Bear H.S. Understanding Calculus

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Answers

6.27 (i) w =

t = 3 sec

(ii)

= 0 when t = 1,

-30

in.lsec at t = 2;

= 0 at t = 3

6.28

fad/sec

6.29 At

t = 3,

(20~~~22)

fad/sec

6.33

~,3;

6.34 0, n , 2rr

6.35

6.36 0,

n ,

,2rr

6.37 0,

n , 2rr

CHAPTER

7.11.16,

1.105, 1.099, 1.099

7.2 1.05, 1.005, 1.0004, 1

7.3

= 2e

7.4 dy = e

x 2

(2x + 1)

dx .

7.5

X 2

(_ 2x )

7.6 dy = 2(eJX +

1)(_1

eJX)

2JX

dy 1 1 .r 1 I .r

7.7 dx = 2

x 2e2

+ 2

2e2

7.8

= 3

log 3

7.9

= 2

x 2

•

2x log 2

7.10

= 2xe

+ 3x

+cosx - x

sinx

7.11

= e' cosx -

sinx

cosx

7.12

logx +

~ex

7.13

= 2xe

log(l +x

)

e;~~~2

dy - 1

. dx - x

7.15 dv

2x+1

= x

2+x

7.16 dy = 1 + 1

dx X

7.17 dy

1_ +

_1_

dx - x-l-I

x+2

7.18 dy =

- log 3

dx x

7.19 log y = x log

= log x + 1; *= XX (log x + 1)

7.20

= (2x + I)X [log(2x + 1) +

2;~1]

7.21

dy = (sin x +cos x)X [IOg(sinx +cos

X(~OSX-SinX)]

dx smx+cosx

7.22 A = 1000; k =

!log3

log,J3;

y(4) = 9000

7.24 k =

log2

7 25

N 1

2"":"

69 d 1 (1 + )....:...c II

....:..

.69

. log(l.06)· ote og

an og x - x lor sma x so log(l+x) - X

(70 divided by the interest rate is the usual approximation).

7.26 n = log

log(l +

.70/x

277

278

7.29 (b)

(~,

e);

1)/a

7.30

-~)

CHAPTERS

2_1

. dx = 3

8 2

+ 5

. dx = - 4x 4 4x 4

8 3

llx~

+ !1

-¥

• dx - 5 5

~x!

_llx~

• dx - 2 2

8.5

_~(x2

1)-j(2x)

8.6

~=[cos(xj+l)]~x-~

8.7

x(1

+ x

)- !

8.8

= 4(cos

X +

1)-!(-2cosxsinx)

8.9

sec2(x~

1)(1x-j)

8.10

-1(secx

I)-~(secxtanx)

811

-II

SIn

= '6

8.12 cos:" f =

8.13 tan-I

8.14

COS-I

1 =

8.15 sin-I ( -1) =

-j

8.16

COS-I

( -

8.17

cos-I(-I)

= n

8.18 sin"! (1) = I

8.19 sin-I (sin zr) = 0

8.20

COS-I

(cos(-j»

= j

821

~-~

• dx -

.J1-x

822

_3_

• dx -

.J1-9x

8.23

- 2

dx -

JI-(2x-I)2

824

~---=L._I_

• dx -

v'I=X

2,jX

~-~

.25 dx -

.J1-e

8.26

- -I

dx -

JI-(x+I)2

827

- 3

• dx - 1+9x

828

- I

• dx - 1+(x+2)2

8.29 dx - 1+(logx)2

8.30

= 1

8.34 y =

!tan-

52)

8.35 (a)

x/v"f=X2

(b)

I/Jl

Answers

(d) x

CHAPTER 9

9.1

f'(x)

= 3x

- lOx + 3;

f"(x)

= 6x - 10

9.2

f'(x)

= 7x

- 6x

- 12x-

;

f"(x)

= 42x

- 12x +

60x-

9.3

f'(x)

!x-4

~x-1;

f"(x)

-*x-~

~x-~

9 4

f'(x)

-2x

f"(x)

-2(l+x

2)2+8x

2(l+x

(l+x

)2 '

O+x

_ 2(3x

- l)

- (T+X2)3

dv 2 d

y 2 2 2

9.5

= 2xe

;

dXi

= 2e

+ 4x e'

9.6

~=2cos2x+3sin3x;

~=-4sin2x+9cos3x

_1_.

=x(l-x2)-~

· dx

Jl-x

dXi

dy = 1

.1.

-4x(4+x

) - 2

· dx

1+:2.

dXi

9.9 local max at x =

-1;

local min at x = 1

9.10 local max at

x =

-2;

local min at x = 0

9.11 local max at

x = -

; local min at x =

9.12 local max at x =

9.13

localminatx

=-1

9.14 local max at x = 1; local min at x = 3

9.17 lOin.

by lOin.

9.18 128 cu in.

