APPENDIX 1: UNITS - Springer978-1-349-11717-8/1.pdf · APPENDIX 1: UNITS The basic SI units ... newton, N joule, J pascal, Pa kgm-3 coulomb, C volts, V Relationship to ... 3 X X+

APPENDIX 1: UNITS

The basic SI units needed in this text are based on the following fundamental quantities shown in Table Al.l

Table Al.l

Quantity

Length

Mass

Time Temperature Angle Electric current

Sf unit

metre, m

kilogram, kg

second, s kelvin, K radian, rad ampere, A

Other common units

1 gram= --kg

1000

oc degree

The units associated with other quantities are based on the fundamental quantities shown in Table A1.2.

Table A1.2

Quantity Sf unit

Velocity ms-1

Acceleration ms-2

Force Work, energy Pressure Density Electric charge Electric potential

newton, N joule, J pascal, Pa kgm- 3

coulomb, C volts, V

Relationship to fundamental units

N = kgms-2

J = Nm = kgm2s-2

Pa = Nm-2 = kgm- 1s-2

C =As V = J/C = kgm2A -ls-3

483

484

APPENDIX 2: MATHEMATICAL FORMULAS

PROPERTIES OF EXPONENTIALS AND LOGARITHMS

log(xy) = log x + logy

log (;) = log x - log y

log(xn) = n log x

QUADRATIC EQUATIONS

If a~ + bx + c = 0 then

x=-----2a

THE BINOMIAL THEOREM

For n a non negative integer

where

n! nc =----r (n - r)!r!

For n a real number and I x I < 1

n(n - 1) n(n- 1)(n- 2) (1 + x)n = 1 + nx + x2 + x3 + ...

2! 3!

ARITHMETIC AND GEOMETRIC PROGRESSIONS

For an arithmetic progression with first term a and common differenced

nth term tn = a + (n - 1)d, and

Sum of n terms Sn = n/2[2a + (n - 1)d)

For a geometric progression with first term a and common ratio p

nth term tn = apn-l, and

a(1 - pn) Sum of first n terms Sn = --'----'-..:....

1 - p

If I p I < 1 then

a Sum to infinity Soo = --

1 - p

DIFFERENTIATION

General rules

The chain rule

dy dy du -=-·-dx du dx

Mathematical Formulas 485

486 Foundation Maths for Engineers

The product rule

dy dv du y = uv ~ - = u- + v-

dx dx dx

The quotient rule

du dv v--u-

u dy y = -~-=

v dx

dx dx

Some useful derivatives

Note in the following table a and b are constants

dy y

dx

a 0

X 1

xn n~-1

axn anxn-1

smx COS X

COS X -sin x

tan x sec2 x

sec x sec x tan x

cosec x -cosec x cot x

cot x -cosec2 x

ex ex

1 In x

X

sin(ax + b) a cos(ax + b)

cos(ax + b) -a sin(ax +b)

eax+b atfx+b

a ln(ax + b)

ax+ b

INTEGRATION

Some Useful Integrals

Note in the following table a and bare constants and cis an arbitrary constant.

Jadx=ax+c

f xn dx = - 1- xn+l + c, provided n ;1'- -1 n + 1

f a~ d.x = _a_ xn+l + c, provided n ¥- -1 n + 1

J sin x dx = -cos x + c

J cos x dx = sin x + c

J; dx =In x + c

J sin(ax + b) dx = -;cos (ax + b) + c

J cos (ax + b) dx = ~ sin (ax + b) + c

f eax+b dx = ; eax+b + c

J --1- dx = _!.In(ax + b) + c ax+ b a



J __ 1_ d.x = _1 tan-1 (-bx) + c a2 + b2x2 ab a

Integration by parts

General logarithmic form

J f'(x) - d.x = ln[f(x)] + c f(x)

Numerical or approximate integration

In the following the interval of integration is from x = a to x = b. This interval [a, b] is subdivided into n equal subintervals/strips.

Each strip has width h = (b - a)/n.

x0 = a, x 1 = a + h, ... , Xn = b

Yo = f(xo), Yt = f(xt), ...

The trapezium rule

rb h Ja f(x) d.x = 2 [yo + Yn + 2(yl + Y2 + · .. + Yn-t)]

Simpson's rule

rb h Ja f(x) d.x = 3 (yo+ Yn + 4(Yt + Y3 + · · · + Yn-l)

+ 2(yz + Y4 + · · · + Yn-z)]

NUMERICAL MATHEMATICS

The Newton-Raphson Iterative Method for approximating to a root of f(x) = 0.

Let the nth approximation be Xn and the (n + 1)th approximation be Xn+b then:

TRIGONOMETRIC IDENTITIES

Basic definitions 1

cosec 9 = -sine

1 sec 9 =-

cos 6

1 cot 9 =-

tan 9

Pythagorean formulas

cos2 6 + sin2 = 1

1 + tan2 6 = sec2 9

cot2 e + 1 = cased 6

Addition formulas

sin (A ± B) = sin A cos B ± cos A sin B

cos (A ± B) =cos A cos B + sin A sinB

( tan A ±tan B tan A ± B) = -----

1 +tan A tan B



Double angle formulas

sin 28 = 2 sin 8 cos 8

cos 28 = 2cos2 8 - 1 {

cos2 8 - sin2 8

1 - 2 sin2 8

2 tan 8 tan 28 = 2 1 - tan 8

1 sin2 8 = - (1 - cos 28)

2

1 cos2 8 = - ( 1 + cos 28)

2

VECTORS

Two dimensional vectors

Magnitude

If a = a1i + a2j then

I a I =a= Yay+ a~

Three dimensional vectors

Magnitude

If a = a1i + a2j + a3k then

I a I =a= Yaj +a~+ a~

MATRICES

2 x 2 matrices

Identity matrix

I= [~ ~]

Determinant

If

then

I ac db 1--ad-bc det A= I A I=

Inverse

If

A= [ ac db] and det A ¥= 0 then

1 [ d -b] A-I---det A -c a


492

ANSWERS TO THE EXERCISES

CHAPTER 1

Exercise 1.2.1 (a) I e I e I . X (d) e I I I. 1. -1 0 1 2 3 4 5 -3-2-1 0 1 2 3

(b) I. I • I • X (e) • I I I I -1 0 1 2 3 4 5 -4-3-2-10 1

(c) • I I • I • X (f) lei I I lei -2 -1 0 1 2 3 -3-2-1 0

2. (a) -3 ~X~ 2 (b) -1~t<5 (c) -0.2 < u < 0.3

Exercise 1.2.2

1. (a) x > -3 and x > 1 so -3 < x < 1 (b) X< -3 and X> -6 SO -6 <X< -3 (C) X ~ -2 and X ~ -3 SO X ~ -2

Exercise 1.3.1

1. Independent variable- resistance; dependent variable - current

Exercise 1.3.2

1

1. (a) Daylight hours (b) Amount of petrol

Jan Dec Distance

Months

(c) Atmospheric pressure

Height

2

1e 2 3

.. X

"' X

..

Time

Answers to the Exercises 493

Exercise 1.3.3

y

(3. 3)

X

(1.5.-3)

Exercise 1. 3.4

80 ~ .. ., 0 -g ~ " 0

E 40 "0 .g "' c..

Distance (thousand million km)

Exercise 1.4.1

v (a) 6 (b) v

(c) Extension

4 2 200

2 100

0 2 X 100 200 300 T 100 200 Mass


(e) Speed

60

(d)

40

20

5 10 Voltage

Exercise 1.4.2

1. (a) 3/3 = 1 (d) 1/4 = 0.25 (g) 1/2 = 0.5

Exercise 1.4.3

1. y

(0,1) 0

2

y

8

6

4

2

0

(b) 1/3 = 0.3 (e) -1/2 = -0.5 (h) -2/5 = -0.4

y

gradient 3

0 X

y

(0, 4) 4

3

2

0 gradient -3

2

2 4

(c) 6/6 = 1 (f) 1/5 = 0.2

6 Time

(i) -1/4 = -0.25

X

X

Exercise 1. 4. 4

1. (a) 1 (b) 3

Exercise 1.4.5

1. Graph Gradient

(a) 2 (b) 0.0033 (c) 0.67 (d) 0.33 (e) -9.8

Exercise 1.4.6

1. Graphs in Figure 1.17

(a) y = x + 4 (b) y = 1/3 X + 4 (c) y = x (d) y = !x + 3 (e) y = -ix + 7 (f) y = 0.2x + 2 (g) y = 0.5x - 2 (h) y = -0.4x + 3 (i) y = -!x- 1

Exercise 1.4.4

(c) 6 (d) 4

Intercept

1.5 0.85 0.0 0.0

70

(a) y = x + 1, y = 2x + 1, y = 3x + 1 (b) y = !x + 3, y = ~X + 3, y = ~X + 3 (c) y = -x + 6, y = -2x + 6, y = -3x + 6 (d) y = -jx + 4, y = -!x + 4, y = -~x + 4

Exercise 1. 4. 7

1. y

-1

-2

3 X

X+ 2y = 1



v 20

10

0

2. (a) 5/3 (d) 5/3

3. (a) y = 5/3 x (d) y = 5/3 X - 1/3

Exercise 1.5.1

1. (a) . 0 Ttme 0.5 Amount 0 7.5

(b) 50

140 ~ 30

g 20 E ~ 10

2

Time(min)

2 t

(b) 2 (e) 12/4 = 3 (b) y = 2x (e) y = 3x- 41

1.0 1.5 15 22.5

e

(c) 2/3 (f) 6/3 = 2 (c) y = 2/3 x + 7/3 (f) y = 2x- 1

2.0 2.5 3.0 30 37.5 45

2. Rate of filling = 6 gallons per 100 sec = 0.06 gaVsec

Exercise 1.5.2

1. (a) Gradient = -5.8 gal/min (b) The tank is emptying

2. (a) Town driving is AB and EF. Motorway driving is BC and DE. Usually less fuel consumption on a motorway

(b) CD is petrol refilling

Exercise 1.6.1

Straight line graphs through the origin for (1) and (5) Note (3) and (4) are straight line graphs not through the origin

Exercise I.6.2

Direct proportionality for (r, q) graph only

Exercise I. 6.3

1. Current = 17.3 amperes when voltage = 8 volts; voltage = 7.4 volts when current = 16 amperes

2. (a) and (c); V = A x D where A is the area of the base

Exercise 1. 6. 4

1. When volume is 250 cm3, the mass is 200 grams. Mass= k x volume. k = 0.8 is called the density.

2. Yes, direct proportionality is a good model. F = 0.41 N

Exercise I. 7.I

1. (a) W = 3 when L = 15 (b) W = 1.125 when L = 40 (c) L = 19.6 when W = 2.3

2. If an inverse proportionality law is appropriate then the product of resistance and current is the same for each pair of data. It is a reasonable model in this case.

5 current = ---

resistance

3. (a) 33j% decrease (b) 100% increase (c) 16~% decrease (d) 25o/o increase

Exercise I.8.I

1. (a) d = 176.4 m (b) d = 11.03 m. When tis multiplied by k, dis multiplied by /2-.

