Fomula Maths

7232019 Fomula Maths

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NATIONAL CENTRE FOR MARITIME ENGINEERING

amp HYDRODYNAMICS

TABLES OF MATHEMATICAL

FORMULAE

January 2013



Copyright c⃝ Dr Irene Penesis University of Tasmania AMC National Centre For

Maritime Engineering amp Hydrodynamics January 2013

Unannotated copies of the AMC ldquoTables of Mathematical

Formulaerdquo may be taken into the examination room

This document has been produced using LATEX 2ε

1



1 Algebra

Factoring Polynomials

x2 minus y2 = (x + y)(x minus y)

x3 + y3 = (x + y)(x2 minus xy + y2)

x3 minus y3 = (x minus y)(x2 + xy + y2)

Binomial Theorem

(x + y)2 = x2 + 2xy + y2

(x minus y)2 = x2 minus 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x minus y)3 = x3 minus 3x2y + 3xy2 minus y3

(x + y)n = xn + nxnminus1y + n(n minus 1)

2 xnminus2y2 + middot middot middot + nxynminus1 + yn

Quadratic Formula

If ax2 + bx + c = 0 then x = minusb plusmn radic

b2 minus 4ac

2a

Exponents and Logarithms

xmxn = xm+n xm

xn = xmminusn

(xy)n = xnyn (xm)n = xmn

xminusn = 1

xn x1n = n

radic x

xmn = nradic

xm

ln(xy) = ln x + ln y ln(xy) = ln x minus ln y

ln(xn) = n ln x lnn m = ln m

ln n

2



2 Geometry

Geometric Formulae

Area and Circumference

Area of a triangle is 12

bh or 12

bc sin A

Area of a circle is πr2

Circumference of a circle is 2πr

Area of a sector of a circle is 1

2r2θ (arc length is rθ)

Area of a sphere is 4πr2

Curved suface area of a cylinder is 2πrh

Volume

Volume of a sphere is 4

3πr3

Volume of a cylinder is πr2h

Volume of a cone is 1

3πr2h

Distance

Distance between P 1(x1 y1) and P 2(x2 y2) d =

991770 (x2 minus x1)2 + (y2 minus y1)2

Lines

Slope of line through P 1(x1 y1) and P 2(x2 y2) m = y2 minus y1x2 minus x1

Equation of line through P 1(x1 y1) with slope m y minus y1 = m(x minus x1)

Circles

Equation of a circle with centre (h k) and radius r (x minus h)2 + (y minus k)2 = r2

Equations of Other Curves

Ellipse x2

a2 +

y2

b2 = 1

Hyperbola x2

a2 minus y2

b2 = 1

Sphere (x minus a)2 + (y minus b)2 + (z minus c)2 = r2 with centre (abc) and radius r

Paraboloid z = x2 + y2

Cone z2 = x2 + y2

Plane ax + by + cz = d

3



3 Trigonometry

bull π radians = 180

bull 1 = π

180 rad 1 rad =

180

π

Important Angles

θ radians sin θ cos θ tan θ

0 0 0 1 0

30 π

6

1

2

radic 3

2

1radic 3

45 π4 1radic 2

1radic 2

1

60 π

3

radic 3

2

1

2

radic 3

90 π

2 1 0 minus

Fundamental Identities

tan θ = sin θcos θ

cosec θ = 1

sin θ sec θ =

1

cos θ

cot θ = 1

tan θ =

cos θ

sin θ sin2 θ + cos2 θ = 1

1 + tan2 θ = sec2 θ 1 + cot2 θ = cosec2 θ

sin(minus

θ) =minus

sin(θ) cos(minus

θ) = cos(θ)

sin(x + y) = sin x cos y + cos x sin y sin(x minus y) = sin x cos y minus cos x sin y

cos(x + y) = cos x cos y minus sin x sin y cos(x minus y) = cos x cos y + sin x sin y

tan(x + y) = tan x + tan y

1 minus tan x tan y tan(x minus y) =

tan x minus tan y

1 + tan x tan y

4



sin x sin y = 1

2 [cos(x minus y) minus cos(x + y)] sin x cos y =

1

2 [sin(x + y) + sin(x minus y)]

cos x cos y = 1

2 [cos(x + y) + cos(x minus y)] cos x sin y =

1

2 [sin(x + y) minus sin(x minus y)]

sin x + sin y = 2 sin983080x + y

2 983081 cos983080x minus y

2 983081 sin xminus

sin y = 2cos983080x + y

2 983081 sin983080x minus y

2 983081cos x + cos y = 2cos

983080x + y

2

983081cos

983080x minus y

2

983081 cos x minus cos y = minus2sin

983080x + y

2

983081sin

983080x minus y

2

983081

sin2x = 2 sin x cos x

cos2x = cos2 x minus sin2 x = 2 cos2 x minus 1 = 1 minus 2sin2 x

sin2 x = 1

2(1 minus cos2x)

