Answers

Chapter 1

1. Vectors are: acceleration, centripetal force,velocity, momentum, magnetic intensity.

2. The order in which the vectors are added isimmaterial.

Fig. 1.36

Fig. 1.37

3.

Fig. 1.38

Fig. 1.39

558 Answers

4.

Fig. 1.40

Fig. 1.41

5.

Fig. 1.42

Fig. 1.43

6. (a) |ab | = |a| cos 60◦ = 5× 12

= 2.5(b) |ab | = 0(c) |ab | = 4

(d) |ab | = | 32

cos(� − �3

)|= | 3

2(−cos �

3)| = 3

4

7. P=(10,−2) Steps in the computation:

a =−−−→P1P2 = P2 −P1 = (5, 2)

b = P3 −P1 = (3, −5)P4 = P1 +a+b = (10, −2)

8. a =−−−→P1P2 = (x2 −x1, y2 −y1),

b=(x3 −x2, y3 −y2),c =(x4 −x3, y4 −y3),d =(x1 −x4, y1 −y4),S =(x2 −x1 +x3 −x2 +x4 −x3

+x1 −x4,y2 −y1 +y3 −y2 +y4 −y3

+y1 −y4)=(0,0) = 0

9. F = (90N, 10N)

10. (a) (4, 3, 0) (b) (5, 3, 7)

11. (a) a = (−14 12, −1, 0)

(b) a = (−7, −22, 12)

12. (a) ea = 3√14

, −1√14

, 2√14

(b) ea = 23, −1

3, −2

3

13. (a) |a| = √24 = 4.90

(b) |a| = √53 = 7.28

Answers 559

14. (a) V1 +V2 = (0,250)km/h(b) V1 +V3 = (50,300)km/h(c) V1 +V4 = (0,350)km/h(d) |V1 +V2| = 250km/h(e) |V1 +V3| = 304km/h(f) |V1 +V4| = 350km/h

Chapter 2

1. (a) a ·b = ab cos ˛ = 3×2× 12

= 3(b) a ·b = 10(c) a ·b = 2×√

2 ≈ 2.828(d) a ·b = −3.75

2. (a) ˛ = �2

Vectors are perpendic-ular

(b) ˛ = 0 Vectors are parallel(c) ˛ = �

3(d) �

2< ˛ < �

3. (a) a ·b = −3−2+20 = 15(b) a ·b = −1.25(c) a ·b = −11

12(d) a ·b = 4

4. (a) a ·b = 0 Vectors are perpendic-ular to each other or one vector-is zero

(b) a ·b = −12 a is not perpendic-ular to b

(c) a ·b = 0 a is perpendicular to b

(d) a ·b = 0 a is also perpendic-ular to b

(e) a ·b = −1 a is not perpendic-ular to b

(f) a ·b = 0 a is also perpendic-ular to b

5. cos ˛ = a·bab

(a) cos ˛ = −1 thus ˛ = �(b) cos ˛ = 2

3thus ˛ ≈ 48◦

6. U = F · s(a) 15 N m (b) 5 N m (c) 0

7. (a) c is parallel to the z-axis(b) c is parallel to the x-axis

8. |a×b| = ab sin ˛

(a) 2×3×√

32

≈ 5.196

(b) 0 (c) 6

9. (a) −83

c (b) 32

b (c) −32

b

(d) −6a (e) 0 (f) 6a

10. c = (aybz −azby , azbx −axbz ,axby −aybx)

(a) c = (10, −9, 7)(b) c = (3, 6, −9)

Chapter 3

1. (a)

(b)

2. (a)

Zeros: x = −1,x = 3Poles: noneAsymptote: none

560 Answers

(b)

Zeros: nonePoles: x = 0Asymptote: x-axis

(c)

Zeros: nonePoles: noneAsymptote: x-axis

(d)

Zeros: nonePoles: x = 0Asymptote: y = x (This is the streightline. It intersects the first and thirdquadrant.)

3. (a) 0.017(c) 0.785

(b) 2.09(d) 7.19

4. (a) 5.73◦(c) 12.61◦(e) 54.43◦

(b) 102.56◦(d) 130.06◦(f) 180◦

5. (a)

(b)

(c)

6. y = 1.5sin

(x +�

3

)

7. (a) 4�

(c)8

3�

(b)2

3�

(d)1

2

8. y = 4sin(4x +c)

9. (a) sin(

u+�

2

)

(c)1

2�

(b)1

2�

10. (a) sin79◦

(c) sin(�

4

) (b) sin3◦

(d) sin

(1

6�

)

11. (a) 1

(c) − tan2 '

(b) tan'

(d)2

cos2 �

12. (a)2sin!1 cos!2

2sin!1 cos!2

= tan!1

(b) 2cos45◦ cosa =√

2cos˛

(c)cos2 '

2sin' cos'=

1

2cot'

13. sin�

2+ sin

�

6=

3

2; 2sin

�

3cos

�

6=

3

2

Answers 561

Chapter 4

1. (a)1

an

(d) 1

(g) 102

(b) 3

(e) y6

(h)1

27

(c) n√

a

(f)1√x3

2. (a) 4√

2(d)

√35

(b) 10√

e(e) 4

(c) 1

(f)√

24 =2√

6

3. (a) 2(d) 6

(b) −3(e) 10

(c) 10(f) 1

4. (a) 3(d) a6

(g) a

(b) −1(e) a6

(h) 2

(c) 5(f) 1

5. (a) e(d) 1

(b) 57(e) e4

(c) 3(f) 5

6. (a) x(d) lg(n

√a)

(b) −x(e) 2n

(c) 7x(f) ld5m

7. (a) lna + lnb

(c) 6

(e) 8x

(b) 2 lgx

(d)1

2ldx

(f) x−3

8. (a) y−1 =x +5

2

(c) y−1 =ex

2

(b) y−1=3√

x−1

2

9. (a) y = (x−1)3

(c) y = x3 +1

(e) f (g(2)) = 2

(b) y =x2 +1

x2 −1

(d) f (g(1))= −3

2

(f) f (g(1)) = −1

Chapter 5

1. (a) 0 (b)1

2(c) −1

(d) 1 (e)1

2(f) 2

(g) 6

2. (a) −1 (b)1

2(c) 5

(d) 0 (e) 1 (f) 0

3. (a) At x = 0 the function is continuousbut not differentiable. It is shown inFig. 5.40.

Fig. 5.40

(b) The function is shown in Fig. 5.41.It is discontinuous at the points x =1, 2, 3, . . .

Fig. 5.41

(c) The function is discontinuous at thepoints x = 2, 4, 6, 8, . . .

4. (a) S5 = 2+3

2+

4

3+

5

4+

6

5= 7

17

60≈ 7.28

(b) S10 = 3× 1− (1/2)g

1/2= 6× 511

512≈ 5.988

(c) S = 3× 1

1/2= 6

5. (a) End points of the secant: P1(1, −1),

P2

(3

2,

3

8

)

Slope of the secant: ms =Δy

Δx=2.75

Slope of the tangent: mT =y′(1)=1

(b) v(t) =ds

dt=6t −8 , v(3) = 10m/s

(c) (i) dy = (2x +7) dx

(ii) dy = (5x4 −8x3) dx(iii) dy = 4x dx

562 Answers

6. (a) 15x4 (b) 8

(c)7

3x4/3 (d) 21x2 −6

√x

(e)x2 +2

5x2

7. (a) 6x2 (b)1

33√

x2

(c) − 2

x3(d)

8

(4+x)2

(e) 6x(x2 +2)2 (f) 4x3 − 1

x2

(g)x√

1+x2(h)

3b

x2

(a− b

x

)2

8. (a) −18 sin 6x (b) 8� cos (2�x)

(c) Ae−x [2� cos (2�x)− sin (2�x)]

(d)1

x +1(e) cos2 x− sin2 x

(f) 2x cos x2 (g) 12x(3x2 +2)

(h) ab cos (bx +c) (i) 6x2 e(2x2−4)

9. (a) y′ = − c√1− (cx)2

(b) y′ =A

1+(x +2)2

(c) y′ =2x√

1−x4

(d) y′ =1

2√

x (1+x)

10. (a) y′ = 0.1 C cosh (0.1x)

(b) u′ = � [1− tanh2(v +1)]

(c) �′ = tanh

(d) s = tanh t

(e) y′ = 0 (since y = −1 = constant)

(f) y′ = 2 coth x (1−x coth x)

11. (a) y′ =10A√

1+100x2

(b) u′ = − C

�2 +2�

(c) �′ =1

cos

(d) y′ =1√

x4 +x2(x −1)2

12. (a) a cos � +1

cos2 �

(b) eu(2+u)

(c) − 1

x2

(d) 120x

13. (a) x1 = 2 (√

2, −8), minimumx2 = −2 (0, 0), maximumx3 = x4 = 0 (

√2, −8), minimum

(b) x = k� ,

x = ±�

2,±5�

2,±9�

2, . . .maximum

(k = 0,±1,±2, . . .) ,

x =±3�

2,±7�

2,±11�

2, . . .minimum

(c) x = 2�k ,

x =±� ,±5� ,±9� , . . . maximum

(k = 0, ±1, ±2, . . .) ,

x =±3� ,±7� , ±11� , . . . minimum

(d) x = 3√−4, none

(e) x = (2k +1)�

2−2 ,

x = 2k� −2, maximum

(k = 0,±1,±2, . . .) ,

x = (2k +1)� −2, minimum

(f) x1 = 4.85,

(−1, 3

1

3

), maximum

x2 = 1.85, (3, −18), minimum

x3 = 0,

14. (a) y = 11 (b) y = −8.89

15.

