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Essential formulae

Number and Algebra

Laws of indices:

am × an = am+n am

an= am−n (am)n = amn

amn = n

√am a−n = 1

ana0 = 1

Factor theorem

If x = a is a root of the equation f (x)= 0,then (x − a) is a factor of f (x)

Remainder theorem

If (ax2 + bx + c) is divided by (x − p), the remainderwill be: ap2 + bp + c

or if (ax3 + bx2 + cx + d) is divided by (x − p), theremainder will be: ap3 + bp2 + cp + d

Partial fractions

Provided that the numerator f (x) is of less degreethan the relevant denominator, the following identitiesare typical examples of the form of partial fractionsused:

f (x)

(x + a)(x + b)(x + c)

≡ A

(x + a)+ B

(x + b)+ C

(x + c)

f (x)

(x + a)3(x + b)

≡ A

(x + a)+ B

(x + a)2+ C

(x + a)3+ D

(x + b)

f (x)

(ax2 + bx + c)(x + d)

≡ Ax + B

(ax2 + bx + c)+ C

(x + d)

Quadratic formula:

If ax2 + bx + c = 0 then x = −b ± √b2 − 4ac

2a

Definition of a logarithm:

If y = ax then x = loga y

Laws of logarithms:

log(A × B)= log A + log B

log

(A

B

)= log A − log B

log An = n × log A

Exponential series:

ex = 1 + x + x2

2!+ x3

3!+ . . . .

(valid for all values of x)

Arithmetic progression:If a = first term and d = common difference, then thearithmetic progression is: a,a + d,a + 2d, . . . ..

The n’th term is : a + (n − 1)d

Sum of n terms, Sn = n

2[2a + (n − 1)d]

Engineering Mathematics. 978-0-415-66280-2, © 2014 John Bird. Published by Taylor & Francis. All rights reserved.

646 Engineering Mathematics

Geometric progression:

If a = first term and r = common ratio, then thegeometric progression is: a,ar,ar2, . . . .

The n’th term is: arn−1

Sum of n terms, Sn = a(1 − rn)

(1 − r)or

a(rn − 1)

(r − 1)

If −1< r < 1, S∞ = a

(1 − r)

Binomial series:

(a + b)n = an + nan−1b + n(n − 1)

2!an−2b2

+ n(n − 1)(n − 2)

3!an−3b3 + . . . .

(1 + x)n = 1 + nx + n(n − 1)

2!x2

+ n(n − 1)(n − 2)

3!x3 + . . . .

Newton Raphson iterative method

If r1 is the approximate value for a real root of the equa-tion f (x)= 0, then a closer approximation to the root,r2, is given by:

r2 = r1 − f (r1)

f ′(r1)

Areas and Volumes

Areas of plane figures:

Rectangle Area = l × b

b

l

Parallelogram Area = b × h

b

h

Trapezium Area = 12 (a + b)h

a

h

b

Triangle Area = 12 × b × h

h

b

Circle Area = πr2 Circumference = 2πr

r

r

su

Radian measure: 2π radians = 360 degreesFor a sector of circle:

arc length, s = θ◦

360(2πr)= rθ (θ in rad)

shaded area = θ◦

360(πr2)= 1

2r2θ (θ in rad)

Equation of a circle, centre at origin, radius r :

x2 + y2 = r2

Essential formulae 647

Equation of a circle, centre at (a,b), radius r :(x − a)2 + (y − b)2 = r2

Ellipse Area = πab Perimeter = π(a + b)

ba

Volumes and surface areas of regularsolids:

Rectangular prism (or cuboid)

h

b

l

Volume = l × b × hSurface area = 2(bh + hl + lb)

Cylinder

r

h

Volume = πr2hTotal surface area = 2πrh + 2πr2

Pyramid

h

A

If area of base = A and perpendicular height = h then:

Volume = 1

3× A × h

Total surface area = sum of areas of triangles formingsides + area of base

Cone

lh

r

Volume = 1

3πr2h

Curved surface area = πrlTotal surface area = πrl +πr2

Sphere

r

Volume = 4

3πr3 Surface area = 4πr2

Frustum of sphere

P Q

R

r

h S

r1

r2

Surface area of zone of a sphere = 2πrh

Volume of frustum of sphere = πh

6

(h2 + 3r2

1 + 3r22

)Prismoidal rule

x

x2

x2

A1 A2 A3

Volume = x

6(A1 + 4A2 + A3)