9.19

!! = 2

9.20

(~)2

9.21

x =

R, h =

12 - 21

• X - 13' Y -

9.23 x = 1, y = 0

2 2

9.24l={h

3+b

9.25 v = 7.5 mph

9.26 A is minimum for x = loorr/(4 + rr)

44 in.

A is maximum for x = 100 in.

CHAPTER

10.1 sin

I~O)

-1·

I~O

= .4849

10.2 J4.02

2 +

• (.02) = 2.005

10.3

eO.

. (0.2) = 1.2

10.4 tan" (1.04)

tan-II

111

•

(.04) = .8054

10.5 log(I.002)

log 1 + t .(.002) = .002

10.6 2.01

+ 3

(.01) = 8.12

1 1

10.7 305 = 325 + 5(32) 5 .

(-2)

= 2 - 5.16 = 1.975

10.8

f(3.1)

f(3)

f'(3)

. (.1) = 25 + 2(.1) = 25.2

279

280

10.9

d;'

10.10

10.11 !

10.12 !

10.13 1

10.14

10.15 !

10.16 0

10.17 0

10.18

= 2.25;

= 2.236

10.19

= 1.444;

= 1.4423

10.20 .6823

10.21 .567

CHAPTER

11.1

~X4

11.2

-lX-

11.3 lOx 3

11.4

2x!

11.5

-10x-!

11.6

4x~

11.7

1x~

~X4

11.8

~x4

11.9

-1cos3x

11.10 !sin 5x

11.11 - 2 cos

sin 2x

11.12

sin x

11.13 - cos(1 +

11.14 !sin(1 + x

)

11.15

~(4+x3)~

11.16 i2(1 + x

) 1l

11.17

~x2

~e4X

11.18

x + 2e

4e2x

11.19

4ex2

11.20 sine"

11.21

x+31oglxl

11.22 j log

+ x

11.23 log

+ X +

11.24 4

sin-I

(2x)

11.25 sin"! (

)

Answers

11.26

tan"! (3x)

11.27 1tan-I 1

11.28 log

11.29

y=x

3+!x

2-7x+¥

11.30

y =e' + log

+ x I+ 1

11.31

y=tan-lx-~

11.32

y = sin-I x + 5

11.33

k = 22

ft/sec?

CHAPTER 12

12.1 ¥

12.2 2

12.3

1~2

12.4

~[3i

2i]

+ log

12.5

12.6 2 - 2 sin !

12.7

12.8 1

12.9

!(e

- 1)

12.10

12.11 %

12.12 !log 2

12.13 36

12.14 2

12.15

12.16

12.17 1

12.18

12.19 1 -

12.20

¥-

12.22 (i) e

x 2

;

(ii).J[+X3;

(iii) sin x

;

(iv)

-log(1

+ x

)

12.23

j(g(x»g'(x)

CHAPTER

13.1 12 ft lbs

13.2 35 ft lbs

13.3 2250 ft lbs

13.4

! .62 . n . 16 . 36 ft lbs

13.5 2852 ft lbs

13.6 18 cubic units

281

282

13.7

~Jra3

cubic units

13.8

~Jra3

cubic units

13.9

cubic units

13.10

I cubic units

13.11

Tt;

cubic units

13.12

~Jrab2;

~Jrba2

cubic units

13.14 264 ft

13.15 99201bs

13.16 225 ft Ibs

13.17 25 ft lbs

CHAPTER

Answers

14.1 The curve is the arc of the parabola y = x

between

(0,0)

and

(1,1).

14.2 The curve is the arc of the parabola (y + 1)2 = 4x between (0,

-1)

and

(1,1).

14.3 The curve is the top half of the ellipse

(~)2

+ y2 = 1.

14.4 The curve is the arc of the hyperbola

y =

from

(4,2)

(1,1).

14.5

x = _y2 + 2; 1

14.6

y =2x

- 1; x

14.7

y = (x - 1)2;

-00

< X < 1

14.8

y = 1 - x; 0

14.9

y - 4 =

~(x

- 15); curve lies under the tangent line

14.10

y = 1; curve lies over the tangent line

14.11

y = x +2; curve is the tangent line

14.12 y - 1 =

2(x

- 1); curve lies over tangent line

14.13

= 0 when t =

.J2

sec and y = 32 ft

14.14

t = vsinOj32 for max. height. t = vsinOj16 when body hits the ground. When

the curve hits the ground,

= - tan 0, so the angle of impact is n -

14.15

Zn a

14 16

728

- 2!

. 24 - 3

14.17 5

14.18 log

1J2

14.19

J2(e

14.20 sinh

14.21

x=asecO,

y=btanO

14.22

x = am, y = am

14.24

x = acosO, y = bsinO

14.25

x = acosO, y = bsinO

CHAPTER

15.1 12(4

)

15 2

1 log 1

. 2 5

Answers

15.3

15.4 0

15.5

15.6

15.7

(log 2)2

15.8

15.94+2Iog5

15.10

~(e

- 1)

15.112(e

2-e)

15.12

15.13 ! ( tan

-I

15.14

!log

15.15 tan"

e -

15.16 sin

-I

15.17

~(J5

v'2)

15.18 0

15.19

( tan

-1

- tan

-I

15.20 1

sin-

15 21 .i.