Answers 563

Fig. 5.42

16. (a) 2.44%, 1.63% (b) 12.95 m

17. (a) 5.27(b) 14.14 in magnitude

18. (a) 1 (b) 0 (c) 1

(d) −1

2

Hint:1

x

(1

sinhx− 1

tanhx

)=

1−coshx

x sinhx(e) e−2/�

Hint: xtan(�/2)x = e(ln x)/cot(�/2)x

(f) 0 (g)b2 −a2

2

19. (a) y′ =−2x

3y

(b) y′ = − 9x2y2 + cos y

6x3y−x sin y

(c) y′ = −2(x +y)+2

2(x +y)+1= −2

20. (a) y′ = 30x −29

(b) y′ = xsin x

(sin x

x+ cos x ln x

)

(c) y′ = xx(1+ ln x)

21. (a) y′ = (vu−gx)/u2

(b) y′ = tan t

22. x(t) = R cos 3�ty(t) = R sin 3�t

23. (a) It is a straight line in three-dimen-sional space (Fig. 5.43)

Fig. 5.43

(b) The curve is an ellipse with axesof lengths 2a and 2b, respectively.b2x2 +a2y2 = a2b2

x2

a2+

y2

b2= 1

24. (a) ax(t) = −!�0 cos !tay(t) = −!�0 sin !t ora(t)=(−!�0 cos !t , −!�0 sin !t)

(b) v(t)=(−R! sin !t , R! cos !t , 1)

v

(2�

!

)= (0, R!, 1)

(c) v(t) = (v0, 0, −gt)

Chapter 6

1. (a) F (x) =3

2x2 +C ; C =

1

2,

F (x) =3

2x2 +

1

2

(b) F (x) = x2 +3x +C ; C = −4,

F (x) = x2 +3x−4

564 Answers

2. (a) 3 (b) 6 (c) 0

3. (a)

[x2

2−2x

]0

−2

=−6, F = |−6|= 6

(b) F = |−2| = 2

(c) F =[

x2

2−2x

]2

0

+[

x2

2−2x

]4

2

= 4

4. (a)d

dx

(x−1

x +1

)=

2

(x +1)2

(b)d

dx

{x − 1

8sin (8x −2)

}

= 1− cos {2(4x −1)}= 1− cos2 (4x −1)+ sin2 (4x −1)= sin2 (4x −1)+ sin2 (4x −1)= 2 sin2 (4x −1)

(c)d

dx

(x

1+x2

)=

1−x2

(1+x2)2

5. (a) ln(x −a)+C(b) tan x +C

(c) a ln (x +√

x2 +a2)+C

(d)1

2(˛− sin ˛ cos ˛)+C

(e)at

ln a+C

(f)3

10x10/3 =

3

10

3√

x10

(g)5

3x3 +

5

4x4 +C

(h)3

8t4 +2t2

6. (a)x2

4(2 ln x−1)+C

(b) x2 sin x +2x cos x−2 sin x +C

(c)x3

9(3 ln x−1)+C

(d) a sinhx

a(x2+2a2)−2axcosh

x

a+C

(e)1

nsin x cosn−1 x+

n−1

n

∫cosn−2 x dx

(f)xn+1

(n+1)[(n+1) ln x−1]+C

7. (a) − 1

�cos(�x)+C

(b) e3x−6 +C

(c)1

2ln |2x +a|+C

(d)1

6a(ax +b)6 +C

8. (a) ln√| sin 2x|+C

(b) ln |a +x2|+C

(c)1

40ln |x40 +21|+C

(d) − 1

cosh u+C

9. (a)1

5sin5 x +2 sin4 x +

1

2sin2 x +C

(b)2

45(3x5 −1)3/2 +C

(c)√

a−x2 +C

(d)1

2sin (x2)+C

10. (a) ln |ex +1|+C

(b) sin(

x− �

2

)+C = cos x +C

(c) sin x − 1

3sin3 x +C

(d) ln | ln x|+C

(e) ln |x3 −x|+C

(f) ln | tan−1 x|+C

11. (a) ln

∣∣∣∣∣(

x +2

1−x

)1/3∣∣∣∣∣+C

(b) ln

∣∣∣∣∣(x−1)5/3

x3/2(x +2)1/6

∣∣∣∣∣+C

(c) ln

∣∣∣∣∣(x−1)1/2(x −3)9/2

(x −2)4

∣∣∣∣∣+C

(d) ln

∣∣∣∣∣(

x2 −2

x2 +1

)1/6∣∣∣∣∣+C

(e)1

3(x +2)+ ln

∣∣∣∣∣(

x−1

x +2

)1/9∣∣∣∣∣

(f) ln

∣∣∣∣∣(

x2 −x +1

x2 +x +1

)1/2∣∣∣∣∣

(g) ln

∣∣∣∣(x−1)2

(x2 +2x +5)1/2

∣∣∣∣

−2 tan−1

[1

2(x +1)

]

Answers 565

Note:x2 +15

(x−1)(x2 +2x +1)

=2

x −1− x +5

x2 +2x +5

12. (a) −32.5 (b) ln 2(c) 1.416 (d) 75

13. (a)255

64(b) 4.5 (c)

27

16

14. (a)1

4(b) ∞ (c) �

1

r0

(d) ∞ (e)1

2(f) ∞

(g) 1 (h) ∞

15. U = {2(x0 +2−x0)+6(y0 −y0)+1(z0 −z0)} N m = 4N m

16. U =∫ P

0F ·dr =

∫ 5

0x dxNm=

25

2Nm

17. Force and path element are perpendic-ular. Thus the scalar product F · dr

vanishes. The line integral is zero.

18. The path element:

dr(t) =(−√

2sin t , −2sin2t ,2

�

)dt

F (t) =(

0,−2t

�, cos2t

)

U =∫ �/2

0F (t)dr(t)

=∫ �/2

0

(4t

�sin2t +

2

�cos2t

)dt

Since∫

t sin2tdt =sin2t

4− t cos2t

2+C

U =4

�

[sin2t

4− t cos2t

2

]�/2

0

+1

�

[sin2t

]�/2

0

=4

�

�

4= 1

Chapter 7

1. 720 square units

2. 9� square units

3. 2.14 square units

4. 0.693 square units

5. 100 square units

6. 2.97 square units

7. 19.24 square units

8. (a) 8 square units(b) 10.67 square units(c) 36 square units

9. (1) 98.12 units(2) 9.42 units(3) 4.064 units

10. 0.1109 square units

11. (a) 20.47 square units(b) 8.34 cubic units

12. (a) 67.02 square units(b) 49.35 cubic units

13. 314.16 cubic units

14. x =4a

3�, y =

4b

3�

15. 236.81 mm

16. 5/6 from the vertex

17. 46.875 mm

18. (a) MR2

(b) M

(R2

2+

L2

3

)

(c)M

2

(R2 +

L2

6

)

19. (a) 32.98kg/m2

(b) 131.9kg/m2

20. I = 1.52kg/m2

21. I = 76.04×106 mm4, k = 74.36mmNote that the centroid is 77.27mm fromthe bottom.

22. 230kN, 6.33m below the surface.

23. 5.23×105 N, 4m

Chapter 8

1. (a) f (x) =√

1−x

= 1− 1

2x− 1

22

x2

2!− 3

23

x3

3!−·· ·

566 Answers

These are two ways of arriving ata solution:(i) with the help of (8.2);

(ii) by using the binomial series

(n =1

2; a = 1).