Irregular areas

Trapezoidal rule

Area ≈(

width ofinterval

)[1

2

(first + lastordinates

)

+ sum of remaining ordinates

]

Mid-ordinate rule

Area ≈(

width ofinterval

)(sum of

mid-ordinates

)

Simpson’s rule

Area ≈ 1

3

(width ofinterval

)[(first + lastordinate

)

+4

(sum of even

ordinates

)

+2

(sum of remaining

odd ordinates

)]

Volume of irregular solids

dddddd

A6 A7A2A1 A3 A4 A5

Volume = d

3{(A1 + A7)+ 4 (A2 + A4 + A6)

+2 (A3 + A5)}

For a sine wave:

over half a cycle, mean value = 2π

× maximum value

of a full-wave mean value = 2π

× maximum value

rectified waveform,

of a half-wave mean value = 1π

× maximum value

rectified waveform,

Trigonometry

Theorem of Pythagoras:

b2 = a2 + c2

A

B Ca

cb

sin C = c

bcosC = a

btan C = c

asecC = b

a

cosecC = b

ccot C = a

c

Identities:

secθ = 1

cosθcosecθ = 1

sin θcot θ = 1

tanθ

tanθ = sinθ

cosθcos2 θ + sin2 θ = 1 1 + tan2 θ = sec2 θ

cot2 θ + 1 = cosec 2θ

Triangle formulae:

Sine rulea

sin A= b

sin B= c

sin C

Cosine rule a2 = b2 + c2 − 2bc cos A

A

CB a

c b

Area of any triangle

(i)1

2× base × perpendicular height

(ii)1

2ab sinC or

1

2ac sin B or

1

2bc sin A

(iii)√

[s(s − a)(s − b)(s − c)] where s = a + b + c

2


Compound angle formulae

sin(A ± B)= sin A cos B ± cos A sin B

cos(A ± B)= cos A cos B ∓ sin A sin B

tan(A ± B)= tan A ± tan B

1 ∓ tan A tan B

If R sin(ωt + α)= a sinωt + b cosωt,then a = R cosα, b = R sinα,

R =√(a2 + b2) and α = tan−1 b

a

Double angles

sin 2A = 2 sin A cos A

cos2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A

tan 2A = 2 tan A

1 − tan2 A

Products of sines and cosines into sums ordifferences:

sin A cos B = 1

2[sin(A + B)+ sin(A − B)]

cos A sin B = 1

2[sin(A + B)− sin(A − B)]

cos A cos B = 1

2[cos(A + B)+ cos(A − B)]

sin A sin B = −1

2[cos(A + B)− cos(A − B)]

Sums or differences of sines and cosinesinto products:

sin x + sin y = 2 sin

(x + y

2

)cos

(x − y

2

)

sin x − sin y = 2 cos

(x + y

2

)sin

(x − y

2

)

cos x + cos y = 2 cos

(x + y

2

)cos

(x − y

2

)

cos x − cos y = −2 sin

(x + y

2

)sin

(x − y

2

)

For a general sinusoidal function y = A sin(ωt ± α),thenA = amplitude ω = angular velocity = 2πf rad/s2π

ω=periodic time T seconds

ω

2π= frequency, f hertz

α = angle of lead or lag (compared with y = A sinωt)

Cartesian and polar co-ordinates

If co-ordinate (x, y)= (r,θ) then r =√

x2 + y2 and

θ = tan−1 y

x

If co-ordinate (r,θ)= (x, y) then

x = r cos θ and y = r sinθ

Graphs

Equations of functions

Equation of a straight line: y = mx + c where mis the gradient and c is the y-axis intercept

If y = axn then lg y = n lg x + lg aIf y = a bx then lg y = (lg b)x + lg aIf y = a ekx then ln y = k x + ln a

Equation of a parabola: y = ax2 + bx + c

Circle, centre (a, b), radius r: (x − a)2 + (y− b)2 = r2

Equation of an ellipse, centre at origin, semi-axes a

and b:x2

a2 + y2

b2 = 1

Equation of a hyperbola:x2

a2 − y2

b2 = 1

Equation of a rectangular hyperbola: xy = c

Odd and even functions

A function y = f (x) is odd if f (−x)= − f (x) for allvalues of xGraphs of odd functions are always symmetrical aboutthe origin.A function y = f (x) is even if f (−x)= f (x) for allvalues of xGraphs of even functions are always symmetrical aboutthe y-axis.