cos2 x = 12

(1 + cos 2x)

tan2x = 2tan x

1 minus tan2 x

sin A

a =

sin B

b =

sin C

c

a2 = b2 + c2 minus 2bc cos A

b

2

= a

2

+ c

2

minus 2ac cos B

c2 = a2 + b2 minus 2ab cos C

5



4 Hyperbolic Functions


sinh x = 12 (ex minus eminusx) cosh x = 12 (ex + eminusx)

tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x

sinh(minusx) = minus sinh(x) cosh(minusx) = cosh(x)

cosh2 x

minussinh2 x = 1 tanh2 x + sech2x = 1

coth2 x minus cosech2x = 1

sinh(x + y) = sinh x cosh y + cosh x sinh y sinh(x minus y) = sinh x cosh y minus cosh x sinh y

cosh(x + y) = cosh x cosh y + sinh x sinh y cosh(x minus y) = cosh x cosh y minus sinh x sinh y

tanh(x + y) = tanh x + tanh y

1 + tanh x tanh y tanh(x minus y) =

tanh x minus tanh y

1 minus tanh x tanh y

sinh2x = 2sinh x cosh x

cosh 2x = cosh2 x + sinh2 x = 2 cosh2 x minus 1 = 1 + 2 sinh2 x

sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x

arcsinh x = ln852008

x +radic

x2 + 1852009

arccosh x = ln852008

x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers

bull z = a + ib where a and b are real

bull reale

z

= a and

imagem

z

= b

bull i =radic minus1 i2 = minus1 i3 = minusi

bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2

bull z = rcis θ = r(cos θ + i sin θ) where r = |z| =radic

a2 + b2 and arg z = θ

bull |zn| = |z|n and arg zn = n arg z plusmn 2kπ

bull z1z2 = r1r2 [cis(θ1 + θ2)]

bull z1z2

= r1r2

[cis(θ1 minus θ2)]

Eulerrsquos Formula eiθ = cos θ + i sin θ

De Moivrersquos Theorem

If z = rcis θ and n is a positive integer

zn = rncis nθ

Roots of a Complex Number

Let z = rcis θ and n be a positive integer Then z has n distinct nth roots

zk = r1ncis

983080θ + 2kπ

n

983081

where k = 0 1 2 n minus 1

7



6 Vectors

Given the vectors a = a1i + a2 j + a3k and b = b1i + b2 j + b3k

bull Length

|a

|= 991770 a21 + a22 + a23

bull Unit Vector a = a

|a|bull Dot (Scalar) product a middot b = a1b1 + a2b2 + a3b3

bull Cross (vector) product a times b =

i j k

a1 a2 a3

b1 b2 b3

= (a2b3 minus a3b2)i minus (a1b3 minus a3b1) j + (a1b2 minus a2b1)k

bull Angle θ between two vectors a and b a middot b = |a||b| cos θ

bull Scalar projection of a in the direction of b is a middot bbull Vector projection of a in the direction of b is (a middot b) bbull Scalar triple product a middot b times c = a times b middot c

Other important properties

minus a middot b times c = b middot c times a

minus a middot b times c = minusb times a middot c

minus a times (b times c) = (a middot c)b minus (a middot b)c

bull Vector equation of a line through the point (x0 y0 z0) parallel to the vector (abc) is

(xy z) = (x0 y0 z0) + t(abc)

bull Equation of the plane with normal (abc) is ax + by + cz = d

bull Velocity and acceleration of a particle r(t) is given by

v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8



bull Length of a space curve between t = tA and t = tB is

l =

int tBtA

|r(t)| dt

7 Series

Geometric Series

a + ar + ar2 + ar3 + middot middot middot + arnminus1 = a(1 minus rn)

1 minus r r = 1

S infin = a

1 minus r if |r| lt 1

Important Maclaurin Series

bull 1

1 minus t = 1 + t + t2 + t3 + t4 + middot middot middot valid for |t| lt 1

bull ln(1 + t) = t minus t2

2 +

t3

3 minus t4

4 +

t5

5 minus middot middot middot valid for |t| lt 1

bull arctan t = t minus t3

3 +

t5

5 minus t7

7 +

t9


bull et = 1 + t + t2

2 +

t3

3 +

t4

4 + middot middot middot valid for all t

bull sin t = t minus t3

3 +

t5

5 minus t7

7 +

t9

9 minus middot middot middot valid for all t

bull cos t = 1 minus t2

2 +

t4

4 minus t6

6 +

t8


bull sinh t = t + t3

3 +

t5

5 +

t7

7 +

t9


bull cosh t = 1 + t2

2 +

t4

4 +

t6

6 +

t8


bull (1 + t)α = 1 + αt + α(α minus 1)