(b) f (t) = sin (!t +�) = −!t +!3t3

3!

− !5t5

5!+

!7t7

7!−+ · · ·

(c) f (x) = ln[(1+x)5]

= 5x−5

2x2+

5

3x3−5

4x4 + · · ·

Derivatives Values

f ′(x)=d

dx[ln(1+x)5]

=5

1+x

f ′(0) = 5

f ′′(x)=−5(1+x)−2 f ′′(0) = −5

f ′′′(x)=5×2(1+x)−3 f ′′′(0) = 5×2

f (4)(x) = −5×2

×3(1+x)−4 f (4)(0)=−5×3×2

(d) f (x)=cos x =1−x2

2!+

x4

4!−x6

6!+−·· ·

(e) f (x)= tan x =x+1

3x3+

2

15x5

+17

315x7+· · ·

Derivatives Values

f (x)= tan x f (0)=0

f ′(x)=1

cos2 xf ′(0)=1

f ′′(x)=2 sin x

cos x

1

cos2 x=2ff ′ f ′′(0)=0

f ′′′(x)=2(ff ′′+f ′2) f ′′′(0)=2

f (4)(x)=2(ff ′′′+f ′f ′′+2f ′f ′′)

=2(f f ′′′+3f ′f ′′)f (4)(0)=0

f (5)(x)=2(ff (4)+4f ′f ′′′+f ′′2) f (5)(0)=16

(f) f (x) = cosh x = 1+x2

2!+

x6

6!+ · · ·

2. (a) The formula (8.7a) cannot be appliedbecause every other coefficient van-ishes (an = 0 if n is even). Instead weuse the formula (8.7b):

R =1

limn→∞

n√|an|

.

Since an =1

n!, if n is odd we find

R = limn→∞

n√

n! = ∞

(b)

∣∣∣∣an

an+1

∣∣∣∣=3n

3n+1=

1

3

Therefore R = limn→∞

∣∣∣∣an

an+1

∣∣∣∣=1

33. (a) P1(x) = x

P2(x) = 0

P3(x) = x +1

3x3

For details of the solution see 1(e).

Fig. 8.8

(b) y =x

4−x=

4

4−x−1=

1

1−x/4−1

(geometric series)

P1(x) =1

4x, P2(x) =

1

4x +

1

16x2,

P3(x) =1

4x +

1

16x2 +

1

64x3

P3 is a parabola of the third degree

with a point of inflection at

(−4

3, − 7

27

).

4. (a) y =sin x

=−(x−�)+(x−�)3

3!−(x−�)5

5!+ · · ·

Answers 567

Fig. 8.9

Derivatives Values

y′ = cos x y′(�) = −1y′′ = −sin x y′′(�) = 0y′′′ = −cos x y′′′(�) = 1

y(4) = sin x y(4)(�) = 0

y(5) = cos x y(5)(�) = −1

(b) y = cos x = −1+1

2!(x−�)2

− 1

4!(x−�)4 + · · ·

5. f (x)= ln x

=(x−1)− (x−1)2

2+

(x−1)3

3−+· · ·

Derivatives Values

f ′(x) = x−1 f ′(1) = 1

f ′′(x) = (−1)x−2 f ′′(1) = −1

f ′′′(x) = 2x−3 f ′′′(1) = 2

6. f (x)=4

1−3x= −4

5+

12

25(x −2)− 36

125

(x−2)2 +108

625(x−2)3 −+ · · ·

Derivatives Values

f ′(x) =4×3

(1−3x)2f ′(2) =

12

25

f ′′(x) =72

(1−3x)3f ′′(2) = − 72

125

f ′′′(x) =648

(1−3x)4f ′′′(2) =

648

625

7. An intersection at

(1,

5

3

)

f1(x) = ex −1 ≈ x +x2

2+

x3

6,

f2(x) = 2 sin x ≈ 2x− x3

3

x +x2

2+

x3

6= 2x − x3

3

x3 +x2 −2x = 0x1 = 0, y1 = 0

x2 = 1, y2 =5

3

Fig. 8.10

8.√

42 ≈ 6.4807Solution√

36+6 =

√36

(1+

6

36

)

= 6×√

1+1

6

We set x =1

6and apply the expansion

into a binomial series for√

1+x ≈ 1+x

2−x2

8+

x3

16−5x4

128+−·· ·

(c.f. exercise 1(a))Hence

6×√

1+1

6≈ 6

(1+

1

2× 1

6− 1

8× 1

36

+1

16× 1

216− 5

128× 1

1296+−·· ·

)

568 Answers

= 6(1+0.08333−0.00347

+0.00029−0.00003)≈ 6.4807

Note that an alternative approach is to

use√

72 −7 = 7

√1− 1

7.

9. (a) ln (1+x) ≈ x− x2

2

(b)1√

1+x≈ 1− x

2+

3

8x2

10. (a) e0.25 ≈ 1+0.25 = 1.25

(b) ln1.25≈ 1

4−1

2

(1

4

)2

=7

32≈ 0.219

(c)√

1.25 ≈ 1+1

2× 1

4=

9

8= 1.125

11. (a)∫

dx

1+x= x− x2

2+

x3

3− x4

4+ · · ·

= ln (1+x)+CSolution:∫

dx

1+x=∫

(1−x+x2−x3+x4 · · ·)dx

=x−x2

2+

x3

3−x4

4+

x5

5· · ·+C

= ln(1+x)+C

(b)∫

cos x dx = x− x3

3!+

x5

5!− x7

7!· · ·

= sin x +CSolution:∫

cosx dx=∫ (

1−x2

2!+

x4

4!−x6

6!+· · ·)

dx

= x−x3

3!+

x5

5!−x7

7!+· · ·+C

= sinx +C

12. (a)∫ 0.58

0

√1+x2 dx ≈ 0.6111

SolutionExpand the integrand by means ofthe binomial series.

Put√

1+x2 =(1+x2)1/2 and obtain√

1+x2 dx =∫ (

1+1

2x2−1

8x4

+1

16x6− 5

128x8

− 5

128x8 + · · ·

)dx

∫ x

0

√1+ t2 dt =x+

x3

6−x5

40+

x7

112

− 5x9

1152+· · ·(|x|<1)

Substituting the limits x = 0.58 andx = 0, we find the value 0.6111.

(b)∫ x

0

sin t

tdt = x− x3

3×3!+

x5

5×5!

− x7

7×7!+ · · ·

Solution:The integrand can be represented bythe series

sin t

t=

1

tsin t

=1

t

(t − t3

3!+

t5

5!− t7

7!+ · · ·)

= 1− t2

3!+

t4

5!− t6

7!+ · · ·

∫ x

0

sin t

tdt =x− x3

3×3!+

x5

5×5!

− x7

7×7!+· · ·

13. (a)1√

1−x2= sin−1 x = x +

1×x3

2×3

+(1×3×5)x7

2×4×6×7+

(1×3×5×7)x9

2×4×6×8×9

+· · ·(|x| ≤ 1)

(b)�

2= 1+

1

6+

3

40+

5

112+

35

1152

+ · · · ≈ 1.3167+ · · ·Comparing the approximation by thefirst five terms with the numer-

value 1.5707 . . ., we find that the ap-proximation is wrong by more than16%! Although the sequence con-verges, it does so very slowly indeed.This series has no significance for nu-merical purposes.

Chapter 9

1. (a) j√

3 (b) 12j

(c)

√5

2j(d) 10j

Answers 569

2. (a) 1 (b) −j(c) j (d) j

3. (a) 6j√

3

(b) (2√

3−2√

2+√

0.6)j(c) −3 (d) j

√ab

(e) 10j (f) −j(g) 4 (h) j(i) −2

√3 (j) 0

(k) j(a−b) for a > b ;j(b−a) for b > a

(l) ±6j

a

4. (a) 7 (b) 15

5. (a) z∗ = 5−2j (b) z∗ =1

2+ j

√3

6. (a) z1 = −2+3j (b) z1 = −3

4+ j

z2 = −2−3j z2 = −3

4− j

7. (a) 10+3j (b)3

2

8. (a) w = −1−5j (b) w = 1−13j

9. (a) w = 2 (b) w = 23+2j

10. (a) 4√

2+1

2j (b) −1

2

√3−2j

(c) −0.4+0.7j

(d)1

2− 1

2j (e) 2j

(f) 8+4j√

3

11. (a) (2x +3jy)(2x −3jy)

(b) (√

a + j√

b)(√

a− j√

b)

12.