2. P = cu3. Multiply u by 2 ~multiply P by 8. Multiply u by k ~ multiply P by ~. P is increased by 137%.

Exercise 1.8.2

1. p = kT/V

2.P=cv~

Exercise 1. 9.1

1. (a) R = 0.17T + 44.8 (b) (i) R = 48.2 ohms (ii) T = -263.5 °C



2. I= 0.01T + 7.2 (a) (i) 7.45 em (ii) 7.75 em (iii) 7.95 em (b) (i) at 0 °C, I= 7.2 em

(ii) at 200 °C, I = 9.2 em (iii) when l = 14.8 em, T = 760 oc

Exercise 1.9.2

1. In Exercise 1, extrapolation is being used. In Exercise 2(a), interpolation is being used. In Exercise 2(b), extrapolation is being used.

Objective Test 1

1. Gradient = 3.1, intercept = -4.2, x = 3.1t - 4.2

2. Gradient (i) 2

(ii) 3 (iii) -4 (iv) -0.5

v y=2x+5

3. Q,R,A,D

4. (a) T vs x2

(c) W vs 1/P 5. p 2.4

P 2.78 x to-5

P = 8.6 X 104p

6. c 5 1000 32

y-intercept 5

-4 8 1.5

3.1 3.59 x w-5

R 0. 2 0.001 0.03125 C = 1/R

7. V = c~~

x-intercept -2.5

1.33 2 3

(b) p vs ljV (d) Tvs VL

4.33 5.01 x w-5

8. (a)

E .c:: 2 5 c:

"' 1;; 'ii a: 20

100 200 300

Temperature (°C)

R = 0.21T + 31.1 (b) 0.21 ohm/°C (c) (i) 52.1 ohm (ii) 115.1 ohm (d) -148 oc (e) c(i) because it uses interpolation.

CHAPTER 2

Exercise 2 .1.1

1. Quadratics Coefficients

a= b=

2x2 + 3x + 1 2 3 3x2 - 5x- 3 3 -5 7 + 4x + x2 1 4 -2x2 + 5 -2 0 -9x2 + x -9 1 -x2 - 3x + 4 -1 -3 4 - 20t + 9.8t 2 9.8 -20 1 + v + 0.7v2 0.7 1 u- 3u2 -3 1


c=

1 -3

7 5 0 4 4 1 0


Exercise 2.2.1

y = 2x2 + 3x + 1 y

-4 y =-x2 - 3x +4

y

-1

Exercise 2.2.2

1. (a) -L -1 (c) -4, 1 (e) -1

2. (a) 0.85, -2.35 (c) -1, -2

1 X

y

2 X

v

y = -2x2 + 5

-4 4 X

v

y = 3x2 - 5x- 3

-2

y

-4 -2 2

(b) -1.6, 1.6 (d) 2.1, -0.5 (f) no roots (b) -1.4, 1.4 {d) -2, 1

4 X

X

3. (i) 2 m (ii) 2.04 s (iii) 4.18 s

Exercise 2.3.1

1. (a) x2 + 4x + 3 (c) x2 + 2x + 1 (e) 4x2 - 12x + 9 (g) x2 - 2x - 35 (i) x2 - u2

(k) 6x2 - 12x

(b) x2 - 3x + 2 (d) 6x2 + x- 2 (f) 2x2 + 7x - 4 (h) x2 - (u - v)x - uv U) r + 2x


Exercise 2.3.2

1. (a) x2 - 2x + 1 (b) x2 + 8x + 16 (c) 4x2 - 12x + 9 (d) x2 - 1 (e) x2 - 16 (f) 4x2 - 9

Exercise 2.4.1

1. (a) (x + 4)(x + 1) (b) (x- 4)(x + 1) (c) (x + 4)(x- 3) (d) (x + 5)(x + 2) (e) (2x + 5)(x + 1) (f) (3x - 1)(3x + 1) (g) (x + 3)2 (h) (x - 4)2

(i) (x - u)(x + u) (j) (2x + u)2

(k) x(x + 4) (l) x(3x - 10)

Exercise 2.5.1

l.(a)-1,-4 (b) -1, 4 (c) -4, 3 (d) -2, -5 (e) -2.5, -1 (t) L -~ (g) -3, -3 (h) 4, 4 (i) u, -u (j) -u/2, -u/2 (k) 0, -4 (I) 0, 10/3

Exercise 2.6.1

1. (i) (a) (x + 3)2 - 9 (b) (x + 5)2 - 25 (b) (x + 5/2)2 - 25/4 (d) (x + 7/2)2 - 49/4

2. (a) -4, -2 (b) -1, -9 -5 V21 -7 V61

(c)-±- (d)-±-2 2 2 2

Exercise 2.6.2

1. (a) 2(x + 2)2 - 8 ( 7r 49 (b) 3 X+- --6 12

(c) 4(x + %r- 12 ( 5r 17 (d) 2 X- 4 - g

3 5 V17 2. (a) x = -- ± Y3 (b) X=-±-

2 4 4

Exercise 2.6.3

1. (a) -4, -2 (b) -1, -9 5 V21

(c) --±-2 2

7 V61 (d)--±-

2 2

-1 ± V13 -1 ± V33 2. (a) (b)

2 4


(c) L 1

(e) ± \15

Exercise 2.6.4

(d) -2 -5 ± \137

(f) 2

1. (a) 40- two roots (b) -8- no roots (c) 24- two roots (d) -16- no roots (e) 16- two roots (f) 49- two roots

2. K = ± V12 gives one repeated root K < -V12 or K > Vl2 gives two roots

3. (i) 1.58 m or S = 8.42 m (ii) 2 m or 8 m zero bending moment

(i) S = 0 m and 10 m 4. T = 307.5 oc 5. (a) w2 - 3w - 4 = 0

(b) 600w2 + 3200w - 1 = 0

Exercise 2. 7.1

(ii) S = 0 m and 10 m

w = 4 (must be +ve) w = 0.00031

1. ---------------------------Polynomial

4x5 + 3x2 + 2x + 1 9x7 + 8x6 - 4x3 + 2x ~- 3t2 + 4t + 1 u4 + 3u3 - 2u2 + u + 7 0.1x2 + 1.2x - 3.7 3.lx11 + 4x7

Degree

5 7 3 4 2

11

Exercise 2. 7.2

1. v V = x3 + 2x2 + 2x - 1

Exercise 2. 7.3

1. -1, -0.5, 0.3, 2 2. -0.33, 0.33, 1.50

Exercise 2.8.1

1. 2 2. 3.732

Exercise 2. 8.3

1. (a), (c) are no good near x = 3; (d) is quickest 2. (a) 0.38, 2.62

(b) 1.21 (c) 0.43

Exercise 2.9.1

1. 1. 2.18 s 2. 243.7 k



3. 1612 K 2(M + m)g

4. 10 + --'------'-"'-k

5. V = 0 when T = 0 or T = 571.4 °C Vis a maximum when T = 285.7°C

6. (a) i = 1 or i = 4 (b) i = 0.58 or i = 1.56 7. R = 5.73 ohms orR= 39.27 ohms

r = 39.27 ohms orr= 5.73 ohms 8. x = 0.6 m or x = 1.639 m (to three decimal places) 9. 0, 3, 6, 10

Objective Test 2

1. (a) 2 (c) 1

2. (b), (d), (e)

(b) 8 (d) 3

3. (a) (x - 4)(x + 4); x = -4 or x = 4 {b) (x - 2)(x - 1); x = 1 or x = 2 (c) (3x- 1)(x + 2);x = -2orx = 1/3

4. x = 1.4 or x = 3.6

y

X

5. (a) R = -0.573 orR = 0.873 (to three decimal places) (b) x = -2.679 or x = 1.679 (to three decimal places)

6. (c), (a), (b)

3 X

-8

7. x = -1.180,x = 2.000

CHAPTER 3

Exercise 3.1.1

Exercise 3.1.2

1. r = G:Y/3 Exercise 3.1.3

1. f = 0.316R- 1/4

Exercise 3.2.1

1. ----------------------Number Base Index

7 21 a 2 a 5

3 13 4 n m 7



Exercise 3.2.2

1. (a) 29 (b) 35 (c) a6 (d) x5

Exercise 3.2.3

1. (a) x7 (b) x 12 (e) 3k (f) x6

(i) aBb12 2. (a) x5 + a7

(c) -m2 - 5m3

3. (a) a5

(c) en+m-p

4. (a) 1 (c) 1

Exercise 3.2.4

1. (a) 1/4 (d) 1/27

2. (a) a-5

(d) p2 (g) e3x + e-x

(j) a3x-2 y-1

Exercise 3.2.5

1. (a) x1/2 (d) x5/2

1. (a) 2 (b) 8 (e) 32 (f) 512 (i) i U) 1

Exercise 3.2. 7

1. (a) k = 4 (d) a= 2 (g) a = 2/3 (j) a = -5/2

(b) 1/16 (e) 1/4 (b) x-3

(e) 6 (h) 9

(d) a2

(h) x15

(b) 6x5 - 10x11

(d) a3b + a2b2 - ab3

(b) 216x (d) 4a2bkt5

(b) 2 (d) 24

(c) 1/25

(c) m-2

(f) ab-1

(k) ~c2a2b-2 (i) 3x-1 y2 (I) m-3p x3n

(b) x3/2 (c) x1/2 (e) x1/2 (f) x- 1

(c) 3 (d) 2 (g) 100 000 (h) 4x3

(k) 2 (l) 10 000

(b) k = 2 (c) r = 5/3 (e) x = 0 (f) X= 3 (h) X= 1/2 (i) a = 4 (k) X= 3/2 (I) X = 0


Exercise 3.3.1

Physical Variables Rule Domain Range

Spring-mass oscillator

T, m T = 1.13m112 O<m<co O<T<co

Radiation heat E, T E = K(f4- To4) To~ T <co loss rate

Newton's Law F, r GMm O<r<co of gravitation F=--

r2

Exercise 3.3.2

1. (i) F(1) = 1 F(3) = 79.601 F(0.61) = 0.997 (ii) F(3.1) = 19.747 F(O) = 0

Exercise 3.3.3

1. g[f(x)] = 3x - 3 f[g(t)] = 3(t - 3) f[f(x)] = 9x g[g(t)] = t - 6

2. f[g(x)] = (x - 2)3

g[h(x)] = vx-::2 f{g[h(x)]} = (vx-::2)3

3. (a) Range of fis 0 < x < co; range of g is -co < x < co (b) f[g(x)] = v'(X=4") domain [4, co) range [0, co) (c) g[f(x)] = Vx- 4 domain [0, co) range [-4, co)

Exercise 3.3.4

(X- 4) 1. r 1 < x) = ...:..____:_

3

2. r 1(x) = (x + 3)113 - 1

Exercise 3.3.5

1. (a) x = 2.5 (c) no asymptotes

(b) X= -15/7 (d) t = -2, t = 2

O~E<co

O<F<co


Exercise 3.3.6

X

y

X

Exercise 3.4.1

l y LL I

_j!, 7 I

I I I I I I

: (x=-!f)

y

1. ----------------------------------Function

Exercise 3.4.2

Rate of change

t = 1

approx 3.3 approx 2.3

t = 2

approx 9.9 approx 5.7

X

l

1. rate of change (i) when t = 1 is approx - 0.3 and (ii) when t = 3 is approx - 0.02