2 t2 +

α(α minus 1)(α minus 2)

3 t3 + middot middot middot

+ α(α minus 1)(α minus 2)(α minus n + 1)n tn + middot middot middot

valid for |t| lt 1

Taylor Series

f (x) = f (x0)+(xminusx0)f prime(x0)+ 1

2(xminusx0)2f primeprime(x0)+

1

3(xminusx0)3f primeprimeprime(x0)+middot middot middot+ 1

n(xminusx0)nf (n)(x0)+middot middot middot

9



8 Useful Formulae for Differentiation and Integration

bull d

dx (u middot v) = uprimev + uvprime

bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d

dx [f (g(x))] = f prime (g(x)) gprime(x) or

dy

dx =

dy

du

du

dx

bull If y = f (x) then y asymp dy

dxx

bullint

f (g(x)) gprime(x) dx =

int f (u) du by setting u = g(x)

bullint

u dv = uv minusint

v du

bull Area between the two curves y = f (x) and y = g(x) on the interval [a b] is

A =

int ba

|f (x) minus g(x)| dx

bull Volume of solid obtained by rotating about the x-axis is

V = π

int ba

R2 minus r2 dx

bull Volume of solid obtained by rotating about the y -axis is

V = π

int dc

R2 minus r2 dy

81 Chain Rules

bull If w = w(x y) and both x = x(t) and y = y(t) then

dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt

bull If w = w(x y) and both x = x(u v) and y = y(u v) then

partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10



82 Numerical Techniques

Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +

middot middot middot+ ynminus1) + yn)

Simpsonrsquos Ruleint ba

f (x) dx = h

3 (y0 + 4(y1 + y3 + middot middot middot + y2nminus1) + 2(y2 + y4 + middot middot middot + y2nminus2) + y2n)

Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method

xn+1 = xn minus f (xn)(xn minus xnminus1)

f (xn) minus f (xnminus1)

Jacobi Iterative Method

Given an initial estimate x(0)

x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1

Gauss-Seidel Iterative Method


x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method

Given y prime = f (x y) y(x0) = y0

yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method


yn+1 = yn + hyprimen + h2

2 yprimeprimen n = 0 1 2

821 Runge-Kutta Methods


Improved Eulerrsquos Method (2nd order R-K)

k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)

Classical Runge-Kutta Method (4th order R-K)

k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)

Improved Eulerrsquos Method for a First Order System

By setting u = y and v = y prime we can express a second order BVP as a system of first order DErsquos

uprime = f (xuv) u(x0) = u0

vprime = g(xuv) v(x0) = v0

These can be solved approximately by using the following second order Runge-Kutta scheme

k1 = f (xn un vn) l1 = g(xn un vn)

k2 = f (xn + h un + hk1 vn + hl1) l2 = g(xn + h un + hk1 vn + hl1)

un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12



822 Polynomial Interpolation

Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)

Newtonrsquos Divided Difference Formula

pn(x) = f (x0) + (x minus x0)f [x0 x1] + (x minus x0)(x minus x1)f [x0 x1 x2]

+ middot middot middot + (x minus x0)(x minus x1)(x minus x2) (x minus xnminus1)f [x0 x1 x2 xn]

Error asymp (x minus x0)(x minus x1)(x minus x2) (x minus xn)f [x0 x1 xn+1]

823 Least Squares Approximation

If φ(x a0 a1 an) = a0 + a1x + a2x2 + middot middot middot + anxn then the normal equations are given by

(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13



83 Table of Integrals

Function Integral

xn 1n + 1 xn+1 n = minus1

ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x

tan x minus ln | cos x|

cosec2 x minus cot x

sec2 x tan x

cot x ln | sin x|

sec x tan x sec x

cosec x cot x minuscosec x

sec x 1

2 ln

1 + sin x

1 minus sin x = ln | sec x + tan x|

cosec x 12

ln 1 minus cos x1 + cos x

= ln |cosec x minus cot x|

arcsin x x arcsin x +radic

1 minus x2

arccos x x arccos x minusradic

1 minus x2

arctan x x arctan x minus 1

2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x

cosech2 x minus coth x

sech x tanh x minussech x

cosech x coth x minuscosech x

sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a

x minus a if |x| gt a

1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x

sinm+1 x cosnminus1 x

m + n +

n minus 1

m + n

int sinm x cosnminus2 x dx

minussinmminus1 x cosn+1 x

m + n +

m minus 1

m + n

int sinmminus2 x cosn x dx

sin ax sin bx sin(a minus b)x

2(a minus b) minus sin(a + b)x

2(a + b)