Fig. 9.21

13. z1 = 2+ j z2 = j−1 z3 = −3z4 = −2− j z5 = −2j z6 = 2− j

14. (a) z =√

2

(cos

3�

4+ j sin

3�

4

)

(b) z =√

2

(cos

5�

4+ j sin

5�

4

)

15. (a) z =5

2− 5

2j√

3

(b) z = −2√

2−2j√

2

16. (a) z = 6(cos 60◦ + j sin 60◦)= 3+3j

√3

(b) z = 5(cos 120◦ + j sin 120◦)

= −5

2+

5

2j√

3

17. (a) z = cos 45◦ + j sin 45◦

=1

2

√2+

1

2j√

2

(b) z = cos 330◦ + j sin 330◦

=1

2

√3− 1

2j

18. Multiplication by −j = j3 means an anti-clockwise rotation of 270◦.Division by −j = j3 means a clockwise ro-tation of 270◦.

19. (a)√

2(−4+4j) (b) −120. (a) (cos 50◦ − j sin 50◦)4

= [cos(−50◦)+ j sin (−50◦)]4= cos(−200◦)+ j sin (−200◦)= cos200◦ − j sin 200◦

(b) (cos ˛ + j sin ˛)n

=cos (n˛)+ j sin(n˛)

21. (a) z1 = 2+3jz2 = −2−3j

(b) z1 = 0.966+0.259jz2 = −0.259+0.966jz3 = −0.966−0.259jz4 = 0.259−0.966j

22. (a) ej�/2 = cos�

2+ j sin

�

2= j

(b)1

2+

1

2j√

3

23. (a) cos˛ = 1, ˛ = 0sin ˛ = 0

(b) cos ˛ = −1, ˛ = �sin ˛ = 0

(c) cos ˛ = 0, ˛ = −�

2

sin ˛ = −1

570 Answers

(d) cos ˛ =1

2

√3, ˛ =

�

6

sin˛ =1

2

24. (a) r = e3 (b) r = e2

˛ = 2 ˛ = −1

2

25. (a) w = ez = −√e

(r =√

e, ˛ = �)(b) w = −

√e3

(r =√

e3, ˛ = −�)

(c) w =1

ej

(r =

1

e, ˛ =

�

2

)

(d) w = e3 (cos 1− j sin 1)≈ e3 ×0.54− e3 ×0.841j

26. (a) (i) Re[w(t)] = e−t cos 2�t

(ii) Period = 1

(iii) Amplitude = e−2×1=1

e2�0.135

(b) (i) Re[w(t)] = e2t cos

(−3

2t

)

= e2t cos

(3

2t

)

(ii) Period =4

3�

(iii) Amplitude = e4 cos 3≈ e4(−0.99)≈−54.0

27. (a) ej� = −1 (b)3

4e−j�/2 = −3

4j

28. (a) z1∗/z2 = 4 e−j� = −4

(b)1

3ej�/2 =

j3

29. (a) z5 = 32 ej� = −32

(b) z3 =1

8ej3�/4

30. (a) z1/5 = 2 ej2� = 2

(b) z1/4 =1

2ej3�/2 = − j

2

31. (a) z = 3ej� (b) z =1

2ej2�/3

32. (a) z = 7.0711 e−0.7854j

or z = 7.0711 e−j45◦

(b) z = 19.8494 e−0.71413j

or z = 19.8494 e−j40◦55′

33. (a) z = 1.8134+1.7209j(b) z = 0.4384+0.8988j

34. (a) x =−1√

2, y =

1√2

(b) z = (cos 135◦ + j sin 135◦)(= ej3�/4)

35. (a) −√3+ j (b) 2 ej5/6�

Chapter 10

1. The linear first- and second-order DEs withconstant coefficients are (b), (e) and (f).

2. (a) non-homogeneous, second order(b) homogeneous, second order(c) homogeneous, first order(d) non-homogeneous, second order(e) homogeneous, second order

3. (a) y = C1e5x +C2ex

Auxiliary equation: 2r2−12r+10=0Roots: r1 = 5, r2 = 1

(b) y = e1.5x(C1 +C2x)Auxiliary equation: 4y2−12r+9=0Roots: r1 = r2 = 1.5

(c) Complex solution:y = e−x [(C1 +C2) cos 2x

+j(C1 −C2) sin 2x]Auxiliary equation: r2+2r+5=0Roots: r1 = −1+2j, r2 = −1−2jReal solution: y = e−x(A cos 2x +B sin 2x)

(d) Complex solution:y = e0.25x [(C1 +C2) cos 0.75x

+j(C1 −C2) sin 0.75x]Real solution:y=e0.25x(Acos0.75x+B sin0.75x)

(e) y = C1 e2x +C2 e−4x

(f) Complex solution:y = e0.2x(C ′ cos0.4x+jC ′′ sin 0.4x)Real solution:y = e0.2x(A cos 0.4x+B sin 0.4x)

4. (a) y(x)= C e−4x (b) y(x)= C e30x

(c) y(x) = C e2x

5. (a) S(t) =t3

3+C1t +C2

(b) x(t) = cos !t +C1t +C2

6. (a) yp = 2x +1 (b) yp = x− 1

2

Answers 571

7. (a) y = C1 ex +C2 e−3x/7 −2

(b) y = C1 e9x +C2 ex +x +10

9

(c) y = C1 e4x/3 + C2 e−x − 1

4x2 +

1

8x− 13

32

(d) y = e−x(A cos 2x +B sin 2x)

+1

17(cos 2x +4 sin 2x)

8. y(x)=C1 e5x +C2 ex+3

10x2+

18

25x

+43

125

9. (a) y = 3 e−4x (b) y = e−21 e2.1x

10. (a) C = 0, y(x) = 0(b) C = −2, y(x) = −2 e2x

(c) C = e2, y(x) = e2 e2x

(d) C = 1, y(x) = e2x

11. y(x) = C1 cos 2x +C2 sin 2x

(a) y(x) = +sin 2x

(b) y(x) = cos 2x− 1

2sin 2x

(c) y(x) =1

2sin 2x

(d) y(x) =−b

4cos 2x +a sin 2x

12. y = ex −x ex = ex(1−x)

13. (a) y = x +Cx2

(b) y = x2 +Cx(c) y = −2 cos2 x +C cos x

(d) y = e−x

(x

2+

C

x

)

14. (a) u = y−2, u′ −2xu = −2x,

y =1√

1+C ex2

(b) u = y−1, u′ +2

x2 −1u = 1,

y =x−1

x +1

1

x− ln(x +1)2 +C

(c) u = y3, x2u′ +3xu = 3,

y = 3

√3

2x+

C

x3

(d) u = y2, u′ +2

xu = −2(x +1),

y =1

x

√C − x4

2− 2x3

3

15. (a) y =1

2ln |2 ex +C |

(b) y2 = ln |C (x +1)2|−2x

(c) y =2

(ln |x|)2 +C(d) y1 = −x +C

y2 = x + ex(1−x)+C

16. (a)∂

∂x

(2y

x

)=−2y

x2=

∂∂y

(4− y2

x2

),

F =y2

x+4x, y2 = Cx −4x2

(b)∂

∂x(1−x e−y) =−e−y =

∂∂y

e−y ,

F = y +x e−y , y +x e−y = C

(c)∂

∂x(2y −x2 sin 2y) = −2x sin2y

=∂

∂y(2x cos2 y), F =y2+x2 cos2 y,

y2 +x2 cos2 y = C

(d)∂

∂x(2x −3)=2=

∂∂y

(3x2+2y),

F =x3+2xy−3y, y=x3+C

3−2x

17. (a) Special case 1: = ex ,e3x −3ex cosy = C

(b) Special case 2: = e−y ,y +x e−y = C

18. (a) x = e6t (Acos t +B sin t)y = e6t [(A−B)cos t+(A+B)sin t ]

(b) x = Acos t +B sin t

y =1

2(B−3A)cos t−1

2(A+3B)sin t

(c) x = (A+Bt) et +(E +F t) e−t

y =1

2(B−A−Bt)et−1

2(E+F

+F t)e−t

Chapter 11

1. (a)3

2s4(b)

5

s +2

(c)4s

s2 +9(d)

2

s(s2 +4)

2. (a)1

2sin

1

2t (b)

1

4(1− e−4t )

(c)2

9(1− cos 3t) (d) −6 sinh t

(e) t − sin t (f)1

2e4t − e2t +

1

2

572 Answers

3. (a) y =2

3(e−t − e−4t )

(b) y =1

5sin 2t − 7

15sin 3t + cos 3t

(c) y =1

5sin t +

2

5cos t +

3

5e−2t

4. y = 2−3 ex +3 e2x

5. y =1

2et − 1

2e−t +

1

6t3 − t

6. y = 25−9 et +5t et −16 e−t/4

7. y = e−t/6 − e−t/2 ,

x = 1− 1

2(e−t/6 + e−t/2)

8. Q=4×10−4(1.12sin447t− sin500t)

Chapter 12

1.x -2 -1 0 1

y

−2 −2 4 6 4−1 2 5 6 5

0 6 6 6 61 10 7 6 72 14 8 6 8

2. (a) The function z = −x −2y + 2 repre-sents a plane. The intersecting curvesof the surface are(1) with the x−y plane: y =−x

2+1

(2) with the x−z plane: z =−x+2

(3) with the y−z plane: z=−2y+2

(b) The function z = x2 + y2 representsa hyperboloid of revolution about thez-axis. Intersecting curves with planesparallel to the z-axis are parabolas.Intersecting curves with planes parallelto the x−y plane are circles.