Exercise 3.4.3

1. n

1 1.5 2 3 4 5 10 100 1000 10000

Exercise 3.4.4

1. (a) 1.0000 (d) 5.4739 (g) 2.7183

(1 + 1/n)n

2 2.151657 2.25 2.370370 2.441406 2.488320 2.593743 2.704814 2.716924 2.718146

(b) 0.1353 (e) 1.3634 (h) 1.6487

3. It is not a reflection in any line! 4. 0.567

Exercise 3.5.1

1. (a) 4 (e) 1

2. (a) x = log3 7 (d) X= loge 4.1

Exercise 3.5.2

1. (a) 1.3979 (d) 3.0000 (g) 1.0000 U) 0.0000

3. (a) 0.4314 (d) -1.0459 (g) 4.4817

Exercise 3.5.3

1. 60 dB

(b) -2 (c) 1 (f) -2 (g) 2

(b) x = log6 4.1 (e) x = log2 8.3

(b) 1.0000 (e) 3.2189 (h) 1.9879 (k) 0.0000 (b) 1.4183 (e) 1.0492 (h) 37.9038

2. 0.100 wm-2 to three decimal places

(c) 20.0855 (f) 0.6570

(d) 3 (h) 6

(c) x = log10 12.2 (f) x = log5 7.6

(c) 0.8633 (f) 2.3026 (i) 0.3222 (I) 0.6931 (c) 0.6534 (f) 1584.8932 (i) 6.3246



Exercise 3.5.4

1. T = 500e -o.41

2. t = 0.0805 seconds 3. t = 22.65 minutes 4. p 2 = 348.51 pascals 5. (a) 86071, 74082, 47237, 22313

(b) h = 4.62 km (c) h = 15.35 km

Exercise 3.5.5

1. (a) 2 log p + 3 log v (b) In p - In v (c) logs 6 + logs p + logs(a + b) (d) In 0.21 + In t- 1 (e) log s + log t - 1/2 log 3 (f) In 0.1 + 2 In v

2. (a) 1 (d) In (u3/v2)

(f) 0

(b) In 3 (e) Iog(xy2jp3)

(5wv) (c) log --;_;-

3. (a) x = 2.7381 (c) x = 0.5438

(b) X = -0.5217

Exercise 3.6.1

1. k = 0.0133 s- 1

2. T = 10.60 h 3. (a) x = 0.1652 min- 1 (b) 4.20 min 4. A,D,E 5. (a) 7 (b) 1/27 = 0.0078 (c) 0.9999 (almost all of it!)

Exercise 3. 7.1

1. (a) y = 3.4 x2·7

(b) v = 1.7 u-1.3 3. 1-L = 47.2 T-2·06

4. C = 0.12 x-2

Exercise 3. 7.2

3. H = 16.7 M 0·92

Exercise 3.8.1

1. v = 13.2 e-0-11

2. Graph of ln(k) against 1/T. k = 5.1 x 1012 x e-30300/T

Objective Test 3

1. (a) l/x2 1

(b)-ab4

1 (c)-

2xy2

2. Domain [3, oo], range [0, oo] 3. r 1(x) = j ln(x), domain [1, 2.117], range [0, 0.25]

(1 - x) g- 1(x) = , domain [0, 1], range [0, 0.25]

4

f[g(x)] = e3(1-4x)

4. (a) X= 0.4720 (b) X= 0 (c) X= 20.09 (d) X= 626.48 (e) X= 7.5546 (f) X= 2.1133

5. Half-life = 5599 yr, age = 3480 yr 6. y = 3 e-4x, s = 4.2 tl. 7

CHAPTER 4

Exercise 4.2.1

1. (a) 140° (b) 290° (c) 50° (d) 233° (e) 300° 2. (a) -160° (b) -161° (c) -22° (d) -45° (e) -80°

Exercise 4.3.1

2. 0.5, -0.8660; -0.5, -0.8660; -0.5, 0.8660; -0.5, 0.8660 3. 0, -1; -1, 0; 0, 1

Exercise 4.5.1

1. (b) 57.3, 573.0, 5729.6; -5729.6, -573.0, -57.3

Exercise 4.6.1

4. -0.342 5. 0.458 6. -1.192 7. -0.7 8. -0.1 9. 0.39

Exercise 4. 7.1

1. (a) ±98.2° (b) 7.2°, 172.8° (c) ±76.7° (d) -5.2°, -174.8° (e) -30°, -150° (f) ±90°

2. (a) 46°48', 133°12' (b) ±66°44' (c) -23°35', -156°25' (d) ±112°1'

Exercise 4.8.1

1. (a) 14.SO + 360°n; 165.SO + 360°n (b) 42° + 180°n (c) ±131.8° + 360°n (d) ±99.6° + 360°n (e) -41.8° + 360°n; 221.8° + 360°n (f) 45° + 180°n



Exercise 4.9.1

3. 360° 4. 180° 5. 180°, 480°, 5

Exercise 4.9.2

1. 2, 4; 360°, 180° 2. 2, 1; 360°, 720°; 1, 0.5 3. 24°, 66°, 204°, 246° 4. 240°, 3/2

5. (a) 72° 5 (b) 1440° 0.25 (c) 600° 3/5 (d) 360/k k (e) 360/k k

Exercise 4.9.3

1. ------------------------------------------------

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Amplitude Max Min y = 0 at

1 1 -1 at -270°, 90° at -90°, 270°

2 2 -2 at -270°, 90° at -90°, 270°

0.5 0.5 -0.5

1

3

I 4

00

at -270°, 90° at -90°, 270°

1 -1 at 0°, ±360° at ±180°

3 -3 at 0°, ±360° at ±180° I I 4 4 at 0°, ±360° at ±180°

nla nla

2. (a) L 360° (b) 5, 360° 3. (a) 5, 360° (b) 10, 180° (c) 7, 480° 4. (a) 2, 3, 120° (b) 4, 2, 180°

(d) 0.73, 0.21, 1714°

Exercise 4. 9.4

±90°, ±270°

(c) oo, 180° (d) 1, 90°

(c) 4.3, L 480°

1. Shift 10° to the left. Period 360°, amplitude 1 2. Shift 40° to the right. Period 360°, amplitude 1. sin(6 + 50°) 3. 0°, 180°, 360°; 155°, 335°; 65°, 245°

4. tan 6: asymptotes 6 = ±90°; zero at 6 = 0°, ±180° tan(6 + Z0°): asymptotes 6 = -110°, 70°; zero at 6 = -zoo, 160°

5. (a) Z40°, Z, 30°; y = Z sinGe + 30°) (b) 120°, 1.7, 180°; y = 1.7 sin(3e + 180°)

Exercise 4.10.1

1. (a) 73.4°, 106.6°, 253.4°, 286.6° (b) 53.7°, 126.3°, 233.7°, 306.3° (c) 35.3°, 144.7°, Z15.3°, 3Z4.7° (d) 65.9°, 114.1°, Z45.9°, 294.1° (e) no solutions (f) 40.9°, 139.1°, Z20.9°, 319.1°

Z. (a) 45°, 135°, ZZ5°, 315° (b) 60°' 120°' 240°' 300° (c) 0°, 180°, 360° (d) 60°' 120°' 240°' 300°

3. (a) ±30° + 1800n (c) ±53.1° + 180°n (e) ±54.r + 180°n (g) ±50.8° + 180°n (i) ±62.7° + 180°n

Exercise 4.10.2

1. (a) Z4.3° + 180°n, 65.7° + 180°n (b) Z47.4° + 360°n (c) ±46.Z0 + l20°n

Z. (a) Z9.Z0 , Z40.8° (c) 45°, Z25°

3. (a) -104.9, 1Z6.9 (c) -40.0°, 80.0°

Exercise 4.11.1

(b) ±48.Z0 + 180°n (d) ±54.7° + 180°n (f) ±21.SO + 180°n (h) 45° + 90°n

(b) no solutions

(b) -159.0°,Zl.0° (d) -133.9°, 46.1°

1. (a) 7.2 (b) 96.4° (c) 81.9° (d) 130.9° (e) -Z3.6° (f) 34.3°

2. (a) 84°11', 264°11' (b) 116°34', 243~6' (c) 142°44', 32Z0 44' (d) 35°16', 114°44' (e) 211°5', 328°55' (f) 66~5', 293°35'

3. (a) ±48.2° {b) -154.2°, -Z5.8° (c) -36.9°, 143.1° {d) ±131.8° (e) 16.6°, 163.4° (f) -135.0°, 45.0°

4. (a) 360°; asymptotes at x = 90° and 270° (b) 360°; asymptotes at x = 0°, 180° and 360° (c) 180°; tan x has asymptotes at x = 90°, 270° and cot x at

X = 0°, 180°, 360°



Exercise 4.12.1

1. (a) 420° (b) -30° (c) 225° (d) 540° 5 5 1T 5

2. (a) - 1T (b) - 1T (c) -- (d) - 1T 12 3 6 2

3. Period Amplitude Zero at

2 1T 21T 41T 51T (a) -1T 2 0,3':3'1T':3':3'21T 3

1T 31T (b) 21T 0.5

2' 2

1T 31T (c) 1T 5 0, 2' 1T, 2' 21T

4. (a) 31T/2 (d) 31T/4, 71T/4

(b) 1T/3, 51T/3 (e) 1T/3, 41T/3

5. ------------------Period Amplitude

21T (a)

3 1

(b) 1T 10

81T (c)

3 0.5

(d) 31T Y2

21T (e)

k 1

21T (f)

k a

(g) 1 b

2 (h) 1

n

(c) 1T/3, 21T/3

Exercise 4.13.1

1. Period (seconds) Frequency (hertz)

1 (a)

1T

(b) 2 O.S

211' k (c)

k 211'

2 k (d) -

k 2

2. 4 em, 1Hz 3. a = 30 mm, w = 411' Hz 4. i = 7S sin(1201Tt) rnA 7. (c) (e) are true 8. Amplitude = so, period = 3 seconds, phase angle= 0.7Sc,

e = s cos c; t) e = so when t = 3n seconds, n = 0, 1, 2, ...

e = 2.s cos ( 4; t)

Objective Test 4

1. 140° 2. -12.7° + 180°n, 102.7° + 180°n 3. 0.0717, l.SO, 2.17, 3.S9, 4.26, 5.69 4. 50.2° + 180°n 5. 1080°

811' 6. 5'2 7. Translate 30° to the right

n1T 8. X=±-

2

s 9. -Hz

211'



CHAPTER 5

Exercise 5.2.1

cos e 1. (a)-.