(tan-

II - tan"

• 10 2 2

15.22 log 61 - log 5

15.23

15.24

) -

)

15.25

- 1) + 2(24 - 1)

15.26

15.27

¥J3

~v'2

CHAPTER

16.1 3 log Ix +

16.2 -!

(X~3)2

16.3

log

14x

16.4 x + 2 log Ix -

16.5

+ x+ log [x]

16.6

2-4x+16Ioglx+41

16.7

log I

x-2

16.8 1log

I~~i

16.9 log I

16.10 log I

~~~

16.11 1 log I

x+21

x-2

16.12

tan-

(x + 1)

16.13

tan-

(x + 2)

283

284

16.14

log Ix

16.15 log

tan-

16.16

!log

+ !

tan"!

16.17 2 log Ix

+ 6x +

101-

12tan" (x + 3)

16.18 log

~~~

16.19

log Ix

+ 4x +

81-

tan-

(Xi

)

16.20 3 log Ix +

+ log Ix

16.21 4 log Ixl + log Ix

+ 2x +

+ tan" (x + 1)

16.22 log I

(X-

~~3+2)21

16.23 log I

x2;~~1)

16.24 log Ixl -

!log

CHAPTER

17.1

XCOS-lX+~

17.2 x

tan"!

x -

!logll

17.3

-!xe-

~e-2x

17.4 x

- 2xe

+ 2e

17 5

_!x

3e-

~x2e-2x

~xe-2x

~e-2x

. 2 4 4 8

17.6

-xcosx+sinx

17.7

~xsin3x+bcos3x

17.8

-!x

2cos5x+fsxsin5x+

1;5cos5x

17.9 4x2logx -

~x2

17.10

~x~

logx

~x~

17.11

kx8logx - f1x

17.12 x (log x)2 - 2x log x + 2x

17.13

x(logx)3 - 3x(logx)2 + 6x log x - 6x

17.14

~x(1

+x)~

15(1

+x)~

17.15

x log(1 + x

- 2x + 2

tan-

17.16 4

(sin x - cos x)

17.17

2JX+1[log(x

+ 1) - 2]

17.18 -

!e-

+ 1)

17.19

I-I

17.20

17.21

15[32

16J2]

17.22

+ f - 4

17.23 I

+ 1)

17.24 2rr

17.25

~rra3

Answers

CHAPTER

! sin" x

. 4

18.2

sin~

18.3

_~cos6x

18.4loglsinxl

18.5 sin x -

sirr' x

18.6

cos" x

18.7 -

cosx

cos' x

18.8

cos! x

18.9

cos' x - !cos'' x

18.10

sin x

cosx

18.11

kx - k sin x cos' x + k sirr' x

cosx

18.12

sin2x +

sin4x

18.13

ftx

- ii sin 2x +

sirr' 2x

18.14 ! tan 2x

18.15

tan? x

18.16 tan x - x

18.17

sec" 3x

18.18

-cosx

18.19

tan~

18.20 ! tan

x + log Icos x]

18.21 log

I sec x + tan x I

-log

Icos x]

18.22 k[sec4x tan4x + log

Isec4x + tan4xll

18.23

~[sec

2x tan 2x + log Isec 2x + tan 2xll

18.24 secx

18.26 (ii) -

sirr' x cos x -

sin x cos x +

18.27

18.28

~1l'2

18.29

18.32 (i) -

cos 5x - ! cos x

(ii) -

cos 6x + k cos 4x

CHAPTER

19.1 I

19.2 !

sin-

!x~

19.3

2sin-I~+!xJ4-x2

19.4

19.5

285

286

19.6

_x_

- sin-I x

Ji=X2

19.7

v'9~x2

19 8

-I

. sin J3

19.9

log

la+~1

19.10

n ab

19.11

log

IJI

+ x

+ xl

19.12

19.13

v'1+x

19.14

Jx2-4-2sec-l~

19.15

4sec-

19.16

4xJx2

- 9 +

log Ix + Jx

19.17

!x~-!loglx+~1

19.18

![ tan-

x +

1:x

19.19

-4

l~x2

tan-

x +

l:x

19.21

y =a log

la-¥I-

19.26

cosh"! u = log lu +

.JU2=11

CHAPTER

20.1

.69325;

.693150

20.2 5

.7853982

20.3 5

= .7818

20.4

(i)

~[~

+log(~+I)]

(ii) 3.0546645

(iii) 5

3.0546645

20.5

.8285°;

= .8282

CHAPTER

21.1

-1

21.2 0

21.3 0

21.4 0

21.5 0

21.6 0

21.7 0

21.8 0

21.9 0

21.10 0

21.11

21.12 1

Answers