(c) The function z =

√1− x2

4− y2

9re-

presents one half of an ellipsoid abovethe x−y plane. The intersecting curveswith the x−z plane and the y−z planeare semi-ellipses.

3. (a) fx = cos x, fy = −sin y

Fig. 12.31

Fig. 12.32

Fig. 12.33

(b) fx = 2x√

1−y2, fy =−x2y√1−y2

(c) fx = −2x e−(x2+y2),

fy = −2y e−(x2+y2)

(d) fx = yz +y, fy = xz +x,fz = xy +1

Answers 573

(e) fx = ex ln y, fy =ex

y, fz = 4z3

(f) fx =esin x cosx−ecos (x+y) sin(x+y),fy =−ecos (x+y) sin(x+y)

4. Tangent in x-direction: 2x

Slope in x-direction at point P: 0Tangent in y-direction 2ySlope in y-direction at point P: 2

5.fxx = −2 fyx = 0fxy = 0 fyy = −2

6. Withfx=2x

y2e(x/y)2

andfy=−2x2

y3e(x/y)2

the statement follows.7. (a) dz =

−x dx√1−x2 −y2

− y dx√1−x2 −y2

(b) dz = 2x dx +2y dy

(c) dz =−2

(x2+y2+z2)2

×(xdx +ydy +zdz)

8. (a) 7.6% (b) 5.13%

9. (a) The contour lines are straight lines.

y = −x

2+1−C

grad f = (−1,−2)(b) The contour lines are ellipses.

x2

4+

y2

9= 1−C 2

grad f =−1√

1−x2/4−y2/9

(x

4,y

9

)

(c) The contour lines are circlescentered at the origin.

grad f =−10√

(x2 +y2)3(x,y)

10. (a) The surfaces of constant functionvalue are planes.

z =x

3+

y

3+C

gradf = (1,1,−3)(b) The surfaces of constant function

value are cylinders centered at theorigin.C = x2 +y2

grad f = (2x, 2y, 0)

(c) The surfaces of constant functionvalue are spheres centered at theorigin.C = x2 +y2 +z2

gradf =3(x2+y2+z2)1/2(x, y, z)

11. (a) sin 2t −3 cos 2t

(b) 3t2 +4+3

t2

12. (a) 2x +2ay (b) 1

(c) −�

8(d) 0

13. (a) Minimum at x = 1, y = 1(b) Maximum at x = 10, y = 8(c) � = 30◦, h = 0.32m, l = 0.20m

14. Cable (a)f (x, t)

= 0.5cos

(2�×5×t−2�x

1.2+�

)m

or

f (x, t) = 0.5cos2�(5t − x

1.2+�1

)m

or

f (x, t)=0.5cos2�

1.2(12.5t −x +�2)m

Cable (b)

f (x, t) = 0.2cos2�(

0.8t − x

4.0+�)

The wave velocities are unequal:ca = 6m/s cb = 3.2m/s

15.∂f

∂ t= −2v(vt −x)e−(vt−x)2

∂ 2f

∂ t2= −2v2e−(vt−x)2

+(2v)2(vt −x)2e−(vt−x)2

∂ 2f

∂x2= −2e−(vt−x)2

+4(vt −x)2e−(vt−x)2

Consequently∂ 2f

∂ t2= v2 ∂ 2f

∂x2

Chapter 13

1. (a) ab (b)2

3

(c) 4 (d) 12

574 Answers

(e) 4 (f)1

a(eaz1 −eaz0 )(y1 −y0)

2. (a) 102

3(b)

4

3

(c) ab� ;

(4a

3�, 0

)

3. (a) r = 3√

2, � =�

4

(b) R2 = x2 +y2, r = R,in polar coordinates

(c) r =�

2�(d)

a3

3√

2

4. (a) V = �h(R22 −R1

2)

(b) V =1

3�R2h

I =3

10MR2 (M = total mass = V)

5. I =2

3MR2, M =

4

3�R3

Chapter 14

1. x′ = x, y′ = y, z′ = z−2

2. The transformations arex = x′ −2 , y = y′ +3

Substitution in the equation gives

y′ = −3x′ +8

3. r ′ =

(1+2

√3

−√3+2

)

4. The transformation equations are

x = x′√

3

2−y′ 1

2, y = x′ 1

2+y′

√3

2

Substitution in the equation gives

y′ =6√3− 3√

3x′

5. The transformation equations are

x′ = 3cos 30◦ +3sin30◦= 4.0981 to 4d.p.

y′ = −3sin30◦ +3cos30◦= 1.0981 to 4d.p.

z′ = 3

Hence r ′ = (4.0981, 1.0981, 3)

6. (a) A +B =

⎛⎝

3 31 8

−1 9

⎞⎠

(b) A−B =

⎛⎝−1 3

3 21 5

⎞⎠

7. (a) 6A =

⎛⎝

12 4218 054 −6

⎞⎠

(b) AB =

⎛⎝

25 −8 2827 9 080 29 −4

⎞⎠

BA =(

27 6332 3

)

Hence AB �= BA.

8. No matrix multiplication is possible in thiscase.

9.

(x′y′

)=(

x−2y5x +7y

)

10. (a) AT =(

1 4 32 −3 0

)

(b) (AT)T =

⎛⎝

1 24 −33 0

⎞⎠= A

11. 3.

12.

⎛⎝

54 0.5 4.50.5 26 524.5 52 9

⎞⎠+

⎛⎝

0 0.5 −3.5−0.5 0 −323.5 32 0

⎞⎠

13.AA−1 =

1

13

⎛⎝

−8+0+21 6+0−6−16+9+7 12+3−2−8−6+14 6−2−4

9+0−918−15−3

9+10−6

⎞⎠

=1

13

⎛⎝

13 0 00 13 00 0 13

⎞⎠= I

Similarly, A−1A = I.

Chapter 15

1. (a) x1 = −1, x2 = 6, x3 = −5

(b) The second and third equation are lin-

Answers 575

early dependent. Thus the solution con-tains z as parameter free to

x =21−10z

25, y =

−79+65z

25

(c) x1 = 13, x2 = 15, x3 = −20

(d) x =0.42−1.5z

0.12, y =

0.24−1.8z

0.12The first and third equations are lin-early dependent.

2. (a)

⎛⎜⎜⎜⎜⎜⎜⎝

1

4

1

2−1

4

1

2−1

1

2

3

8−3

4

1

8

⎞⎟⎟⎟⎟⎟⎟⎠

(b)1

20

(7 −8

−6 −4

)

3. (a) x1 = x2 = x3 = 0

(b) x = −7z

20, y =

z

10

4. (a) −9 (b) 0 (c) −322(d) −186 (e) 22

5. (a) r = 2 (b) r = 3

6. (a) detA = −104 �= 0; hence unique so-lution.

(b) detA = 0; no unique solution exists.(c) detA �= 0; unique solution exists.(d) detA = 0; first and third equation are

dependent.(e) detA = 0; third equation is a linear

combination of the first two equations.

Chapter 16

1. (a) The characteristic equation is

det

(4−� 2

1 3−�

)

= (4−�)(3−2�)−2

= �2 −7�+10 = 0�1 = 2, �2 = 5

For � = 2, solve(2 21 1

)(x1

y1

)= 0,

i.e. 2x1 +2y1 = 01x1 +1y1 = 0

This reduces to x1 + y1 = 0. A con-venient solution is

r1 =(

1−1

)

For �2 = 5, solve(−1 21 −2

)(x2

y2

)= 0,

i.e.−1x2 +2y2 = 01x2 −2y2 = 0

This reduces to x2 −2y2 = 0. A con-venient solution is

r2 =(

2−1

)

Fig. 16.4

2. No. The characteristic equation is a realpolynomial equation of degree 2z. We knowfrom algebra that if z is a complex rootthen z∗ is a root as well, i.e. this character-istic equation has either two complex rootsor two real roots.