sm e

(d) sine cos e

1 (b)-.-

sm e

(e) cos e

1 (c)--

cos2 9

1 (f) -sin-e -co-s -e

sine (g)-

cos e (h) sin2 e - cos2 e

Exercise 5.2.2

1. (a) 77/85 (d) 84/85

(b) 13/85 (e) -77/36

(A + B) is in quadrant II (A - B) is in quadrant I

(c) -36/85 (f) 13/84

2. (a) 0.8912 (d) 0.8820

(b) 0.4712 (e) -1.9652

(c) -0.4535 (f) 0.5343

(A + B) is in quadrant II (A - B) is in quadrant I

Exercise 5.2.4

2. (a) 5 sin(x + 53.1°) (c) V2 sin(x- 45°) (e) v106 sin(x + 150.9°)

Exercise 5.2.5

1. (a) 14.SO, 90°, 165.SO, 270° (c) 90°, 180°, 270° (e) 53.8°, 159.9°, 233.8°, 339.9° (g) 0°, 360°, 126.9°

(b) 13 sin(x + 22.6°) (d) Vl3 sin(x + 146.3°) (f) v666 sin(x + 54.SO)

(b) 68.SO, 291.SO (d) 0°, 45°, 180°, 225°, 360° (f) 96.8°, 263.2°

2. (a) 103.3°, 330.SO (b) 157.4° (c) 0°' 30°' 120°' 150°' 240°' 270°' 360° (d) 39.0°, 162.8°, 219.0°, 342.8°

Exercise 5.3.1

1. y

X

Exercise 5.3.2

1. (a) f(x) = 30° or 7r/6 (b) f(x) = -30° or -7r/6 (c) f(x) = 90° or 7r/2

X

g(x) = 60° or 7rl3 g(x = 120° or 27r/3 g(x) = oo or 0

(d) f(x) = 45° or 7r/4 (e) f(x) = -38.3° or -0.67 (f) f(x) = -71.8° or -1.25 (g) no function values

g(x) = 45° or 7r/4 g(x) = 128.3° or 2.24 g(x) = 161.8° or 2.82

(h) f(x) = 11° or 0.19

2. (a) 1.7321 (c) -0.4364 (e) 0.2316

Exercise 5.4.1

g(x) = 79° or 1.38

(b) 0.7141 (d) 0.8660 (f) no value

1. 2a sin(SOOt) cos(St), period 27r/5, a is amplitude of original waves 2. 2a sin 2ft cos ft 3. 2a sin(wt + 'P/2) cos('P/2)

Period is same as original wave. Amplitude is 2a cos('fJ/2) and phase 'fJ/2. (a) 'fJ = 0 : wave is twice original amplitude. (b) <p = 7r/2: wave is amplitude Y2a, and 7r/4 out of phase with

original wave. (c) <p = 7r : two waves destroy each other.

Exercise 5.4.2

1. x =A sin wt 5. P = E2 + J2(R2 + X 2) + 2EJ VR2 + X2 sin(2t + a)

a = arctan(R/X) where



6. x = 5.63 sin(4t + 0.99), amplitude = 5.63 em and period = 1.57 s. t = 0.40 s.

7. 50 sin(51Tt + 0.64), amplitude =50, period = 0.4, phase= 0.64, t = 0.059.

8. 45° 9. y = -ix + 2

Exercise 5.5.1

1. (a) C = 75°, b = 4.90, c = 6.69 (b) c = 45°, b = 2.02, c = 3.39 (c) A = 69.9°, b = 11.90, c = 6.36 (d) B = 42.1°, C = 82.9°, c = 9.22 (e) C = 22.0°, A = 117.0°, a = 0.29 (f) C = 19.4°, B = 115.6°, b = 20.40 (g) A = 48.2°, B = 15.8°, b = 1.24 (h) B = 45.3°, C = 74.7°, c = 293

2. F2 = 13.7 newtons, angle = 42.6° 3. 104.6 metres 4. 036.9° 5. 18.7 metres.

Exercise 5.6.1

1. (a) c = 6.29, B = 67.1°, A = 37.9° (b) c = 3.39, B = 107.7°, A = 29.2° (c) c = 8.58, B = 10.2°, A = 42.8° (d) b = 337, A = 30.1°, C = 39.9° (e) C = 116°, A = 22.7°, B = 41.3° (f) C = 21.3°, A = 4.8°, B = 153.9° (g) a= 17.7, B = 40.4°, C = 66.6° (h) b = 13.33, A = 32°, B = 43o

2. 11.7 newtons, angle = 24.3° 3. Speed = 1119.5 knots, bearing = 267.1° 4. Distance = 62.6 miles, bearing = 078.2°

Objective Test 5

1. e = 'TT/3, 2.3 or e = 60°, 131.8°

b cos a + a sin 13 . a cos 13 - b sin a 2. cos x = , sm x =

cos( a - 13) cos( a - 13)

3. x + y = 6 sin 20t cos 5t

(x + y) ~ 6 sin (20t) cos (5r)

5. 5 sin(2x + 36.9°); x = 1S, 51.7° 6. 29.6°, 87.4°, 6.3; 3.8, 45.2°, 94.8°

CHAPTER 6

Exercise 6.1.1

1. 1/10, 1/100, 1/1000 3. 2.1, 2.01, 2.001 5. 3/4, 3/16, 3/64 7. 1' -1/10, 1/100 9. 3, 5, 7

11. Gr 13. Gr-1 15. 3(-1t-l

2. 1, 1/10, 1/100 4. 4, 3.1, 3.01 6. -1, 1, -1 8. 4, 2.9, 3.01

10. 2!, 2, 1~

( 1)n-1 12. -

4

14. 2( -1)n

16. c~r or (0.1Y

17. 5 + c~r 18. 1 _ c~r 19. £6650, £4655, £3258.50; £9500(0.7t



Exercise 6.1.2

1. Gr-1. 0

3. (-l)n-l, no limit

1 n 5. 2--- nolimit

2 2'

2. 3 + Gr· 3

4. 5n, no limit

( 1 )n-1

6. 10 + 10 , 10

7. 0.8nCinit where Cinit = initial charge

( 1 )n-3

8. V2i, V 2 I, the squares shrink to a point

Exercise 6.1.3

1. ± (J..)n-1 n=l 10

5

3. L (2n- 1) n=l

31

5. ~ xn-l

n=l

Exercise 6.1. 4

1. 5050 4. 15 150

Exercise 6.2.1

1. 29 5. 10

Exercise 6.2.2

1. 5050

2. 2870 5. 2185

2. 25.25 6. 101

2. 318.5 4. 860, 1890, 1030

Exercise 6.3.1

1. 1.2, 13.437 (three decimal places

3. 15th term 5. £8192

2. tGr 10

4. L (-lt-12n n=l

3. -7 7. 51

3. 18 496 6. 180 125

4. 15, -2

3. 7.5 5. 825

2. -0.8, 13.422 (three decimal places)

4. 7 seconds 6. 21.5%

Exercise 6.3.2

1. 453.320 (three decimal places)


5. 26/21, 4

Exercise 6.4.1

5 1.-

9

18 4. 5-

99

Exercise 6.4.2

1. 75 3. Does not exist 5. Does not exist

1 10

37 2.-

99

17 5. 5-

90

7. 3 - m, - m from A 3 17


4. 1/4

2. 5 4. -3j 6. 8

417 3.-

999

442 6. 3-

495

8. 20 units, (8, 4)

25 9. 1.316 m (three decimal place),- (1 - 0.62n), n = 11

19

10. 60'1T em

Exercise 6.5.1

1. q = x2 - x - 1; r = 2 3. q = 6x2 - 2x + 5; r = 8 5. q = - t2 - t - 1; r = 1

Exercise 6.6.1

1. 5 2. 8 5. -48 6. Yes 9. Yes 10. No

12. (t + 1)(t + 2)(t + 6)

Exercise 6.6.2

2. q = 2x2 + x - 3; r = 0 4. q = 3x - 4; r = 20x - 21 6. q = !v - ~; r = 2!

3. 0 4. -1 7. Yes 8. No

11. (x- 2)(x + 3)(x- 4)

1. 2~ 2. 2 3. -1i 4. No 5. Yes 6. Yes 7. (x - 2)(2x + 1)2

8. (x + l)(x - 3)(2x - 1) 9. (t + 1)(t - 2)(3! - 2)(! + 4) 10. (y - l)(y - 2)(2y + 1)(3y - 2)



Exercise 6. 7.1

l.x2 +9 3. (x - 1)(x2 + x + 1) 5. (x - 3)(x2 + 2) 7. (y + 3)(2y2 + y + 3)

Exercise 6.8.1

2. (x + 3)(x - 3) 4. (x + 1)(x2 - x + 1) 6. (1 + 1)(1- 1)2(t + 2) 8. (3z - 1)(z2 + 2z + 3)

1. Improper 4. Proper

2. Improper 5. Proper

3. Improper 6. Improper

2 7. 4+-

x- 1

3 9. s + 2 + -

s- 1

Exercise 6.8.2

1 1 1. --+-

x+1 x-1

4 3 3.-----

1-1 1+2

1 1 5.---

2(x - 1) 2(x + 1)

Exercise 6.8.3

1 1 1 1. --+------

1 - 3 I + 2 (1 + 2i 2 5 1

3. -+-+-X x2 x-1

3 1 1 1 5 ---+ ---

. x2 x (x - 1)2 x - 1

Exercise 6.8.4

X 4 1. --+-

x2+2 x+1

2 1 1- s 3. --+--+-

s + 1 s + 2 s2 + 1

1 4 3 5 ---+--.1 12 12 +1

8. 3 - (31- 3) 12 + 31 + 1

1- y 10. 2y- 1 + 2

1 + 2y- y

2 3 2.--+--

1-21 t+1

2 4 1 4.-----+-

x+1 x-1 x-2

1 1 2. 2+---

2(2x + 1) 2(2x + 3)

1 2 1 4.- +-+--

t3 t t + 1

3 1 + 2t 2----

. t t2 + 3

1 2v - 3 4. -+--

v v2 + 1

Exercise 6.8.5

3 1 1. 2v - 1 + - - --

v v + 1

1- X 3 3. X+--+-

x2 + 2 x

Exercise 6.8.6

1 l.---

1

6(x- 3) 6(x + 3)

3 1 3.--+--

x-2 x+2

1 1 1 5.-- +---

2x 3(x + 1) 6(x + 2)

2 t 7----

. t t 2 + 1

Exercise 6. 9.1

1 3 2. 2+------

t + 1 (t + 1)2

2 1 4. 5 +-----

x-2 x-1

1 2.---

1

2(v + 1) 2( v + 3)

2 1 4.-----

v + 3 2v + 1

2 3 6. 1--+-

t t- 1

1-2x 1 2 8. x 2 + 1 + (x + 1)2 - x + 1

1. x6 - 6x5y + 15x4y2 - 20x3y3 + l5x2l - 6xy5 + l 2. a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7

3. 243p5 + 405p4q + 270p3q2 + 90p2q3 + 15pq4 + q5

4. 81x4 - 216x3y + 216x2y2 - 96xy3 + 16/ 5. 1 - 5x2 + l0x4 - 10x6 + 5x8 - x10

6. 81 v4 - 54v3 + 27/2 v2 - 3/2 v + 1/16 7. 1 - 3/x + 3/x2 - 1/x3

8. t6 + 6t4 + 15t2 + 20 + 15/t2 + 6/t4 + 1/t6

Exercise 6.9.2

1. 24 4. 840 7. 11.10

2. 1 5. 30 240 8. 11.5

10. n(n - 1)(n - 2)

Exercise 6. 9. 3

1. 15 4. 12

Exercise 6.9.4

2. 56 5. 3

1. u3 + 6u2v + 12uv2 + 8v3

3 1 1 2. x4 - 2x3 + - x 2 - - x + -

2 2 16

3. 3 628 800 6. 9 9. n

3. 252 6. 21



10 1 3. 32t10 + soP + sor4 + 40t + - + -

r2 ts

4. 11520s8r2 21

7. -p5 32

Exercise 6.10.1

1. 1 + 4x + 12x2 + 32x3; I x I < !