cos ax cos bx sin(a minus b)x

2(a minus b) +

sin(a + b)x

2(a + b)

sin ax cos bx minuscos(a minus b)x2(a minus b)

minus cos(a + b)x2(a + b)

eax sin bx eax

a2 + b2 (a sin bx minus b cos bx)

eax cos bx eax

a2 + b2 (a cos bx + b sin bx)

xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1

tannminus1 x minus int tannminus2 x dx

secn x 1

n minus 1 secnminus2 x tan x +

n minus 2

n minus 1

int secnminus2 x dx

16



9 Laplace Transforms

Function Laplace Transform

f (t) F (s) = int infin

0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2

sin at minus at cos at 2a3

(s2 + a2)2

sin at + at cos at 2as2

(s2 + a2)2

t sin at 2as(s2 + a2)2

sin at sinh at 2a2s

s4 + 4a4

cos at sinh at as2 minus 2a3

s4 + 4a4

sin at cosh at as2 + 2a3

s4 + 4a4

cos at cosh at s3

s4 + 4a4

H (t minus a) eminusas

s

δ (t minus a) eminusas

a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17



91 Laplace Transforms General Formulae

Definition of a Laplace Transform

F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt

Inverse Laplace Transform

f (t) = Lminus1 F (s)

Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem

L[eatf (t)] = F (s minus a)

Second Shift Theorem

L[f (t minus τ )H (t minus τ )] = eminusτsF (s)

Differentiation of a Transform

L[tnf (t)] = (minus1)ndnF (s)

dsn

Differentiation of a Function

L983131

dnf (t)

dtn

983133 = snF (s) minus snminus1f (0) minus snminus2f prime(0) minus middot middot middot minus sf (nminus2)(0) minus f (nminus1)(0)

bull L[f prime(t)] = sF (s) minus f (0)

bull L[f primeprime(t)] = s2F (s) minus sf (0) minus f prime(0)

18



Integration of a Function

L983131int t

0f (u) du

983133 =

F (s)

s

Integration of a Transform

L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem

L[f (t) lowast g(t)] = L983131int t

0f (τ )g(t minus τ ) dτ

983133 = F (s)G(s)

10 Fourier Series

A function f satisfying f (x + 2l) = f (x) of period 2l can be represented as a Fourier series by

f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l

983081with Fourier coefficients defined by

an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx

An even periodic function is represented by a Fourier Cosine series

f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2

An odd periodic function is represented by a Fourier Sine series

f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient

The gradient of a scalar function φ(xy z) is given by

nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence

The divergence of a vector function F(xy z) = F 1(xy z)i + F 2(xy z) j + F 3(xy z)k is given

by

nabla middot F = partF 1

partx +

part F 2party

+ part F 3

partz

Curl

The curl of a vector function F(xy z) = F 1(xy z)i + F 2(xy z) j + F 3(xy z)k is given by

nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz

)i minus (partF 3partx

minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k

Directional Derivative

The directional derivative of f in the direction of the vector u is given by

df

du =

nablaf middot u

Surface Normals

The normal to the surface F (xy z) = 0 is parallel to the vector

nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz

Surface Integral int int S

F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy

Greenrsquos Theorem in the plane

int int R 983080partQ

partx minus part P

party 983081 dxdy =

C P dx + Q dy

where C is a simple closed curve traversed anticlockwise which bounds the region R in the xy

plane

Gaussrsquo Divergence Theoremint int S

F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV

where V is a region bounded by the closed surface S

(Note n is the outward unit normal to S )

Stokesrsquo Theorem int int S

nabla times F middot n dS =

C

F middot dr

where C is a simple closed curve which bounds the open surface S

(Note If

n = k then C is traversed in an anticlockwise direction)

Element of area in plane polar coordinates dA = dxdy = r dr dθ

Element of surface area dS for a sphere (radius a) dS = a2 sin φdθdφ

Element of volume dV in cylindrical and spherical polar coordinates

bull Cylindrical dV = r dr dθ dz bull Spherical dV = r2 sin φdrdθdφ

21



12 Statistics

bull P (A cap B) = P (A)P (B | A)

bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)

bull Binomial probability function p(x) = P (X = x) =

1048616n

x

1048617 px(1 minus p)nminusx

micro = np σ2 = np(1 minus p)

bull Poisson probability function p(x) = P (X = x) = λx

xeminusλ

micro = λ σ2 = λ

bull Continuous probability function F (x) = P (X le x) =

int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)

bull Mean (continuous density function) micro =

int infinminusinfin

xf (x) dx

bull Variance (continuous density function) σ2 =

int infinminusinfin

(x minus micro)2f (x) dx

bull Density of normal distribution f (x) = 1

σradic

2πeminus

1

2(xminusmicroσ )