3. The characteristic equation is

(3−�)(1−�)+4 = �2 −4�+7 = 0

There are no real roots, since�1,2 = 2±√

4−7 are complex numbers.

4. (a) The characteristic equation is

det

⎛⎝

−1−� −1 1−4 2−� 4−1 1 5−�

⎞⎠

= −�3 +6�2 +4�−24 = 0

If � is an integral root, then it must di-vide into 24, the last coefficient.�1 = 2,�2 = −2,�3 = 6.

576 Answers

(b) For �1 = 2; solve

− 3x1 − y1 + z1 = 0− 4x1 + 4z1 = 0− x1 + y1 + 3z1 = 0

This reduces to x1 = z1 and y1 =−2x1. x1 = 1 gives the particular so-lution:

r1 =

⎛⎝

1−2

1

⎞⎠

For �2 = −2, solve

x2 − y2 + z2 = 0− 4x2 + 4y2 + 4z2 = 0− x2 + y2 + 7z2 = 0

This reduces to x2 −y2 +z2 = 0 andx2−y2−z2 = 0. Hence x2 = y2 andz2 = 0.Choosing x2 = 1 gives the particularsolution:

r2 =

⎛⎝

110

⎞⎠

For �3 = 6, solve

− 7x3 − y3 + z3 = 0− 4x3 − 4y3 + 4z3 = 0− x3 + y3 − z3 = 0

This reduces to x3 +y3 −z3 = 0 andx3 −y3 +z3 = 0.Hence x3 = 0 and y3 = z3. Choosingy3 = 1 gives the particular solution:

r3 =

⎛⎝

011

⎞⎠

5. �1 = 1, �2 = 1.For the first eigenvalue an eigenvector canbe quickly found:

r1 =(

10

)

But for �2 we should like to have anothereigenvector which is truly different (i.e. notmerely a multiple of r1). Unfortunately, nosuch vector exists.

Chapter 17

1. (a) A = 4(0,0,1)(b) A = 4(0,1,0)(c) A = 4(1,0,0)−A would be a proper solution in eachcase also.

2. A =a ·b√

2(0,1,1)

3. (a) F ·A = 5+3 = 8(b) F ·A = 10(c) F ·A = 9

4. A1 = 6(0,0,1) = −A2

A3 = 8(0,1,0) = −A4

A5 = 12(1,0,0) = −A6

5. F = (2,2,4) is a homogeneous vector field.∮F ·dA = 0 for (a) and (b)

6. F (x,y,z) is a spherical symmetric fieldfor (a) and (b). Rule 17.8 tells us∮

F ·dA = 4�R2f (R) for R = 2

(a) F (R) =3R

R2=

3

R∮

F ·dA=4� · 3R2

R= 12�R

(b) F (R) =R√

1+R2

∮F·dA=4�R2 R√

1+R2=

4�R3

√1+R2

7. The differential surface element vector isdA = (dydz,0,0).∫

F ·dA =∫

zdydz =∫ 3

0zdz

∫ 2

0dy

=9

2·2 = 9

8. (a) divF = 3Each point in space is a source.

(b) divF = 2zIn the plane z = 0 no point is a sourceor a sink. All points below are a sink,all points above are a source.

9. (a) curlF = (0,0,1)This vector field has curl.

(b) curlF = (0,0,0)This vector field is curl-free.

Answers 577

10. We know that curlF = (0,0,0). Therefore∮

CF ·ds = 0

11. Because of curlF = (0,0,0) the line in-tegral is independent of the path. Thereforewe chose the path of integrationalong the z-axis, z = 0 to z = 3∮

CF ·ds =

∫ 3

0(0,y,z) · (0,0,dz)

=∫ 3

0zdz =

9

2

Chapter 18

1. Since f (x) is an even function, the coeffi-cients bn vanish.

a0 =1

�

∫�−� f (x) dx = 1

an =1

�

∫�/2

−�/2cosnx dx

=1

�n

[sinnx

]�/2

�/2

=1

�n

(sin

n�

2−sin

(−n�

2

))

=2

�nsin

n�

2

For n even an is zero. Thus the Fourierseries is

f (x)=1

2+

2

�

∞

∑n=1

(−1)n−1

2n−1cos(2n−1)x

2. The function is odd. Thus all coefficientsan are zero.

bn =1

�

∫ �

−�f (x)sinnx dx

=1

�

∫ 0

−�sinnxdx− 1

�

∫ �

0sinnxdx

=−1

�n[1−cos(−n�)]+

1

�n[cosn�−1]

=−2

�n+2

(−1)n

�n

For n even the coefficients bn are zero.Thus we obtain

f (x) = − 4

�

∞

∑n=0

1

2n+1sin(2n+1)x

3. a0 =1

�

(−∫ 0

−�sinx +

∫ �

0sinx

)=

4

�

an =1

�

(−∫ 0

−�sinx cosnx dx

+∫ �

0sinx cosnx dx

)

=1

�× 1

2

[[cos(n+1)x

(n+1)

− cos(n−1)x(n−1)

]0

−�

+[− cos(n+1)x

(n+1)

+cos(n−1)x

(n−1)

]�

0

]

an =

⎧⎨⎩

4

�(n+1)(n−1), n even

0, n odd

bn =1

�

(∫ 0

−�−sinx sinnx dx

+∫ �

0sinx sinnx dx

)= 0

The Fourier series of the rectified wave-form

f (x) =2

�− 4

�

1

1×3cos2x

− 4

�

1

3×5cos4x

− 4

�

1

5×7cos6x −·· ·

4. A similar function, with the period 2� , hasbeen treated in the example on p. 495.

f (x)=1

2+

2

�

∞

∑n=1

(−1)n−1

2n−1cos

(2n−1)2

x

578 Answers

Chapter 19

1. Let A, B, C, D, E be the five dishes. Thesample space consists of the following setsof possible pairs:{AB, AC, AD, AE, BC, BD, BE, CD, CE,DE}

2. P =1

2

3. P =1

3

4. hA =1

30

5. P(blue) = 0.8, P(green) = 0.2Compound probability = 0.16

6. P =1

36× 1

9=

1

324

7. P =1

36+

2

36+

3

36=

1

6

8. Np = 5! = 120

9. N =(

153

)=

15!3!(15−3)!

= 455

Chapter 20

1. The probability distribution for the randomvariable ‘sum of the number of spots’ wasgiven in Sect. 20.1.1.

Random Probability Random Probabilityvariable variable

21

368

5

36

32

369

4

36

43

3610

3

36

54

3611

2

36

65

3612

1

36

76

36

The mean value is

x = 2× 1

36+3× 2

36+4× 3

36+5× 4

36

+6× 5

36+7× 6

36+8× 5

36+9× 4

36

+10× 3

36+11× 2

36+12× 1

36.

Thus x = 7.

2. x =∫ +∞

−∞x f (x) dx =

∫ 2

0x

x

2dx

=[

x3

6

]2

0

=4

3

3. P =(

108

)(0.6)8(0.4)2

= 45×0.016×0.16= 0.12

4. The random variable which is distributedaccording to

f (x) =1

�√

2�e−[(x−�)/� ]2/2

has the mean value . Hence it follows that

(a) x = 2, (b) x = −4.