6. -48384v5

9. 490

2. 21;5 (1 + __!_ y - __!_ l + 2_ l) · I y I < 2 10 50 500 '

3. 32/5 (1 - _i_ t- _i_ t2 - ~ t3) • I t I < 3/2 15 75 3375 '

1 1 1 4. x112 -- x- 112 y -- x-312y2-- x 512l· I Y I < I xI

2 8 16 '

1 1 5. 10 +- x- -- x2• 10.149

20 8000 '

6. 3.936

1 x x 2 x3 7.- +--- +-

2 4 8 16

Objective Test 6

3(-1t 1. 1 2n-

3. 16, 5

1 5. -3-

32

X 1 7.--+-

x2+2 2x+3

t t2 10. 2----

12 288

2. 1

4. 1.92 mm

6. (x - 2)(x + 3)(2x - 1)

1 3 8. 2x+----

x 2x + 1

CHAPTER 7

Exercise 7.1.1

The complete list of average velocities (ms- 1) is

0.7, 2.1, 3.8, 5.7, 8.1, 11.3, 15.1, 20.1.

Exercise 7.1.2

1. Yes. We shall see that the instantaneous rate of change is obtained from the average rate of change as time steps decrease in size.

2. Average decay rates are correct to three ·decimal places.

0.118 0.104 0.092 0.081 0.071 0.063 0.056 0.049 0.043 0.038 0.034 0.030

-'~ "' i 0.12

> .. 0 .,

"0 ., "' ::: ., >

<( 0 120

Time Is)

Exercise 7.3.1

1. Instantaneous velocity is 7 ms- 1•

2. Instantaneous rate of change of current is -0.4 As- 1.

Exercise 7.3.2

2. Gradient of f(t) = t2 is 4; gradient of f(t) = t 5 is 80

Exercise 7.4.1

1. -2t

Exercise 7.6.1

3 5. - t112

2

1 9. -- u-5/4

4

2.

6.

10.

2. 9t2 - 8t 3. X+ 1

3 -4 - X 3. -t-2 4.

1.1 VO.l 7. -1.1v-2.1 8.

lx0 = 1

i cl/2

1 _ u-2/3 3



Exercise 7.6.2

1. 4x3 - 3x2 + 8x - 8

5. 24t-4

4 7 7. - u-2/3 +- u-I/2

3 2

1 9. 6t--

4

Exercise 7. 7.1

1 2. -- x-112

2

1 4. - rl/2 + 3

2

6. -1.2v-2·2

2 8. -- t-1/3 + t-3/2

3

10. 27rt

3 1. - xl/2

2

1 2. -- x-3/2

2

1 3. 1- -x-1/2

2

7 4. 3-2

X

Exercise 7. 7.2

1. 6t

2 6. -- x-5/3

9

10. 0

Exercise 7.8.1

2. -2

7. 2

5. 2.1xl.l - 1.1x0·1

3. 0 2

4. 3 t

8. X+ 1 4

9. - x-7/3 9

1 1.-

6

1 2. -

9 (1 65) (3 59)

4" 3 ' 27 ' 5 ' 25

6. 12x - 12; x = 1 m 7. 1.832 sm-1

8. 250 Jsm-1 9. 6 WA - 1

11. t = 10 s 12. 3y = 17x - 21 13. (2, 6), ( -2, -6), y + 3x = ±12 14. y = x + 3; (0, 3) 15. y = 8x - 21

Exercise 7.9.1

1. -2x3(2 _ x4)-1/2 2. 2(4- x)-3

3. 3(2x + 1)(x2 + x + 1)2 4. 0.3x2(x3 - 1)-0·9

1 5. - (3r2 - 6t + 1) (f - 3rl + t - 1)-1/2

2

3 6. -- (1 + 3t)-l/Z

2

1 8. 2t - - (1 - t)-l/Z

2

2 10. -v-2 +- v(v2 - 2)-213

3

Exercise 7. 9.2

1. 94.2 cm2s- 1

2RI 3 ----

. (t + 1 )2

500 5.-2

1TW

7. 1 - (1 + t)-2

2. 0.00796 cms- 1

4. 0.0111 ms-1

-2 6. 3 ; -0.0741 ms-2, -0.000216 ms-2; -2.46 x 10-7 ms-2

(2s + 1)

Exercise 7./0./

1. -4; decreasing 3. 39; increasing

5. -1/Y2; decreasing

7. <-L -1) 9. (0, 1)

11. None

Exercise 7.10.2

1. ( -1, 2) minimum 3. None

Exercise 7.10.3

1. Max - -( 1 17) 2' 4

3. Max ( -3, -27D, min (3, 27!}

2. -1; decreasing 4. 0; stationary

6. G · 112)

8. (0, 1) 10. (0, 0) 12. (1, -3) and (-1, 5)

2. (0, 1) inflection 4. (1,2),(-1,2)minima

2. Min ( -2, -16), max (2, 16)

4. Min (0, 1) 5. Max ( -1, 3), min (0, 0), max (1, 3)



Exercise 7.11.1

1. 2x(2x + 1)2(5x + 1)

3. - x(1 - x2) 112(1 + Sr)

44 5. 2

(10- 7x)

1 7. - (1 - t)- 112(1 - 6t)

2

1 - t2

9. 1 + 2 2 (1 + t)

1 2. - (x + 1)-If2(3x + 2) + 2x

2

2x2(x + 6) 4. 2

(4 + x)

2x(1 + x2) 6. 3(2 + 2

(1 + 2x)

1 8. - t(6 - t)-7/3(36 - 7t)

3

t112 [ 9 + 2t ] 10. 3t2 +-2 (3 + 2t)2

11. (-1, -!)minimum, (1, D maximum 12. (1, !) maximum

Exercise 7.12.1

1. 1.633 m from A, deflection of 2.2 mm 2. R1 = R2 =50 kfi; P = 0.1 W 3. 62.8 mpg at 51.8 mph 4. 37.5 m x 25m 5. h = 85.4 em, r = 39.8 em 6. One 1 m diameter; other zero diameter

Exercise 7.13.1

1. COS X

2. -sin x (i.e. the sine graph inverted)

Exercise 7.13.3

1. -cosec2 x 3. -cosec x cot x 5. ! sec2(!x) 7. 2 sin x cos x 9. 2x cos(x2)

11. -sin ex sin x 13. 2w cos wt, - 2w2 sin wt

2. sec x tan x 4. -5 sin 5x 6. 8.

10. 12.

-sin x + 2 cos x 6 tan x sec 2x -4 cos 2x sin 2x cos 2x

14. -2w cosec 2wt cot 2wt, 4w cosec 2wt (cot2 2wt + cosec2 2wt) 15. cos wt - wt sin wt, -w (2 sin wt + wt cos wt) 16. -2w sin(2wt + ex), -4w2 cos(2wt +ex) 17. 2w sin(wt- ex) cos(wt- ex) = w sin[2(wt - ex)],

2w2 cos[2(wt- ex)] 18. w cos 2wt, - 2w2 sin 2wt

a 1T 19. e =- +-

2 2

20. e =arctan v'2 (= 54.7°), V = 21Tz3/9v'3

Exercise 7.14.1

3. 1.00 4. 1.10

Exercise 7.14.2

1. -6e-6x 2. ~(ex - e-x) 3. -2xe-x2

4. e2x(l + 2i) 5. -el-x 6. xe-x(2- x) 7. e-21(cos t- 2 sin t) 8. ea1(a cos wt - w sin wt)

10. e-1[wcos(wt + a) - sin(wt + a)]

13. y =ex, ey + x = e2 + 1, A(O, 0), B(e2 + 1, 0), !e(e2 + 1) 14. (-1, -e), minimum 15. 5e-31 (2 cos 2t- 3 sin 2t), 5e-31(5 sin 2t- 12 cos 2t); 0.294 s

Exercise 7.14.3

1. ~In a

Exercise 7.15.1

1 1.

x

3 5.-

X

1 1 9. ---

t2

12. 2t In(2t) + t

15. 1

2 2.

X

1 6.-

2x

2. 2x In 2, 32x In 9

2at 10. 1 + aP

2 3.

X

1 7. 1 +

t

2 - In t2 13.--

(2

6 Int 16.

t

2 4.

X

1 8.--

a2 11.

t

1 + t

2t In t- t 14. (In t)z

18. (0, 0), minimum

Answers to the Exrecises 529


Exercise 7.16.1

(1)

(3)

3 y=--2

(5)

(6)

y

y

---'r:-----~ .-IN/

II I '<

I I

y

(2)

(4) y

(2, 4)

I

:L:=x2-8 1 x-3 I I I I I I (4,8) I

I I IM Ill I'<

/ /

X

X

y = e• cos2x

X

17) y

X

(9) y

Exercise 7. I 7.1

1. 6 cos(2x)

3. 0

(8) y

(10)

I I I I~ I,: I'< I I

2. -l sec2 (2:. - ::) 2 4 2

4. (2x e-zx- 2x 2e-2x) sin(2x) + 2x2 e-zx cos(2x)

-2x2 sin(2x) - 2x cos(2x) 5. 4

X

6. 2 cos(2x) cos(3x) - 3 sin(2x) sin(3x)

2e3x 7. 3 e3x ln(2x + 1) + --

(2x + 1)

8. sec2(x) sin(2x) + 2 tan(x) cos(2x)

9. 12 sin(l - 4x) 10. 20 e41

11. -14 e-21 12. e21 + 2te21

X

X

13. 2t e12 14. 2e21 sin(4t) + 4e21 cos(4t)

15.-----e8r

ds 18. v == - = 2 + lOt

dt

2 16. -

t

4 17.---

(3 - 4t)

(a) t = 0, v = 2 (b) t == 2, v = 22 (c) t = 4, v = 42 19. dT/dt = -500 e-51

(a) t = 0, dT/dt = -500 (b) t = 1, d7/dt = -500 e- 5

20. dM/dt = 4.3(-2.1 e-2·11) = -9.03 e-2 11



21. v = ds/dt = 0.21 cos(0.7t) a= dvjdt = -0.147 sin(0.7t) (a) t = 0, v = 0.21, a = 0 (b) t = 1, v = 0.21 cos(0.7) = 0.161,

a = -0.147 sin(0.7) = -0.095 22. (a) r(t) = 9 em

(b) r = 3 when t = 1 second (c) dr/dt = 9/t2 = 9 when t = 1 (orr= 3) (d) Largest value of r is 12 em [lim,_.oo r(t)]

23. Speed v = dh/dt = 21 - lOt. The table shows the results.

(a) t (s) v (ms-1)

1 11

2 1

3 -9

4 -19

(b) The stone comes to rest when v = 0, i.e. when t = 2.1 seconds. (i) Initially v is positive so that the stone is climbing for

t < 2.1. (ii) For times t > 2.1, the value of vis negative, so that the

stone is falling.