2

bull Standard normal variable Z =

X

minusmicro

σ

bull Uniform density function f (x) =

1

b minus a for a lt x le b

0 otherwise

bull Sample mean x = 1

n

nsumi=1

xi

bull Sample variance s2 = 1

n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics

bull Central Limit Theorem Z = X minus micro

σradic

n

bull Test for single mean t =

x minus micro

sradic n follows a t

nminus1 distribution

bull Test for two means t = x1 minus x2 minus (micro1 minus micro2)

s p991770

1n1

+ 1n2

follows a tn1+n2minus2 distribution where

s2 p = (n1 minus 1)s21 + (n2 minus 1)s22

n1 + n2 minus 2

bull Test for a single variance χ2 = (n minus 1)s2

σ2 follows a χ2


bull Test for proportions z = ˆ p minus p

radic p(1

minus p)n

where ˆ p = X

n

100(1 minus α) Confidence Intervals

bull for true mean micro amp known σ x plusmn zα2σradic

n

bull for true mean micro amp unknown σ x plusmn tnminus1α2sradic n

bull for true variance σ2 amp known micro (n minus 1)s2χ2nminus1α2

(n minus 1)s2χ2nminus11minusα2

bull for true variance σ2 amp unknown micro (n minus 1)s2

χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2

bull for true mean difference micro1 minus micro2 amp known σ (x1 minus x2) plusmn zα2 σ

860698 1

n1+

1

n2

bull for true mean difference micro1 minus micro2 amp unknown σ (x1 minus x2) plusmn tn1+n2minus2α2 s p

860698 1

n1+

1

n2

100(1 minus α) Prediction Intervals

bull if σ is known x plusmn zα2 σ

1057306 1 +

1

n

bull if σ is unknown x plusmn tnminus1α2 s

1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990

Table 1 The Cumulative Distribution Function P (Z le z) for the Standard Normal Distribution

- positive values for Z [1]

24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090

Table 2 The Cumulative Distribution Function for the t Distribution [1]

25



References

[1] Vining GG and Kowalski SM Statistical Methods for Engineers 3rd ed BrooksCole

2011

[2] Stewart J Calculus 7th ed (International Metric Version) BrooksCole 2012

[3] Burden RL and Faires JD Numerical Analysis 8th ed BrooksCole Pacific Grove

2005

26





Copyright c⃝ Dr Irene Penesis University of Tasmania AMC National Centre For

Maritime Engineering amp Hydrodynamics January 2013

Unannotated copies of the AMC ldquoTables of Mathematical

Formulaerdquo may be taken into the examination room

This document has been produced using LATEX 2ε

1



1 Algebra



x3 + y3 = (x + y)(x2 minus xy + y2)


Binomial Theorem

(x + y)2 = x2 + 2xy + y2


(x + y)3 = x3 + 3x2y + 3xy2 + y3




Quadratic Formula


b2 minus 4ac

2a


xmxn = xm+n xm

xn = xmminusn


xminusn = 1

xn x1n = n

radic x

xmn = nradic

xm



ln n

2



2 Geometry

Geometric Formulae



bh or 12

bc sin A







Volume


3πr3



3πr2h

Distance


991770 (x2 minus x1)2 + (y2 minus y1)2

Lines



Circles



Ellipse x2

a2 +

y2

b2 = 1

Hyperbola x2

a2 minus y2

b2 = 1



Cone z2 = x2 + y2


3



3 Trigonometry


bull 1 = π

180 rad 1 rad =

180

π

Important Angles


0 0 0 1 0

30 π

6

1

2

radic 3

2

1radic 3

45 π4 1radic 2

1radic 2

1

60 π

3

radic 3

2

1

2

radic 3

90 π

2 1 0 minus



cosec θ = 1

sin θ sec θ =

1

cos θ

cot θ = 1

tan θ =

cos θ



sin(minus

θ) =minus

sin(θ) cos(minus

θ) = cos(θ)





tan x minus tan y

1 + tan x tan y

4



sin x sin y = 1


1


cos x cos y = 1


1



2 983081 cos983080x minus y

2 983081 sin xminus

sin y = 2cos983080x + y

2 983081 sin983080x minus y

2 983081cos x + cos y = 2cos

983080x + y

2

983081cos

983080x minus y

2


983080x + y

2

983081sin

983080x minus y

2

983081



sin2 x = 1

2(1 minus cos2x)

cos2 x = 12

(1 + cos 2x)

tan2x = 2tan x

1 minus tan2 x

sin A

a =

sin B

b =

sin C

c


b

2

= a

2

+ c

2

minus 2ac cos B


5






tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x


cosh2 x







tanh x minus tanh y




sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x


x +radic

x2 + 1852009


x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





1 Algebra



x3 + y3 = (x + y)(x2 minus xy + y2)