Chapter 21

1. (a1) S (a2) R (b) S (c) S

2. (a)

i i − ( i − )2

(g/cm3) (g/cm3) (g/cm3)2

3.6 0.4 0.163.3 0.1 0.013.2 0 03.0 −0.2 0.043.2 0 03.1 −0.1 0.013.0 −0.2 0.043.1 −0.1 0.013.3 0.1 0.01

Sum 28.8 0 0.28

�2 =0.28

8= 0.035(g/cm3)2

� = 0.19g/cm3

= 3.2g/cm3

Answers 579

(b) Mean value:

� = ∑�i

n=

12.80

10m/s = 1.28m/s

Variance:

�2 = ∑(�i −�)2

N −1=

0.011

9(m/s)2

= 0.00122(m/s)2

Standard deviation:� = 0.035m/s

3. Mean value:

=∫ 1

0x dx =

1

2

Variance:

�2 =∫ 1

0

(x− 1

2

)2

dx =1

12

4. (a) �M=�√N

=0.19g/cm3

3= 0.06g/cm3

Confidence intervals:

3.14g/cm3 ≤ ≤ 3.26g/cm3

3.08g/cm3 ≤ ≤ 3.32g/cm3

(b) �M =0.035m/s

3.16= 0.01m/s

Confidence intervals:

1.27m/s ≤ v ≤ 1.29m/s

1.26m/s ≤ v ≤ 1.30m/s

5. 16%

6. (a) A=xy=120×90cm2=10 800cm2

Calculation of �MA using Gaussianerror propagation law:

Ax =∂

∂x(xy) = y,

Ay =∂

∂y(xy) = x

Ax(x, y) = 90cm

Ay(x, y) = 120cm

�MA2 = Ax

2�x2 +Ay

2�y2

= 902(0.2)2cm4

+1202(0.1)2cm4

= 468cm4

�MA = 21.63cm2

A = (10 800±21.63)cm2

(b)V =

4

3�

(D

2

)3

= 124.79cm3

=M

V=

1000

124.79g/cm3

= 8.014g/cm3

Calculation of �MV using Gaussianerror propagation law:

∂∂M

(M

V

)=

1

V=

1

124.79cm3

= 0.0081

cm3

∂∂D

(M

V

)=

∂∂D

(6M

�D3

)=−18m

�D4

�M2 = (0.008)2(0.1)2

( gcm3

)2

+(3.88)2(0.01)2( g

cm3

)2

= 0.0015041( g

cm3

)

�M = 0.039g

cm3

= (8.014±0.039)g

cm3

7. a = ∑mi Si −nmS

∑m2i−nm2

=65.6−5×4×3

90−5×42

=5.6

10= 0.56

b = S −am = 3−0.56×4 = 0.76

m m2 S mS(g) (g2) (cm) (g cm)

1 2 4 1.6 3.22 3 9 2.7 8.13 4 16 3.2 12.84 5 25 3.5 17.55 6 36 4.0 24

∑ 20 90 15 65.6

m = 4g S = 3cm

580 Answers

8. 73◦2′8′′

Fig. 21.7

Index

A

Abscissa 42Absolute error 122Acceleration 99, 134, 158, 273Acoustic wave 371Addition

formulae 62 f.law 511of vectors 4theorems 67, 411

Amplitude 55, 59Angular velocity 131Antisymmetric matrix 423Aperiodic system 297Approximate polynomial 233, 243Approximation 228

first 234second 234third 234

Areabounded by curves 191function 150, 151in polar coordinates 195of a circle 196, 385, 388

Argand diagram 250, 253, 256Argument 41, 252, 256Arithmetic mean value 525Asymptote 46, 119Atmospheric pressure 237Augmented matrix 433, 434, 436Auxiliary equation 281, 284Average velocity 98Axial symmetry 390

B

Base 69, 74Base vector 471Bernoulli DE 306Bernoulli’s equations 306Binomial

coefficient 71, 517distribution 527, 529, 533expansion 517theorem 71

Bound vectors 9Boundary condition 147, 276, 291, 293, 294,

296, 372

C

Cantilever beam 292Cardioid 202Cartesian coordinate system 42, 387Catenary 79Cauchy 232Center of mass 208, 210Center of pressure 208, 222, 223Centroid 208, 210, 398, 550Chain rule 104Characteristic

equation 454–456, 459polynomial 455, 456

Circle, equation in parametric form 131Circular frequency 57, 369Circulation 481 f.Clearing the fractions 172Co-domain 40Cofactor 439, 443Column vector 415, 432, 445, 451Combination 516 ff.

582 Index

Common ratio 86Commutative law 27Complementary

area 193, 194function 277, 278, 285

Complex conjugate 248, 266Complex number 247 ff.

addition and subtraction 249, 251arithmetic form 266division 250exponential form 254, 266graphical representation 250multiplication and division 258periodicity 266polar form 252, 261product 249raising to a power 263roots of a 263summary of operations 266transformation of one form to another 260

Complex root 171, 174Component 10, 43Composition 66 f.Compound

event 513, 514probability 532

Confidence interval 545Conservative field 184, 487Continuity 91Continuous quantity 522Contour line 350, 351, 357, 360Contraflexure 117Coordinate 11

system 42, 386 ff.Correlation 554

coefficient 554Cosine 58

function 58 ff.function, exponential form 255, 266function, integration of 156rule 27

Cotangent 61Cramer’s rule 438, 445 ff.Critical damping 297Cross product 32 ff., 483Curl 480 ff., 483Curl-free 480, 484Curvature 123, 125

centre of 123radius of 123, 125

Curve sketching 45 f., 118Cycloid 137

area of 194

Cylindricalcoordinates 389symmetry 391

D

D’Alembert’s solution 372Damping 296DE see Dfferential equation 273De Moivre’s theorem 263Definite integral 147, 149, 153, 154, 175,

191, 379Derivative 97–99, 114, 145

of a constant 102of a constant factor 102, 138of a cosine function 107of a curve given in parametric form 136of an exponential function 139of a function 97 f., 344of a function of a function 104of a hyperbolic function 110of a hyperbolic trigonometric function

139of an implicit function 127of an inverse function 105 f.of an inverse hyperbolic function 111of an inverse hyperbolic trigonometric

functions 139of an inverse trigonometric function 109,

138of a logarithmic function 139of a parametric function 129, 133partial 347of a position vector 133of a power function 101, 138of a product of two functions 103of a quotient of two functions 104of a sine function 107of a sum 102 f.of a trigonometric function 107, 138

Designation 177Determinant 423, 424, 438 ff.

evaluation of a 440expansion of a 439of a square matrix 439properties of a 442

Diagonal 414, 430Diagonal form 444Diagonal matrix 421Difference

quotient 96, 97vector 6

Differential 99, 100, 351calculus 98

Index 583

coefficient 97, 99, 100total 351

Differential equation (DE) 273 ff.exact 308first-order linear 275general first-order 306general linear first-order 302higher-order 317homogeneous 275, 277homogeneous first-order 279homogeneous second-order 279, 281, 282linear first-order 277linear with constant coefficients 275linear with constant coefficients, solution of

328non-homogeneous 277non-homogeneous linear 285second-order 276simultaneous first- and second-order 313simultaneous with constant coefficients

330solution by substitution 285

Differentiation rules 138Direction 1Dirichlet’s lemma 495Dirichlet, Peter G.L. 495Discriminant 48Distributive law 27Divergence 461Divergence of a vector field 475Domain 340

definition of 39, 64of definition 40

Dot product 24Double integral 383, 398

E

Eigenvalue 452 ff.Eigenvector 452 ff.Electrical field 470Elemental

area 388volume 392

Elementaryerror 545event 508, 509, 514, 532

End term 92Equation

of a line 26of a sphere 406

Equations, linear algebraic 429Error

constant 537

random 537systematic 537

Error propagation 547Estimate

of the arithmetic mean value 541of the standard deviation 541of the variance 541

Euler’sformula 255, 266, 282number 71, 88

Even function 55, 495Event 508

exclusive 512statistically independent 514

Exact DE 308 ff.Expansion

of a function 228, 229, 235of the binomial series 231of the exponential function 230

Experiment 508Exponent 69, 74Exponential function 71 f., 76, 109, 110Extrapolation 39Extreme point 362

value 119, 121value, necessary condition 364 f.value, sufficient condition 364 f.

F

Factorial n 516Favoured number 509First moment 209, 398Flow density 462Flow, partial 467Flux 464Fourier

serie 492spectrum 502

Fourier’s theorem 491Fraction

improper 170proper 170

Frequency 369, 492Function 40, 41

circular 52continuous 91discontinuous 91explicit 127exponential 71fractional rational 170implicit 127inverse 50limits of 125

584 Index

linear 43of a function 66 f., 67of two variables 337, 367periodic 54real 41trigonometric 52

Fundamental harmonic 495Fundamental theorem

of algebra 170of the differential and integral calculus

149, 152

G

Galtonian board 528Gauss 470Gauss complex number plane 250Gauss–Jordan elimination 431, 433–435,

443Gaussian

bell-shaped curve 240elimination method 430, 431normal distribution 545

Gaussian law 470General

solution 276, 279term 85, 92

Geometricprogression 86, 93serie 93, 227

Geometrical addition 4Gradient 483

of a function 357of a line 94

Graph 42Graph plotting 44 ff.

