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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)
648 Engineering Mathematics
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
Essential formulae 649
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.
650 Engineering Mathematics
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
Essential formulae 651
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
652 Engineering Mathematics
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
Essential formulae 653
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
654 Engineering Mathematics
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
Essential formulae 655
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
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