Binomial Theorem

(x + y)2 = x2 + 2xy + y2


(x + y)3 = x3 + 3x2y + 3xy2 + y3




Quadratic Formula


b2 minus 4ac

2a


xmxn = xm+n xm

xn = xmminusn


xminusn = 1

xn x1n = n

radic x

xmn = nradic

xm



ln n

2



2 Geometry

Geometric Formulae



bh or 12

bc sin A







Volume


3πr3



3πr2h

Distance


991770 (x2 minus x1)2 + (y2 minus y1)2

Lines



Circles



Ellipse x2

a2 +

y2

b2 = 1

Hyperbola x2

a2 minus y2

b2 = 1



Cone z2 = x2 + y2


3



3 Trigonometry


bull 1 = π

180 rad 1 rad =

180

π

Important Angles


0 0 0 1 0

30 π

6

1

2

radic 3

2

1radic 3

45 π4 1radic 2

1radic 2

1

60 π

3

radic 3

2

1

2

radic 3

90 π

2 1 0 minus



cosec θ = 1

sin θ sec θ =

1

cos θ

cot θ = 1

tan θ =

cos θ



sin(minus

θ) =minus

sin(θ) cos(minus

θ) = cos(θ)





tan x minus tan y

1 + tan x tan y

4



sin x sin y = 1


1


cos x cos y = 1


1



2 983081 cos983080x minus y

2 983081 sin xminus

sin y = 2cos983080x + y

2 983081 sin983080x minus y

2 983081cos x + cos y = 2cos

983080x + y

2

983081cos

983080x minus y

2


983080x + y

2

983081sin

983080x minus y

2

983081



sin2 x = 1

2(1 minus cos2x)

cos2 x = 12

(1 + cos 2x)

tan2x = 2tan x

1 minus tan2 x

sin A

a =

sin B

b =

sin C

c


b

2

= a

2

+ c

2

minus 2ac cos B


5






tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x


cosh2 x







tanh x minus tanh y




sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x


x +radic

x2 + 1852009


x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





2 Geometry

Geometric Formulae



bh or 12

bc sin A







Volume


3πr3



3πr2h

Distance


991770 (x2 minus x1)2 + (y2 minus y1)2

Lines



Circles



Ellipse x2

a2 +

y2

b2 = 1

Hyperbola x2

a2 minus y2

b2 = 1



Cone z2 = x2 + y2


3



3 Trigonometry


bull 1 = π

180 rad 1 rad =

180

π

Important Angles


0 0 0 1 0

30 π

6

1

2

radic 3

2

1radic 3

45 π4 1radic 2

1radic 2

1

60 π

3

radic 3

2

1

2

radic 3

90 π

2 1 0 minus



cosec θ = 1

sin θ sec θ =

1

cos θ

cot θ = 1

tan θ =

cos θ



sin(minus

θ) =minus

sin(θ) cos(minus

θ) = cos(θ)





tan x minus tan y

1 + tan x tan y

4



sin x sin y = 1


1


cos x cos y = 1


1



2 983081 cos983080x minus y

2 983081 sin xminus

sin y = 2cos983080x + y

2 983081 sin983080x minus y

2 983081cos x + cos y = 2cos

983080x + y

2

983081cos

983080x minus y

2


983080x + y

2

983081sin

983080x minus y

2

983081



sin2 x = 1

2(1 minus cos2x)

cos2 x = 12

(1 + cos 2x)

tan2x = 2tan x

1 minus tan2 x

sin A

a =

sin B

b =

sin C

c


b

2

= a

2

+ c

2

minus 2ac cos B


5






tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x


cosh2 x







tanh x minus tanh y




sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x


x +radic

x2 + 1852009


x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





3 Trigonometry


bull 1 = π

180 rad 1 rad =

180

π

Important Angles


0 0 0 1 0

30 π

6

1

2

radic 3

2

1radic 3

45 π4 1radic 2

1radic 2

1

60 π

3

radic 3

2

1

2

radic 3

90 π

2 1 0 minus



cosec θ = 1

sin θ sec θ =

1

cos θ

cot θ = 1

tan θ =

cos θ



sin(minus

θ) =minus

sin(θ) cos(minus

θ) = cos(θ)