1 - t 2 9 24. 2 2 ; t = 1;- ; it approaches (0, 4) along the axis

(1 + t ) 2

5 1 25. t =- ; v = -- e-5/4 = -0.0358

2 8

26. -8.08 ms-2

27. (3, 1), X + 3y = 6 28. (a) 41 (b) 36 (c) y = 36x- 31 (d) X + 36y = 1478 29. (a) Pis a maximum when X= R

(b) p

X

30. (b) f(a)

31. (a) yv// 1

y=x+X

v (1, 2) /

/

(c)

32. (a) Vo =

(b)

I I I I~

I~ I'<

p

X

"'


(b) y

y = 1

I

"'1~+2 Ill y=--

-~~---x-6 _

X

(d) I y I

~ y = 1

I 3 X ,, -3 II I x-3 y=--'<I x+1


Objective Test 7

1. 15.08, 15.08 (three decimal places)

1 3. - - x - 2x In x

X

5. y = 1

7. -0.081

CHAPTER 8

Exercise 8.1.1

2. -2.297

4. (1-p)-2,2(1-p)-3

6. 36191 mm3s-1

8. ( e-1/2, - ~ e-1)

1 10. -+cot x

X

s should settle down to 35.33 m, but there may be trouble with rounding errors

Exercise 8.1.2

1.3m 2.5jm,-2jm 3. v'12 """ 3.46 s

Exercise 8.3.3

1. ( q6 - p6)j6 5. ~

2. 20 3. 7.5 4. 2(2v'2 - 1)/3 7. 1.5125, 12j, 1~ 6. 12.4

8. -4, 4, 0, 8

Exercise 8.4.1

In each case c is an arbitrary constant.

1. X+ C

5. -cos x + c

Exercise 8.5.1

1. -cos t + e1 + c

1 2. - x3 + c

3

3 4. -x4 + c

16

1 3 6. - x3 + - x2 + c

3 2

u3 3 2. - + - u2 + Bu + c

3 2

3. -1/v + c 5. 2v'x + c 7. tan x + c

Exercise 8.6.1

1. 12.4

5 6.-

6

2 2.-

3

2 7. 26-

3

2 3. 3-

3

2 8. 25-

3

4. 2pl/2 + c or 2 V p + c 6. -e-x+ c 8. -cot x + c

1 4. 1-

8

11 9. 1-

15

1 5. 1-

3

19 10. 3-

24

2 1 11. 21-

3 12. 53.99 (two decimal places) 13. 3- 14. 12

3

1 1 2 15. 12-

3 16. Integral = - - area = -

6 ' 6

1 1 17. Integral = -1- area = 1-

3 ' 3

1 19. 4-

2

1 20. 1-

3

2 21. 10-

3

1 22. 1-

3

25. 2 + e-1 - e-3 = 2.318

Exercise 8. 7.1

1 1. - - cos 3x + c

3

1 3. - (2t - 1)312 + c

3

5. i sin ( 2x - ; ) + c

1 7. - (x2 + 2x - 1)6 + c

12

1 9. - (3 - 2v)-2 + c

4

3 11. - -(1- eY)413 + c

4

5 13. - (1 + 2v)l.2 + c

12

Exercise 8. 7.2

A is a positive arbitrary constant.

26.2 27.e - 1

2 2. - (x - 1)312 + c

3

4. -cos ( x + ; ) + c

1 2 6. -ex + c

2

1 8.- --+ c

1 + t

1 10. - (2t + 1f12 + c

7

5 12. - (1 + v)I.2 + c

6



1. ln(Ax114) 2. ln(Ax2) 3. ln(Ax9)

7 4. - 2ln[A(1 - 2x)) 5. ln[A(3t + 2)113]

6. ln[A(3t - 2)1f3]

1 9. -ln[A(ax + b))

a

1 11. -ln[A(1 + t)]

3

1 7. -ln(Ax)

a

a 10. - ln[A(bx + c)]

b

2 12. - ln[A(4 + tl2)]

3

8. a ln(AX)

Note that in each case the constant of integration could be added on; i.e. in 1 the answer could have been written

ln(x114) + C or ~ ln(x) + C.

Exercise 8. 7.3

1. .!. 2. .!. ln (2.) 3 5 4

1 4. 3 (3v3 - 1) = 1.399

3. ~ ln (~)

5. ~ ln (~) 6. 9

1 7.-

4 8. -0.575 (three decimal places)

1 1 9 -----

. 4 2(1 + e2)

Exercise 8.8.1

1. ln(sin e) + c

4. -ln(1 + cos e) + c

6. ~ ln (I) 1

8. - ln(1 + 2x) + c 2

Exercise 8.8.2

1 1. - arctan(3x) + c

3

v3 10.-

2'7T

2 2. - ln(2 + ~2) + c

3

5. ln(V6)

1 3. - ln(sec 29) + c

2

2. ~arctan (~) + c

1 (3x) 3. 6 arctan 2 + c

5. arcsin x + c

1 (bx) 7. b arcsin -;; + c

1T 9.-

24

Exercise 8. 8.3

1. ln(cosec e - cot e) + c 2. -2(1 + tan e/2)- 1 + c

Exercise 8.8.4

1 1 1. - x - - sin 2x + c

2 4

1 3. - (x + sin x) + c

2

Exercise 8.9.1

1. X sin X + COS X + C

3. !x sin 2x + i cos 2x + c

2 5. - (x + 1)312 (3x - 2) + c

15

4. ~ arctan (bx) + c ab a

6. arcsin (~) + c

1T 8.-

6

1T 10.-

12

1T 1 2.---

8 12

1 1 4. -+-

16 811'

2. (x - 1)ex + c

1 2x 6. - (2 - 9x2) cos 3x + - sin 3x + c

27 9 xz xz

7. -(x2 + 2x + 2)e-x + c 8. -In x-- + c 2 4

Exercise 8.9.2

4 1.- (11\12- 4)

15

1 3. l5 (297 - 56\17)

5. 8(11'- 2) 7. 2 In 2 - 1

2 4.--

9

6. 8(5e2 - 1)



Exercise 8. 9.3

1 1 1. 2 ex (sin X - COS x) + C 2. -e-x (sin x - cos x) + c

2

1 2 3. - e2x (cos 4x + 2 sin 4x) + c 4. - (e, - 2)

10 5

2 1 5. - - - (V3 + 2)e-,16

5 10

1 6. - (3V2e, - 4e,l2)

10

Exercise 8.10.1

In A and c are arbitrary constants.

[A(x + 1)3] 1. In 2 or 3 ln(x + 1) - 2 ln(x - 1) + c

(x- 1)

3 2. ln[A(x + 3)4(2x - 1)312] or 4 ln(x + 3) + -ln(2x - 1) + c

2

1 1 3. -ln[A(2x + 3)(2x - 1)] or -ln[(2x + 3)(2x - 1)] + c

4 4

[Ax\x- 1)]

4. In 2 or ln[x5(x - 1)] - 2 ln(x + 3) + c (x + 3)

2 5. - ln(3x + 1) + 3 arctan x + c

3

6. ln[(x + 3)2Yx2 + 1) - arctan x + c

1 or 2 ln(x + 3) +- - ln(x2 + 1) - arctan x + c

2

1 2 7. - -- + -ln(3x - 2) + c

X + 1 3

8. ln(2x + 3) +~arctan (~) + c

9. In(2.7) 10. In G~)

Exercise 8.11.1

1T 1.-

7

1T2 2.-

2

3 11. In 160 --

2

10161T 4.--

15

11" 5.-

2

9. 11"

Exercise 8.12.1

11" 6.-

2

12811" 10.--

7

2. (~ ~) 5, 7

11" 7.-

5

3. (~ ~) 2 , 8

5. (0.459, 0.402) to three decimal places

Exercise 8.13.1

1. 4

3. 3.1312

5. 1

1 7. - (e4 - 1)

8

2. 16.5

4. ! ln(x2 + 1) + c

1 1 6. - te3t - - e3t + c

3 9

8 . .!_ ln ( 2 + x) + c 2 2-x

1 1 9. - - Vl - 9x2 -- arcsin(3x) + c

9 3

1 10. --cos2x+c

4

14. In -- + c (X- 2) x+3

1 16. -sin 1rx + c

11"

19. (2 - x2) cos x + 2x sin x + c

11. 0.5 + ln(3/4)

13. In(r - 3) + c

15. In (%)

x2 sin2 x x sin x cos x (x) 20. 4 + - 4-- 2 + c 21. i arctan 2 + c

22. ! ln(t2 + 4) + c 23. ! ln [ G ~ ~) ] + c

24. - iJn(t2 - 4) + c 25. u +In u + c



26. u - ln(u + 1) + c

2 28. t-- + c

t

30. ! ln(t2 + 2) + c

32. ! arcsin x + i xV1 - x2 + c

1 34. - -ln(5- 6t) + c

6

1 36. 3(5 - 6t)l/2

1 38. - u3 (3u2 + 5) + c

15

t2 27. 2 ln t + - + c

2

29. t- V2 arctan (~2) + c

2 33. - -(1- x)312 + c

3

1 35. + c

6(5 - 6t)

(u2 + 1)2 37. + c

4

1 39. - [ u(2u2 + 1) V u2 + 1 - u - ln V u2 + 1 ] + c

8

Objective Test 8

1. ? + c

2.4(1-~2) 3. 1

e- 1 4.--

2

6. ~arctan (k) + c

7. _!_ ln ( 8 + t) + c 16 8- t

1 1 8. - x +- sin(2'TTx) + c

2 4'TT

In each case c is an arbitrary constant.

CHAPTER 9

Exercise 9.2.1

1. y = 2x2e 3. y = 5- ze-x 5. X= 4(1 - t)

Exercise 9.3.1

y

I 3

I

/

/

-3

" \

\

3

I I I I

I I I I

I I I

/ / / / /

/ / / / /

"" " " " \ \ \ \ \

\ \ \ \

\ \ \

I -3

y

\ " /3

\ " - /

I I I I I

I I

/-3

3

2. y = ln(2x + 3) 4. y = arctan t + -rr/2

I

I

/ / / / / /

/ / / / / /

3 X

" " " " " " \ \ \ \ \ \

\ \ \

I I I I I

-3 X

/ - ' \ \ \

/ " \


y

\ \3 I

\ \ I I I

\ I I I

\ I I I \ \ I I I

I \ \ I I I

-31X

\ \ I I I I

I \ \ I I I I I \ \ \ I I

\ \ \ I I I

\ \ \ I I I

\ \-3 I

2

y

- - - - - -3

I I I I I I

I I I I

3 -3 X

I I I I I I I

- - - - - --3

4


Exercise 9.4.1

In the following answers, A and Care arbitrary constants.