H

Hadamard 232Half sine wave 498Half-life 73Harmonic 503

analysis 491oscillator 294oscillator, damped 296oscillator, undamped 294wave 371, 373

Helix in parametric form 132Hertz 57Higher derivatives 112Homogeneous

equation 279linear equations 436

Hyperboliccosine function 79cotangent 80function 78 f.sine function 78tangent 79

I

Imaginarynumber 247, 248, 266part 248unit j 266

Implicit function 360Impossible event 509Improper integral 179 ff.

convergent 179divergent 179

Increment 122Indefinite integral 145, 159Indeterminate Expression 125Index 69Infinite series 93Initial condition 291Inner

function 66, 105integral 379product 24 f.

Instantaneous velocity 98Integral

calculus 145 ff.sign 149

Integrand 149Integrating factor 302, 304, 311 ff.Integration

application of 191 ff.by part 161 ff., 186by partial fraction 170 ff., 186by substitution 164 ff., 186constant 276

Intersecting curve 340, 342, 345, 367Interval of convergence 232Inverse 50 f., 432

cosine function 65cotangent function 65function 50, 64, 67, 322hyperbolic cosine 81hyperbolic cotangent 82hyperbolic function 81hyperbolic sine 81hyperbolic tangent 82matrix 424, 432matrix, calculation 433sine function 64

Index 585

tangent function 65trigonometric function 64

Irrotational field 480

L

L’Hôpital’s Rule 125Lagrange 235Laplace transform 321 ff.

of a sum of functions 326of derivates 326of products 324of standard functions 322table of inverse transforms 333table of transforms 332

Last term 92Leading diagonal 414, 421, 423Leading term 92Lengths of a curve 198

in polar coordinates 201Lever law 30Limit

of a function 89, 90of a sequence 86

Limiting value 86Line integral 181 ff., 480Linear

algebra 413factor 456

Linear DE 274Linear independence 280Logarithm 74 ff.

common 75conversion of 76natural 75

Logarithmicdifferentiation 128function 77, 109

Lower limit of integration 149Lower sum 149

M

Maclaurin’s series 229, 232, 237Magnitude 2, 4Matrix 413 ff.

addition and subtraction 415algebra 413column of 414element of a 415equation 432for successive rotation 419multiplication by a scalar 416notation 434, 445

of coefficient 432, 433rectangular 414row of 414

Maximum 114, 115, 119, 120local 113of a function of several variables 361

Maxwell 372Mean value 525 ff., 538

continuous distribution 542continuous random variable 526discrete random variable 525of a function 178theorem 178weighted 548

Mechanical work 25, 180Meridian 391Minimum 115, 119, 120

local 113of a function of several variables 361

Minor 439Modulus 252, 255Moment of inertia 208, 213, 215, 395, 398Monotonic function 51Multiple integral 377 ff., 378

with constant limit 378, 379with variable limit 382, 384

Multiplication of two matrices 417

N

N -order determinant 439Nabla operator 483Natural frequency 294, 301, 316

damped 301undamped 301

Negative vector 6Neutral axis 220Newton–Raphson

approximation formula 239Non-homogeneous equation 278Non-trivial solution 436Normal distribution 73, 529, 530 ff., 545Normal vector 135, 136Normalisation condition 512, 524Null

matrix 422sequence 86vector 6

O

Oblique coordinate system 8Odd function 55, 495Order

586 Index

of a DE 273of integration 383

Ordinate 42Orthogonal matrix 422, 424Orthogonality 458Oscillation

damped 298forced 299

Outcome space 508Outer

function 66, 105integral 379product 32 f.

Over-damped system 297

P

Paired values 41Pappus’ first theorem 211Pappus’ second theorem 212Parallel axis theorem 218, 221Parameter 129–132Parametric

form of an equation 129 ff., 181function 194

Parent population 540, 542Partial

derivative 347derivative, higher 348differential equation 371differentiation 344

Particularintegral 276, 285solution 276, 291, 458

Path element 182, 357Period 54, 56 f., 59, 369, 492Periodic function 491Periodicity 263Permutation 515, 516, 532Perpendicular axis theorem 217Phase 57 f., 59, 370Phase velocity 369Point charge 470Point of inflexion 113–115, 118, 120Polar angle 391Polar coordinate 195, 387 ff.Polar moment of inertia 216, 398Pole 45, 118Polynomial 170

as an approximation 237Position vector 9, 42, 43, 129, 182Postmultiplication 424Potential 485, 486

field 486

Power 69 f., 74Power series 227–229

infinite 227interval of Convergence 232 f.

Premultiplication 424, 433Primitive function 145, 146, 154, 159Principal value 264Principle of verification 159Probability 508 ff., 520 ff.

classical definition 509density 524, 526density function 524, 525distribution 519, 520, 524–526distribution, continuous 522, 523distribution, discrete 519statistical definition 510

Product moment correlation 554Product of a matrix and a vector 416Product rule 103, 161Projection 7, 43, 350Pythagoras’ theorem 27, 60

Q

Quadrant 42Quadratic 47, 363

pure 47Quadratic equation 47 f.

root of 48Quantity

dependent 39independent 39

Quotient rule 104

R

Radian 52Radioactive decay 293Radius

of convergence 232, 233of gyration 218, 398vector 391

Randomexperiment 508, 509, 519, 526, 527sample 538, 540variable 519 ff.variable, discrete 519

Random error 552Range

of definition 40, 116, 120of value 40, 119

Rankof a determinant 444of a matrix 444

Real

Index 587

matrix 413part 248

Rectangular waveform 498Reduction formula 163Regression

curve 552line 549 ff., 554

Relationship 41Relative

error 122frequency 510

Remainder 233, 235Resonance 301Resultant 5Right-hand rule 32Rotation 404, 407, 409

in three-dimensional space 411transformation rules for 409

Row vector 414

S

Saddle point 114, 362Salient features of a function 45Sample space 508Sampling error 544Sawtooth function 496Scalar 1 ff.

product 23 ff., 24, 357, 416, 417quantity 4

Secant 96Second

derivative 112, 113harmonic 495moment of area 213, 220

Second-order determinant, evaluation of 440Separable variable 307Separation of variables 279, 307, 308Sequence 85 ff., 92

convergent 87, 87 f., 88divergent 87, 88

Series 92 ff.Set of linear algebraic equations 429, 445Set of linear equations

existence of solutions 435 ff., 445Shift theorem 323Sine 53

function 53 f., 55function, exponential form 255, 266

Singular matrix 423Singularity 45Skew-symmetric matrix 423Skew-symmetry 423Slope 98, 114, 346

of a line 44, 94Small tolerance 354Space integral 378Spatial polar coordinate 391Special solution 276Sphere, equation of a 343Spherical

coordinate 391symmetry 394wave 371

Square matrix 414, 422, 424, 444Standard deviation 531, 539, 540

of the mean value 544Standard integral 159Static moment 398Stationary wave 373Statistical

mechanics 507probability 510

Steady state 300Steiner’s theorem 218Stokes’ theorem 484, 485Straight line 43, 145

equation in parametric form 131Submatrix 444Substitution 165

of limits of integration 177Successive

elimination of variables 430rotation 411

Sum rule 102Summation sign 92Superposition formula 63, 295Surface area of a solid of revolution 202Surface element 461

vector 462vector differential 471

Surface in space 350Surface integral 464Symmetric matrix 423, 424

T

Tangent 61, 95, 113, 114plane 361vector 135, 136

Taylor’s series 236, 237Techniques of integration 186Term-by-term integration 228Theory of errors 507Third-order determinant, evaluation of 441Torque 30, 31Total

derivative 358, 360

588 Index

differential 351 ff.differential coefficient 359 f.

Trace 456Transformation

equations for a rotation 419matrix for rotation 421rule 405, 407rule for successive rotation 411

Transient phase 300Translation 403, 404Transpose 422 ff., 458Transposed matrix 422Triangle law 5Triangular function 497Trigonometric functions 52Triple integral 378Trivial solution 436

U

Uniformly loaded beam 118Unique solution 435Unit

circle 52matrix 421, 433vector 10 f., 34

Upperlimit of integration 149sum 149

V

Variabledependent 41independent 41of integration 149

Variance 539 ff.explained 554of a continuous distribution 542

Variate 519Variation 518

of parameters 289of the constant 302 ff.

Vector 1 ff., 414addition 4, 12component representation 7magnitude 15multiplication by a scalar 14product 23, 30 ff., 448product, determinant form 35projection of a 7quantities 2representation of a 3subtraction 6, 9

Vector derivatives 488Vector field 461

homogenous 461, 462Velocity 98, 134, 158Vibration 297Vibration equation 108Volume

of a parallelepiped 448of a solid of revolution 202of a sphere 394

W

Wave equation 371, 372Wavelength 368Weight 548Weighted average 548Work done by the gas during expansion 193

Z

Zeroof a function 46