tan x minus tan y

1 + tan x tan y

4



sin x sin y = 1


1


cos x cos y = 1


1



2 983081 cos983080x minus y

2 983081 sin xminus

sin y = 2cos983080x + y

2 983081 sin983080x minus y

2 983081cos x + cos y = 2cos

983080x + y

2

983081cos

983080x minus y

2


983080x + y

2

983081sin

983080x minus y

2

983081



sin2 x = 1

2(1 minus cos2x)

cos2 x = 12

(1 + cos 2x)

tan2x = 2tan x

1 minus tan2 x

sin A

a =

sin B

b =

sin C

c


b

2

= a

2

+ c

2

minus 2ac cos B


5






tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x


cosh2 x







tanh x minus tanh y




sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x


x +radic

x2 + 1852009


x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





sin x sin y = 1


1


cos x cos y = 1


1



2 983081 cos983080x minus y

2 983081 sin xminus

sin y = 2cos983080x + y

2 983081 sin983080x minus y

2 983081cos x + cos y = 2cos

983080x + y

2

983081cos

983080x minus y

2


983080x + y

2

983081sin

983080x minus y

2

983081



sin2 x = 1

2(1 minus cos2x)

cos2 x = 12

(1 + cos 2x)

tan2x = 2tan x

1 minus tan2 x

sin A

a =

sin B

b =

sin C

c


b

2

= a

2

+ c

2

minus 2ac cos B


5






tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x


cosh2 x







tanh x minus tanh y




sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x


x +radic

x2 + 1852009


x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26








tanh x = sinh x

cosh x cosech x =

1

sinh x

sechx = 1

cosh x coth x =

1

tanh x =

cosh x

sinh x


cosh2 x







tanh x minus tanh y




sinh2 x = 1

2(cosh 2x minus 1)

cosh2 x = 1

2(cosh 2x + 1)

tanh2x = 2tanh x

1 + tanh2 x


x +radic

x2 + 1852009


x +radic

x2 minus 1852009

arctanh x = 1

2 ln

1 + x

1 minus x

6



5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





5 Complex Numbers


bull reale

z

= a and

imagem

z

= b


bull |z| =radic

a2 + b2

bull z = a minus ib

bull z + w = z + w

bull zw = z w

bull zn = zn

bull zz = |z|2





bull z1z2

= r1r2





zn = rncis nθ



zk = r1ncis

983080θ + 2kπ

n

983081


7



6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





6 Vectors


bull Length

|a

|= 991770 a21 + a22 + a23




i j k

a1 a2 a3

b1 b2 b3









(xy z) = (x0 y0 z0) + t(abc)



v(t) = r(t) = x(t)i + y(t) j + z(t)k

and

a(t) = r(t) = x(t)i + y(t) j + z(t)k

8




l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26






l =

int tBtA

|r(t)| dt

7 Series

Geometric Series


1 minus r r = 1

S infin = a



bull 1



2 +

t3

3 minus t4

4 +

t5



3 +

t5

5 minus t7

7 +

t9



2 +

t3

3 +

t4



3 +

t5

5 minus t7

7 +

t9



2 +

t4

4 minus t6

6 +

t8



3 +

t5

5 +

t7

7 +

t9



2 +

t4

4 +

t6

6 +

t8



2 t2 +




valid for |t| lt 1

Taylor Series



1



9




bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26






bull d


bull d

dx 983080u

v983081 =

uprimev

minusuvprime

v2

bull d


dy

dx =

dy

du

du

dx


dxx

bullint



bullint

u dv = uv minusint

v du


A =

int ba



V = π

int ba

R2 minus r2 dx


V = π

int dc

R2 minus r2 dy

81 Chain Rules


dw

dt =

partw

partx

dx

dt +

part w

party

dy

dt


partw

partu =

partw

partx

partx

partu +

part w

party

party

partu

and

partw

partv =

partw

partx

partx

partv +

part w

party

party

partv

10




Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26






Trapezoidal Rule

int b

a

f (x) dx = h

2

(y0 + 2(y1 + y2 + y3 +



f (x) dx = h


Newtonrsquos Method

xn+1 = xn

minus f (xn)

f prime(xn)

n = 0 1 2

Secant Method





x(k+1)i =

1

aii

bi minusnsum

j=1j=i

aijx(k) j

for i = 1 2 n k = 0 1



x(k+1)i =

1

aii biminus

iminus1

sum j=1

aijx(k+1) j

minus

n

sum j=i+1

aijx(k) j

for i = 1 2 n k = 0 1

Eulerrsquos Method


yn+1 = yn + hf (xn yn) n = 0 1 2

11



Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





Taylorrsquos Method







k1 = f (xn yn)

k2 = f (xn + h yn + hk1)

yn+1 = yn + h

2 (k1 + k2)


k1 = f (xn yn)

k2 = f (xn + h

2 yn +

h

2k1)

k3 = f (xn + h

2 yn + h

2 k2)

k4 = f (xn + h yn + hk3)

yn+1 = yn + h

6 (k1 + 2k2 + 2k3 + k4)








un+1 = un + h

2

(k1 + k2) vn+1 = vn + h

2

(l1 + l2)