1. y = Ax3

3. y =COS X+ C 5. l =A e1

7. C - COS X = ! t2

1 9. -- = e 1 + C

X

11. y = 4x2 + 1

1 13. y = 1 - - sin 2wt

2w

'TT 15. arctan y = t2 + -

4

17. e2x = 2[2- e-1(1 + t)]

2 4(1 + t)2

19. 1 +X = 4 (1 - t)

Exercise 9.5.1

1. 0.711 m, 147 s

Exercise 9.6.1

1. 3.003

Objective Test 9

1. y=!sin2x+c

3. i = V(e12 + c)

5. 1.136

CHAPTER 10

Exercise 10.2.1

1. (a) 10 gram

2. 1.084

2. y2 - x2 =A 4. y =A ek1

6. ln y = t2 + 4t + C

8. e-x = C - ~ t 3

10. cot y = C - ! sin 2x

12. y = 3e-Zx

14. 3t2 + 2e-3Y = 2

16. x2 = arcsin(2t)

18. sin 2wy + 2 cos wy = 2

20. y sin y + cosy = ex

2. m = e-0.0127t

3. 1.096, 1.095

2. v = Ael.512

4. y = el/(x+l)

(b) 0.1, 10%; 0.01, 1 %; 0.00001, 0.001% (c) 127 < m < 147

2. 3.135 < m < 3.145

Exercise 10.2.2

(a) error I = 0.0015; 5.4975 < n < 5.5005 (b) error I = 0.0056; 23.8914 < n < 23.9026 (c) error I = 0.0090; 2.4099 < n < 2.4278 (d) error I = 0.0058; 6.8007 < n < 6.8123 (e) error I = 0.9 331.50 < n < 333.30 (f) error I = 0.0045; 2.3923 < n < 2.4013

Exercise 10.2.3

(a) E = 0.019 ; e132 = 3.743 ± 0.019; 4.0 to appropriate accuracy (b) E = 0.050 ; cos(l.5) = 0.0707 ± 0.0499; 0.0 (c) e = 0.00019; ln(2.634) = 0.9685 ± 0.00019; 0.97 (d) E = 17.28 ; e<2·1)2 = 82 ± 17; 100 (e) e = 0.08 ; 3.42e132 = 12.80 ± 0.08; 13

Exercise 10.3.1

1. For root near x = 3.4, method (d) only. For root near x = 0.6, methods (a) and (c) could be used. (c) would converge more quickly.

2. Use method (a); x = 0.567 to three decimal places.

( 1- X) 3. (a) Xn+l = sin- 1 T ; x = 0.338

(b) X = V1 + 2e xn · X = 1 253 n+l ' ·

1 + 2x2 - x3

(c) Xn+l = n n ; X= 0.430 3

Exercise 10.3.2

(a) X= 0.5858 (b) X= 0.5671 (c) X= 0.3376 (d) X= 1.2534 (e) x = 0.4302

Exercise 10.4.1

1. (a) 2.927 (b) 0.2438 (c) 0.6932 (d) 3.142 (e) 0.1054 2. 0.62 km 3. 35.8 ms- 1

4. e = 49.03 (simple average = 49.45)

Objective Test 10

1. e = 0.389; appropriate answer 63 2. For the root near 0.3, methods (a) and (c) cannot be used.

Quickest method is (d). For the root near 5, methods (b) and (d) cannot be used. Quickest method is (c). Roots are 4.9571 and 0.3429.

4. (a) -0.39 (b) 0.83



CHAPTER 11

Exercise 11.1.1

force, velocity, displacement.

Exercise 11.1.2

1. N

Derby

Leicester

Scale 10mm = 10km

2. 16N

Stone

30 N Scale 2 mm= 1 N

3.

131 mph

Scale: 1 em =20 mph

N

••--•.,..2N Scale: 1 em= 1 N

(v)

(iii) Oxford 1-------

Scale: 1 em =2 N

F 19.6 N

(ii)

r Scale: 1 cm:oc10mph

Exercise 11.2.1

1. c = k, d = f, e = n 2. (a) e = -2a

(d) m = 2d (g) l = 1¥/

3. (i) = (vi), (ii) = (v)

Scale: 1 cm=20km

Portsmouth

(vii)

70 mph

(b) h = -a (e) g = -d (h) j = -b

(c) (f)

n = -2a i=-2d



4. Vector Magnitude Direction (bearing)

a 1N 270° b 1.4N 135° c 1.1N 117° d 1.4N 045° e 2N 0900

f 1.4N 045° g 1.4N 225° h lN 090°

2.8N 215° j 1.4N 315° k l.lN 117° I 2.1N 045° m 2.8N 045° n 2N 090°

Giving the direction of a vector as a bearing provides a unique answer.

Exercise 11.2.2

1. LJ ... ~ a e

g

m

7\ e


2. This solution depends on your choice of a and b

D

3. 13 newtons; angle = 22.6° to larger force. 4. 15.5 newtons; angle = 29.6° to smaller force. 5. 100.9 miles on bearing 112°

Exercise 11.3 .1

1. [20 cos 30°] = [10v'3] 20 COS·60° 10

[ 250 cos 70°] [ 85.5] 2• 250 cos 20° = 234.9

3. F = [ O J · G = [ 75 J 100 ' 0

Exercise 11.3.2

1. [ =~~ ::: ~~:] -[ -~;~3] 2. r-30 cos 70°J = [-10.26J

-30 cos 20° -28.18


3. Vector Component form

a [ -~] b [ _;J c [ -~.5 J d GJ e [~] f GJ g [ =:J h [~] i [ =~J j [ -:] k [ -~.5 J

[1.5] l 1.5 m GJ n [~]


Exercise 11.3.3

Direction Vector Magnitude bearing a (see p.418)

a 5 036.9° 53.1° b 13 022.6° 67.4° c 5 323.1° 126.9° d Y5 206.6° -116.6° e V2 225° -135°

I Y10 161.6° -71.6°

Exercise 11.3.4

2. Direction

Vector Magnitude bearing or a (see p.418)

a+ b = [ 1!] 8V5 026.6° 63.4°

b + c = [ 1!] 2Y65 007.1° 82.9°

a+c=[~] 8 oo goo

c +I= [ -~] Y5 296.6° 153.4°

d-e=[-~] 1 180° -90°

c- a= [ -~] 6 270° 180°

b+e=[1:] V137 020° 70°

I- e = [ -~J 2V2 135° -45°

3. (a) ( 5 cos 4SO ) 5 cos 45°- 4

3.6N, -7S

(b) c- 3 cos 30°) 3 cos 60°- 2

0.78N, -140.1°


(5.3 cos 30° - 4.7 cos 60° )

(c) 5.3 cos 60° + 4.7 cos 30°- 6

(4 cos 45° + 5 cos 30° - 3)

(d) 4 cos 45° - 5 cos 60°

2.35N, 17.8°

4.17N, 4S

4. ------------------------------------------Direction

Vector Magnitude bearing or a (see p.418)

a+ b = [ -~J v'13 123.7° -33.7°

b+c=[-~] 1 180.0 -90°

a-b=[-~] v'65 352.9° 97.1°

3a+2b=[-~] v'50 098.1° -8.1°

[ -16] - 2a - 3b + 4c = 25 29.7 327.4 123.6°

Exercise 11.3.5

1. 2i, 5i, 4i, I, 7 .3i 2. a= 3i + j

b = -i + 3j c = -3i- 2j d=3i+j

3. I a I = 5, I b I = v'B, I c I = 5

c

II

II X

4. Same as Exercise 11.3.4, Problem 4. 5. (a) (3 - 2.5 COS 30°)i + (2- 2.5 cos 60°)i = 0.83i + 0.75j

(b) (5.3 COS 30°- 4.7 COS 60°)i + (5.3 COS 60° + 4.7 cos 30° - 6)j = 2.24i + 0.72j

(c) (7.1 cos 15°- 5.4 cos 80°- 8cos 60° + 4.1cos 40°)i + (7.1 cos 75° + 5.4 cos 10°- 8 cos 30° - 4.1 cos 50°)j = 5.06i - 2.41j

Exercise 11.4.1

1. a = j b = i c = - 2i + 3j d = -i- 2j

c

b

d

2. c = 3i + 4j; f = 3i + 4j; d = 3i + 4j; c is the position vector of (3, 4)

3. a = i + 2j; b = 5i + j; c = 7i + Sj ~

AB=b-a=4i-j oc = c - b = 2; + 7i ___,. CA =a- c = -6i- 6j

4. a = 4i + 2j; b = 5i + 4j; AB = i + 2j ~ I AB I = V5, direction = 63.4°

Exercise 11.4.2

1. Case Velocity Acceleration

t=O t=l t=O t=1

(a) (b) (c)

3i lOi + lOj wj

6i + Bj lOi + 0.2j (w- g)j

Si -9.8j -gj

6i + Bj -9.8j -gj

speed at timet = 0 is (a) 3 ms- 1 (b) 10Y2 ms- 1 (c) w ms- 1

(3 (3t3 ) 2. v = ri - (3t - 2)j; r = 3 i - 2 - 2t j

3. v = -dw sin wti + dw cos wtj a= -dw2 cos wt i- dw2 sin wtj a = - w2r hence a is parallel to r



Vector

a+b

a-b

2a- 3b

Objective Test 11

1. Scalars: area, temperature, volume, energy, time; vectors: velocity, force, displacement, acceleration

2.

3. a = [~] or a = 3i

[ 2.5 J 5\1'3 b = 5\1'3/ 2 or b = 2.5i + - 2-j

Direction Component form Magnitude bearing or a

[ 5.5 J 5\1'3 5\1'3/ 2 or 5.5i + - 2-j 7 052° 38.2°

[ 0.5 J 5\1'3 y' ( or 0.5i - --j -5 3 2 2

4.36 173.4° -83.4°

[ -1.5 J 15\1'3 -15\1'3/2 or -1.5i- -2-j 13.1 186.6° -96.6°

4. (3 cos 45° - 4 cos 60° - 2)i + (3 cos 45° + 4 cos 30° - 8)j = -1.879i- 2.415j; magnitude= 3.06 N; direction 217.9° or a= -127.9°

5. 18.5 knots bearing 240.3° 6. 12.4 km bearing 108° 7. r(t) = (t3 + t)i- (2t2 + 2t)j, a(t) = 6ti - 4j

CHAPTER 12

Exercise 12.2.1

Exercise 12.3.1

[_:] 2. [ 0.4 -0.1] -0.3 -0.7

4. 3 [ -~ -~J 1 5. - [4 1 -2)

8

6. X [ 3

\ ] a -x

10 2

8 [ ln 10. [a + 2b 2a + 3b 3a + 4b]

[0 -0.5] 11. 2.3 -2.4

13 [ ~] 5

16. a = 6, b = - -3

Exercise 12.4.1

1. [ -~ ~] [:] = [~]

7. [ =:~]

9. [3x +By] 2y- X

[ 0 -5] 12. 23 -24

[ -8 1 0 ] 14.

3 3 0

2 5 0

17. x = 5, m = 2



3. [-~ ~ J [; J = [: J

Exercise 12.5.1

1. 1 2. 2y- 3x 6. 3; (3, 0) 8. 4; -1.25, 3.75

10. 0, no unique solution

Exercise 12.5.2

1. 27 4. 4

Exercise 12.5.3

1. 0

2 8.

