=~,

cos~=~,

cos,,=~

33.3 R(t) = (1 +

5t)1

+ (2 + t)j + (3 + 3t)k; cos

a=~

cos

~=~

cos

1'=~

33.4 R(t) = (3 +

4t)1

+(4 - 3t)j; cos a

=~,

cos

= 0

Answers

293

33.5 (a) If (5, 1, 3) were on the line, then t would have to be zero, and t = 0 gives

y = 4 instead

ofy

= 1

(b)

Q=(6,

1,2)

(c)

IIPQII

=V2

33.6 47.1°; .82rad

33.7 82.6°; 1.44rad

33.8

(x-1)-(y-2)+2(z-3)=0

33.9 2x + 3y - z - 3 = °

33.10

y=O

33.11

y=x

33.12 8i + 8j + 8k

33.13

-3i

+ 2j + 10k

33.14

i-2j

33.15

-7i

+37j + 41k

33.16 2(x - 2) + 8(y - 1) + 3(z - 3) = °

33.17

-8(x

- 1)+ 9y + 5(z - 5) = °

33.18 y = 1

33.19

33.20

3/vO

33.21

2/v14

33.22

8/V2I

33.23

1/v14

CHAPTER

34.1

34.2

34.3

34.4

34.5

34.6

34.7

34.8

34.9

34.10

34.11

34.12

34.13

34.14

34.15

The plane has intercepts (4, 0, 0), (0, 2, 0), (0, 0, 2)

The plane has intercepts (4, 0, 0), (0, 8, 0), (0, 0, 2)

The plane has intercepts (2, 0, 0), (0, 2, 0), (0, 0, 1)

Draw the lines joining the three points

The plane passes throughthe line and is perpendicular to the

xy-plane

A circularcylinderof radius 3 centeredon the z-axis

A circularcylindercenteredon the x-axis

The surface consists of all lines through the curve x = Z2 and parallel to the

y-axis

Circularcylinderof radius one, parallel to the y-axis, with its axis throughthe

point (0, 0, 1)

The surface consists of all lines through the curve

x =

and parallel to the

z-axis

(0,0, 1),2

(2,

-1,

3), 5

(

lo)~

-2'

- 2'

(1,

-2,

4), 2

(a) (1,

-2,

(b) the point (1, 0, 4)

294 Understanding Calculus

34.16 Cylinder centeredon they-axis

34.17 Rotatethe parabolay = x

aboutthey-axis

34.18 The

double

cone obtained by rotatingthe line z = x (or z =y) about the z-axis

34.19 Rotatethe curvez = 3x

aboutthe z-axis

34.20 This is a hyperboloid of one sheet; the cross-sections in the planes z = const.

are circles

34.21 An ellipsoid; rotate the ellipse t +r = 1 about the x-axis

34.22 A hyperboloid of two

sheets;

rotate the hyperbola x

r = -1 about the

y-axis

34.23 An ellipsoid (a closedsurfacesuchthat everycross-section perpendicular to an

axis is an ellipse). The intercepts are (±1, 0, 0), (0, ±2, 0), (0, 0, ±3)

CHAPTER

35.1

35.2

35.3

35.4

35.5

35.6

35.7

35.8

35.9

35.10

35.11

35.12

35.13

35.14

35.15

35.16

: = 2xy +2x cos(x

+r)

=r + 2y

cos(r

+r)

: =y

sin(xy)

xy2cos(xy)

= x sin(xy)+

cos(xy)

sec(x

+y) tan

2(x

+y) + sec'(x +y)

: = e' tan(x+2y) +esec

2(x

+2y)

2ex

sec

(x + 2y)

a'lz a'lz

2x 6

axay

ayax

= +

+y2 + 4x

+y2

: =

2r(1

- cos (J);

sin (J

tPz . (J

=-Slncpcos

27Trh;

~=27Th

g£

Swh

at =-

;

?=se-

c::.

sec

--b

sec?

xxz

yyz

e 2

J;xz

cos(yz)

- yz

sin(yz)

i = 2 cosz

xxy

CHAPTER

36.1 t

= i + 9k; t

=j + 9k; N =

-9i

- 9j + k. z = 18 + 9(x - 3) + 9(y - 1)

36.2

= i + 6k; t

=j + 4k; N =

-6i

- 4j + k. z = 6 + 6(x - 1) + 4(y - 2)

36.3

= i + 2k; t

=j + 2k; N =

-2i

- 2j + k. z = 1 + 2x +2(y - 1)

36.4 t

=i'

t =J.'

N=k'

z=O

'y'

36.5 t = i + k: t =

J.'

N =

-i

+ k: z = 1 - x

'y'

Answers

36.6 t

= i + 6k; t

= j + 4k; N =

-6i

- 4j + k; z = 9 + 6(x - 3) +4y

36.7

(0,-1,0)

36.8 (1, 0,

-1)

36.9 (1, 0,

-2),

(-1,

0, 2)

36.10

(-~,

-~,

-~)

36.11 z = 27.9

36.12 z =l + .050 = .30

36.13 z = 2 -

= 1.825

36.14 z = 3.1

36.15 z = 1.2

36.16

.33%

CHAPTER 37

295

37.1

37.2

37.3

37.4

37.5

37.6

37.7

37.8

37.9

37.12

:=-22

:=2

= 228

tfl:

= 838

=-86

-6

dt -

'fit

= 108

(b)

exVr:.-"l-+-co-s-ex

CHAPTER 38

38.1

38.2

38.3

38.4

38.5

38.6

38.7

38.8

38.9

38.10

38.11

/j(1I

0) - 1 +

TTI3J

l 4' - 2 4

fi(I

2)=~

TTI4J

1 , 2

TT!(I,2)=-8

DTTIJ(I,

0) =

/j(1I

1I\

- 1

-TT13

J l 4' 4J -

-2

- 2

Dof(O,0) = 1

TTIJ(O,

V =

'1-

TTI2!(0,

1) = 2e

16/V'5

-~)/v1O

5/vL6

296

38.12

38.13

38.14

38.15

38.16

38.17

38.18

38.19

3cosh 8

28/\16

~(I

+ log 2)

= i + 2j; T = 2i - j

= 2i + I2j; T =

I2i

- 2j

N =

i + e]: T = ei _.2

N=i+!(e+~)j;

T=!(e+~)i-j

-6

dt -

Understanding Calculus

CHAPTER 39

39.1

J(O,

0) is a relativeminimum

39.2 No relativemaximumor minimum

39.3 There is no relativemaximumor minimumat the critical point (0, 0) (consider

J(x, O));J(I, 1) is a relativeminimum

39.4

J(I,

0) is a relativeminimum

39.5 No relativemaxima or minima

39.6

J(I,

is a relativeminimum;J(-I,

-1)

is a relativemaximum

39.7

J(I,

-2)

is a relativemaximum;J(I, 0) is neither max nor min

39.8 There are no criticalpoints

39.9 No relativemaxima or minima

39.10 (2, 1, 1)

39.11

(-1,

-t,

39.12

J(O,

4) =

-4

is the minimum

39.13 x = y = z = 20/3

CHAPTER 40

40.1

40.2

40.3

40.4

40.5

40.6

40.7

40.8

40.9

40.10

40.11

ll;

+y)dx dy

f~xy

dx dy

?o;

f;1/2

(x + y)dx dy

If;

f~l

f6-

(2 + y)dy dx

f~fi

xy'dx dy

...L

e-I

!e-I

The region is symmetric about they-axis, andI(-x, y) = -/(x, y), so the integral

overthe left half of

R cancels the integral overthe right half of R.