Complex Numbers

z = a + jb = r(cos θ + j sin θ)= r∠ θ = r e j θ wherej2 = −1Modulus r = | z | =

√(a2 + b2)

Argument θ = arg z = tan−1 b

aAddition: (a + jb)+ (c + jd)= (a + c)+ j (b + d)

Subtraction: (a + jb)− (c + jd)= (a − c)+ j (b − d)

Complex equations: If m + jn = p + jq thenm = p and n = q

Multiplication: z1z2 = r1 r2∠(θ1 + θ2)

Division:z1

z2= r1

r2∠(θ1 − θ2)

De Moivre’s theorem:[r∠θ ]n = rn∠nθ = rn(cosnθ + j sin nθ)= re jθ

Statistics and Probability

Mean, median, mode and standarddeviation

If x = variate and f = frequency then :

mean x =∑

f x∑f

The median is the middle term of a ranked set of data.The mode is the most commonly occurring value in aset of data

Standard deviation σ =√√√√

[∑{f (x − _

x)2}

∑f

]for a

population

Binomial probability distribution

If n = number in sample, p = probability of the occur-rence of an event and q = 1 − p, then the probability of0, 1, 2, 3, .. occurrences is given by:

qn, nqn−1 p,n(n − 1)

2!qn−2 p2,

n(n − 1)(n − 2)

3!qn−3 p3, ..

(i.e. successive terms of the (q + p)n expansion)

Normal approximation to a binomial distribution:

Mean = np Standard deviation σ = √(n p q)

Poisson distribution

If λ is the expectation of the occurrence of an event thenthe probability of 0, 1, 2, 3, .. occurrences is given by:

e−λ,λe−λ,λ2 e−λ

2!,λ3 e−λ

3!, ..

Product-moment formula for the linearcorrelation coefficient:

Coefficient of correlation r =∑

xy√[(

∑x2)(

∑y2)]

where x = X − X and y = Y − Y and (X1, Y1),(X2, Y2), .. denote a random sample from a bivariatenormal distribution and X and Y are the means of the Xand Y values respectively

Normal probability distribution – Partial areas underthe standardised normal curve – see Table 41.1 on page408.

Student’s t distribution – Percentile values (tp) forStudent’s t distribution with ν degrees of freedom – seeTable 44.2 on page 436.

Symbols:

Populationnumber of members Np , mean μ, standard deviation σ .

Samplenumber of members N , mean x , standard deviation s.

Sampling distributionsmean of sampling distribution of means μx

standard error of means σx

standard error of the standard deviations σs

Standard error of the means

Standard error of the means of a sample distribution, i.e.the standard deviation of the means of samples, is:

σ_x = σ√

N

√(Np − N

Np − 1

)

for a finite population and/or for sampling withoutreplacement, and

σ_x = σ√

Nfor an infinite population and/or for sampling withreplacement


The relationship between sample meanand population mean

μ_x = μ for all possible samples of size N are drawn

from a population of size Np

Estimating the mean of a population (σknown)

The confidence coefficient for a large sample size,(N ≥ 30) is zc where:

Confidence Confidencelevel % coefficient zc

99 2.58

98 2.33

96 2.05

95 1.96

90 1.645

80 1.28

50 0.6745

The confidence limits of a population mean based onsample data are given by:

x ± zcσ√N

√(Np − N

Np − 1

)for a finite population of size

Np , and by

x ± zcσ√N

for an infinite population

Estimating the mean of a population (σunknown)

The confidence limits of a population mean based onsample data are given by:

μ_x ± zcσ_

x

Estimating the standard deviation of apopulation

The confidence limits of the standard deviation of apopulation based on sample data are given by:

s ± zcσs

Estimating the mean of a population basedon a small sample size

The confidence coefficient for a small sample size(N < 30) is tc which can be determined using Table44.1, page 432. The confidence limits of a populationmean based on sample data given by:

x ± tc s√(N − 1)

Differential Calculus

Standard derivatives

y or f (x)dy

dxor f ′(x)

axn anxn−1

sin ax a cos ax

cos ax −a sin ax

tan ax a sec2 ax

sec ax a sec ax tan ax

cosec ax −a cosec ax cot ax

cot ax −a cosec 2 ax

eax aeax

ln ax 1x

sinh ax a cosh ax

cosh ax a sinh ax

tanh ax a sech 2 ax

sech ax −a sech ax tanh ax

cosech ax −a cosech ax coth ax

coth ax −a cosech 2 ax

Product rule: When y = uv and u and v are functionsof x then:

dy

dx= u

dv

dx+ v

du

dx

Quotient rule: When y = u

vand u and v are functions

of x then:

dy

dx=v

du

dx− u

dv

dxv2


Function of a function:

If u is a function of x then:dy

dx= dy

du× du

dx

Parametric differentiation

If x and y are both functions of θ , then:

dy

dx=

dy

dθdx

dθ

andd2 y

dx2 =d

dθ

(dy

dx

)

dx

dθ

Implicit function:

d

dx[ f (y)] = d

dy[ f (y)] × dy

dx

Maximum and minimum values:

If y = f (x) thendy

dx= 0 for stationary points.

Let a solution ofdy

dx= 0 be x = a; if the value of

d2 y

dx2 when x = a is: positive, the point is a minimum,

negative, the point is a maximum

Points of inflexion

(i) Given y = f (x), determinedy

dxand

d2 y

dx2

(ii) Solve the equationd2 y

dx2 = 0

(iii) Test whether there is a change of sign occurring

ind2 y

dx2 . This is achieved by substituting into the

expression ford2 y

dx2 firstly a value of x just less

than the solution and then a value just greater thanthe solution.

(iv) A point of inflexion has been found ifd2 y

dx2 = 0

and there is a change of sign.

Velocity and acceleration

If distance x = f (t), then velocity v = f ′(t) ordx

dt

and acceleration a = f ′(t) ord2x

dt2

Tangents and normals

Equation of tangent to curve y = f (x) at the point(x1, y1) is:

y − y1 = m(x − x1)

where m = gradient of curve at (x1, y1)

Equation of normal to curve y = f (x) at the point(x1, y1) is:

y − y1 = − 1m (x − x1)

Integral Calculus

Standard integrals

y∫

y dx

axn axn+1

n + 1+ c (except where n = −1)

cos ax1

asin ax + c

sin ax − 1

acos ax + c

sec2 ax1

atan ax + c

cosec 2ax − 1

acot ax + c

cosec ax cot ax − 1

acosec ax + c

sec ax tan ax1

asecax + c

eax 1

aeax + c

1

xln x + c

tan ax1

aln(secax)+ c

cos2 x1

2

(x + sin 2x

2

)+ c

sin2 x1

2

(x − sin 2x

2

)+ c

tan2 x tan x − x + c

cot2 x −cot x − x + c


t= tanθ

2substitution

To determine∫

1

a cosθ + b sinθ + cdθ let

sin θ = 2t(1 + t2

) cosθ = 1 − t2

1 + t2

and dθ = 2 dt(1 + t2

)

Integration by parts If u and v are both functionsof x then:

∫u

dv

dxdx = uv −

∫v

du

dxdx

Numerical integration

Trapezoidal rule

∫y dx ≈

(width ofinterval

)[1

2

(first + lastordinate

)

+ sum of remaining ordinates

]

Mid-ordinate rule

∫y dx ≈

(width ofinterval

)(sum of

mid-ordinates

)

Simpson’s rule

∫y dx ≈ 1

3

(width ofinterval

)[(first + lastordinate

)

+ 4

(sum of even

ordinates

)