15

Exercise 12.5.4

2. 1 5. -18

2. 0

9. 0

1. X= 0, y = 0.7, Z = 0.3

3. 1 4. 0 5. 1 7. -10; (2, 3) 9. 22; (2.7, 1.8)

3. -0.018 6. xy(y- x) 7. 3

3. 0 7. -1860

10. -y(z- xf

2. det M = 0, so no unique solution 3. a = -2, b = 2, c = 1

Exercise 12.6.1

1 [v'3 -1 J 2" 4' 4 1 v'3

-1, [ ~ 1

-~] 3. 0, no inverse 4. -2 -1 1 -1

[ Va 0 0 ] 61.u -k kn-m]

5. abc, ~ 1/b 0 1 -n

0 1/c 0 1

Exercise 12.6.2

1. X = 2, y = 3 2. S = 2, t = 3 3. X = 5, y = -4 4. X= -1, y = 5 5. X= 0, y = 2, Z = -1 6. p = 0.3, q = 0.5, r = 0.2

Exercise 12. 7.1

1. X = 2, y = -1, Z = 4 3. u = 1, v = 0.5, w = 2.5

2. X= -0.7, y = 0.2, Z = 0.5 4. p = 5, q = 1, r = 0

Exercise 12.8.1

1. inconsistent, no solutions

3 + A 2. Infinitely many solutions: u = -- , v = A

6

3. Inconsistent, no solutions

4 + A (1 + 4A) 4. Infinitely many solutions: x = - 6- , y = - 3 , z = A

5. u = 8 - 4A, v = 3(A - 1), w = A

Exercise 12.9.1

1. Currents (A): 0.581, -0.452, 1.032; Voltages (V): 2.903, -0.903, 3.097 across AB, BC and BD.

2. Currents (rnA): -0.243, 0.054, -0.297 Voltages (V): -1.216, 0.216, -1.784 across AB, BC and BD.

3. Currents (rnA): 0.415, -0.418, -0.727, 0.564, 0.309; Voltages (V): 0.873, -0.418, -1.455, 1.127, 1.545 across AB, BC, CD, BF and CE.

4. Currents (A): 0.343, 0.214, 0.129, 1.000; Voltages (V): 1.714, 1.286, 1.288, 4.000 across AB, BC, BD, EF

5. Currents (A): 0.75, 0.25, 0.5,1.5, -0.75; Voltages (V): 1.5, 1.5, 1.5, 3.0, -3.0 across AB, BC, BD, DF, EF.

Objective Test 12

1. (a) [ -0.6 1.2] 0.8 -0.6

[ 1.5 (b) -0.7

-1.2] -0.3

[ 0.13 0.04 J (c) -0.46 0.29

[0.17 (d) 0.20

-0.68] 0.25

u 0 n 2. 1

0



5. p = 11, q = 42 6. -15

[ 0.1 0.3] 7. det M = 0.2; M- 1 = 5 -0.6 0.2

8. X = 1.5, y = 11 9. X= -3, y = 0, Z = 4

3- 5A 10. u = , v =A

2

Index

abscissa 7 adjoint matrix 453 amplitude 108, 124 angle convention 109 angular frequency 122 AP 176 areas 296

negative 304 arithmetic progression 176 associative 415 asymptote 80, 113, 288, 291 average

rate of change 228, 233 speed 228, 229

base 69 beats 143, 158 binomial theorem 218, 221, 222,

484 bound vector 424 boundary conditions 370

calculus, fundamental theorem of 317

centroid 358 chain rule 252, 485 check sums 458 codomain 75 coefficient 35, 217 cofactor 452 combinations 220 common difference 177, 485 common ratio 182, 485 commutative 415 completing the square 46 components 416 composite function 77

differentiation 252 continuous 81 convergence 56, 398, 401 coordinates 6 cosecant 132, 145, 486 cosine

as a projection 110 differentiation 275, 486 general solutions 118, 135

integration 320, 487 period 120, 134 primary and secondary

solutions 115 rule 164

cotangent 132, 145, 486 Cramer's rule 440, 447 cubic 51 current 468, 474 curve sketching 286

decay constant 94 decay process 87, 232

and differential equations 384 half life 94

decimal search 54 definite intergral 313, 319 degree 51 denominator 204 dependent variable 5 derivative 242

second 247 determinant

of a 2x2 matrix 439, 491 of a 3x3 matrix 442 meaning of zero value 465

differentiation applications 248, 258, 269, 286 constant 242 chain rule 252, 485 composite functions 252, 485 differences 244 exponential functions 281, 283,

486 function of a function 252, 485 functional notation 246 limiting process 237 logarithmic functions 284, 486 maxima and minima 258 multiples 244 product rule 267, 486 quotient rule 267, 486 rules 242, 252, 267 substitution 252 sums 244 trigonometric functions 273, 486

557

558 Index

differential equations 367 analytical solution 376 direction fields 372 Euler's method 385 general solution 369 initial conditions 370 numerical solution 384 order of 368 particular solution 369 separation of variables 376 setting up 379

direct proportion 2, 19 direction fields 372 discontinuous 81 discriminant 49 displacement 231, 302, 410 distance 231 divergence 58 domain 75

e 85 electric circuits 468 elimination 458

check sums 458 multipliers 461 pivots 460 valid steps 459

empirical model 476 equation

normal 250 quadratic 34 straight line 9, 14 tangent 250

errors 393 absolute 394 estimation 396 propagation 395 relative 394 rounding 393

Euler's method 385, 387 even function 114 exponent 69

laws 70, 484 exponential equations 74 exponential form 69 exponential functions 83, 85

differentiation 281 integration 320

exponential law 102 extrapolation 29

factors 40 factor theorem 198, 201 factorials 219 factorizing 42, 198, 201 force 409 formula iteration 55

free vector 424 frequency 137 function 76

composite 77 continuous/discontinuous 81 even 114 decay 83, 87 exponential 83 growth 83, 87 inverse 79 logarithmic 88 odd 114 periodic 108, 118, 121 sinusoidal 121

functional notation for derivatives 246

fundamental theorem of calculus 317

general solutions differential equations 369 trigonometric equations 118

geometric progression (GP) 181 geometric sequence 181 gradient 10, 84, 239, 242

normal 250 rate of change 15 straight line 10 tangent 239, 250

graphical model 8 graphs

growth process 87 reduction to straight line 100 sketching 286 straight line 9, 14

half angle substitution 340 half life 94

identities trigonometric 144, 167, 489

identity matrix 435 image 75 improper partial fractions 204, 213 improved Euler method 388 indefinite integral 320 identity

double angle 150 sum and difference 148 trigonometric 145, 167

independent variable 5 index 69

laws 70, 484 inflexion 52, 261 initial conditions 370 instantaneous rate of change 228,

234, 242

instantaneous speed 228 integral

definite 319 indefinite 320 logarithmic form 336 trigonometric substitutions 338,

340 integration

as an area 297 by parts 344, 346, 488 by quadrature 403 by substitution 327, 332, 338,

342 exponential functions 321, 322,

487 fractions 330, 336 numerical 402, 488 partial fractions 351 polynomials 315, 320, 321, 330,

487 relation to differentiation 317 trigonometric functions 320,

322, 487 volumes 355

intercept 12, 38 interference 142 interpolation 29 intervals 3 mverse

function 79, 155 matrix 450, 453, 491 proportion 3, 23 square law 25 trigonometric functions 155, 338

irreducible quadratics 202

Kirchhoff's laws 468, 474

limit of a sequence 172 of sin xlx 273

limiting process 237 line of best fit 27 linear

equation 14 extrapolation 29 factors 40 interpolation 29 law 26 relation 2

local maximum or minimum 52, 261

long division of polynomials 192 logarithm

base 88 base e 89 base ten 89

common 89 differentiation 284 graph paper 100 law 103 natural 89 rules 92, 484

logarithmic integral 336 log-linear graph paper 104 log-log graph paper 100 loop current 468 lower sum 307

magnitude of a vector 410 matrices 430

addition 433 adjoint 453 cofactor 452 identity 435, 453, 491 inverse 450, 453, 491 minor 443, 452 products 434 subtraction 433 transpose 451

maximum 37, 52, 252, 261, 269 mesh current 468 minimum 37, 52,252,261, 269 modelling 26, 473

assumptions 475 empirical 474 exponential laws 102 graphical 8 linear laws 26 polynominals 60 power laws 99 quadratic 60 simplification 475 validation 475

moments 359

negative angles 109 negative areas 302 Newton-Raphson method 400,

489 non- singular systems 465 numerical methods 392

decimal search 54 finding roots 54, 400 for differential equations 384 formula iteration 55 integration 402

numerator 204

Ohm's law 22, 468 odd function 114 ordinate 7

Index 559

560 Index

parabola 35 parallelogram law 414 partial fractions 204, 215

applied to integrals 351 improper 204 proper 204

particular solution 369 Pascal's triangle 216 period 108, 120, 134 phase angle 126 phasor 112 point of inflexion 52, 260 polynomial

coefficient 51 cubic 51 degree 51 division 192 irreducible 202 linear 51 models 60 quadratic 34 quartic. 51

position vector 424 power 67, 69

fractional 67, 72 laws 70, 98, 484 negative 68, 71

primary solution 115 principal values 156 product rule 267, 488 proportionality

constant of 21 direct 2, 18 general 24 inverse 22

quadratic completing the square 46 equation 34 factorizing 42 formation 40 formula 48, 484 graph 36 law 34 roots 38, 45, 48

quartic 51 quotient 193, 195 quotient rule 267, 486

radian 133 range 75 rate of convergence 398 rates of change 15, 84

average 233 gradient 15 instantaneous 234

reciprocal 71

rectangle rule 300 recurrence relation 170 remainder 193, 195

theorem 197, 201 roots 45, 52

by decimal search 54 by formula iteration 55 by graph plotting 53 by Newton-Raphson

method 400 by 'quadratic formula' 48

scalar 410 secant 132, 145, 486 second derivative 247 secondary solution 115 semi-log graph paper 104 separation of variables 376 sequence 170 series 174 setting up differential

equations 379 Simpson's rule 403, 488 simultaneous equations

Cramer's rule 440, 447 elimination 458 infinitely many solutions 464 matrices 437, 455 non-singular solution 465 singular system 465 types of solution 465 unique solution 465

sine as a projection 110 differentiation 274, 486 graph 112 integration 321, 487 p~riod 120 primary and secondary

solutions 115 rule 161

singular systems 465 SI units 483 slope 10, 84 solid of revolution 355 speed 227,231,427 standard index form 68 stationary points 258

classification 261 steady state 87 straight line graphs 9, 14

gradient 10 intercept 12

sum lower 207 of cubes 176 of first n integers 176

Documents

APPENDIX 1: UNITS - Springer978-1-349-11717-8/1.pdf · APPENDIX 1: UNITS The basic SI units ... newton, N joule, J pascal, Pa kgm-3 coulomb, C volts, V Relationship to ... 3 X X+