+2

(sum of remaining

odd ordinates

)]

Area under a curve:

area A =∫ b

ay dx

y

y 5 f(x)

x 5 a x 5 b x0

A

Mean value:

mean value = 1

b − a

∫ b

ay dx

R.m.s. value:

r.m.s. value =√{

1

b − a

∫ b

ay2 dx

}

Volume of solid of revolution:

volume =∫ b

aπ y2 dx about the x-axis

Centroids

y

C

Area A

xy

y 5 f(x)

x 5 a x 5 b x0

x =

∫ b

axy dx

∫ b

ay dx

and y =12

∫ b

ay2 dx

∫ b

ay dx

Theorem of Pappus With reference to the above dia-gram, when the curve is rotated one revolution about thex-axis between the limits x = a and x = b, the volumeV generated is given by:

V = 2π Ay


Second moment of area and radius of gyration

Shape Position of axis Second moment Radius of

of area, I gyration, k

Rectangle (1) Coinciding with bbl3

3

1√3length l

(2) Coinciding with llb3

3

b√3

breadth b

(3) Through centroid, parallel to bbl3

12

1√12

(4) Through centroid, parallel to llb3

12

b√12

Triangle (1) Coinciding with bbh3

12

h√6Perpendicular

(2) Through centroid, parallel to basebh3

36

h√18

height h

(3) Through vertex, parallel to basebh3

4

h√2

base b

Circle (1) Through centre, perpendicularπr4

2

r√2radius r to plane (i.e. polar axis)

(2) Coinciding with diameterπr4

4

r

2

(3) About a tangent5πr4

4

√5

2r

Semicircle Coinciding with diameterπr4

8

r

2radius r

Parallel axis theorem:If C is the centroid of area A in the diagram shown then

A k2B B = A k2

GG + A d2 or k2B B = k2

GG + d2

G B

G B

d

C

Area A

Figure FA19

Perpendicular axis theorem:If O X and OY lie in the plane of area A in the diagramshown then

A k2O Z = A k2

O X + A k2OY or k2

O Z = k2O X + k2

OY

Z

O

Y

X

Area A

Figure FA20


Further Number and Algebra

Boolean algebra

Laws and rules of Boolean algebraCommutative Laws: A + B = B + A

A · B = B · A

Associative Laws: A + B + C = (A + B)+ CA · B · C = (A · B) · C

Distributive Laws: A · (B + C)= A · B + A · CA + (B · C)= (A + B) · (A + C)

Sum rules: A + A = 1A + 1 = 1A + 0 = AA + A = A

Product rules: A · A = 0A · 0 = 0A · 1 = AA · A = A

Absorption rules: A + AB = AA · (A + B)= A

A + A · B = A + B

De Morgan’s Laws: A + B = A · BA · B = A + B

Matrices:

If A =(

a bc d

)and B =

(e fg h

)then

A + B =(

a + e b + fc + g d + h

)

A − B =(

a − e b − fc − g d − h

)

A × B =(

ae + bg a f + bhce + dg cf + dh

)

A−1 = 1

ad − bc

(d −b

−c a

)

If A =⎛⎝ a1 b1 c1

a2 b2 c2a3 b3 c3

⎞⎠ then A−1 = BT

|A| where

BT = transpose of cofactors of matrix A

Determinants:

∣∣∣∣ a bc d

∣∣∣∣ = ad − bc

∣∣∣∣∣∣a1 b1 c1a2 b2 c2a3 b3 c3

∣∣∣∣∣∣ = a1

∣∣∣∣ b2 c2b3 c3

∣∣∣∣ − b1

∣∣∣∣ a2 c2a3 c3

∣∣∣∣

+ c1

∣∣∣∣ a2 b2a3 b3

∣∣∣∣

Differential Equations

First order differential equations

Separation of variables

Ifdy

dx= f (x) then y =

∫f (x)dx

Ifdy

dx= f (y) then

∫dx =

∫dy

f (y)

Ifdy

dx= f (x) · f (y) then

∫dy

f (y)=

∫f (x)dx

These formulae are available for downloading at the website:www.routledge.com/cw/bird

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