12




Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26






Lagrange Form

pn(x) =n

sumi=0

f iLi(x)

where

Li(x) =nprod

j=0j=i

(x minus x j)

(xi minus x j)







(m + 1)msumi=0

xi

msumi=0

xi2 msumi=0

xin

msumi=0

xi

msumi=0

xi2

msumi=0

xi3

msumi=0

xin+1

msumi=0

xi2

msumi=0

xi3

msumi=0

xi4

msumi=0

xin+2

msumi=0

xin

msumi=0

xin+1

msumi=0

xin+2

msumi=0

xi2n

a0

a1

a2

an

=

msumi=0

f i

msumi=0

xif i

msumi=0

xi2f i

msumi=0

xinf i

13




Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26






Function Integral


ex ex

eax+b 1

aeax+b

1

x ln |x|

ax 1

ln aax

ln x x ln x minus x

sin x minus cos x

cos x sin x



sec2 x tan x

cot x ln | sin x|

sec x tan x sec x


sec x 1

2 ln

1 + sin x


cosec x 12




1 minus x2


1 minus x2


2 ln(1 + x2)

sinh x cosh x

cosh x sinh x

14



Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





Continued

Function Integral

tanh x lncosh x

coth x ln | sinh x|

sech2x tanh x




sech x 2 arctan ex

cosech x ln

tanh x

2

= ln

ex minus 1

ex + 1

1

a2 minus x2

1

a

arctanh x

a

= 1

2a

ln a + x

a minus x

if

|x

|lt a

1

a arccoth

x

a =

1

2a ln

x + a


1

x2 + a21

a arctan

x

a

1radic a2 minus x2

arcsin x

a

1radic x2 minus a2

arccosh x

a

1radic x2 + a2

arcsinh xaradic

x2 + a2 1

2xradic

x2 + a2 + 1

2a2 arcsinh

x

aradic x2 minus a2

1

2xradic

x2 minus a2 minus 1

2a2 arccosh

x

aradic a2 minus x2

1

2xradic

a2 minus x2 + 1

2a2 arcsin

x

a

15



Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





Continued

Function Integral

sinm x cosn x


m + n +

n minus 1

m + n



m + n +

m minus 1

m + n




2(a + b)


2(a minus b) +

sin(a + b)x

2(a + b)



eax sin bx eax


eax cos bx eax


xnex xnex minus n

int xnminus1ex dx

tann x 1n minus 1


secn x 1


n minus 2

n minus 1

int secnminus2 x dx

16






0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26








0 eminusst

f (t) dt

1 1

s

tn n

sn+1

eat 1

s minus a

sin at a

s2 + a2

cos at s

s2 + a2

sinh at a

s2 minus a2

cosh at s

s2 minus a2


(s2 + a2)2


(s2 + a2)2


sin at sinh at 2a2s

s4 + 4a4


s4 + 4a4


s4 + 4a4

cos at cosh at s3

s4 + 4a4


s


a

2tradic

πteminusa

24t eminusaradic s

1radic πt

eminusa24t eminusa

radic s

radic s

J 0(2radic

at) 1

s

eminusas

17





F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26







F (s) =

L[f (t)] = int

infin

0

eminusstf (t) dt



Linearity

L[af (t) + bg(t)] = aF (s) + bG(s)

Theorem

L[f (at)] = 1

aF

983080s

a

983081

First Shift Theorem






dsn


L983131

dnf (t)

dtn




18




L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26






L983131int t

0f (u) du

983133 =

F (s)

s


L983131

f (t)

t

983133 =

int infins

F (u) du

Convolution Theorem



983133 = F (s)G(s)

10 Fourier Series


f (x) = 1

2a0 +

infinsumn=1

983080an cos

nπx

l + bn sin

nπx

l


an =

1

l int l

minusl f (x)cos

nπx

l dx

bn = 1

l

int lminusl

f (x)sin nπx

l dx


f (x) = 1

2a0 +

infinsumn=1

an cos nπx

l

where

an = 2l

int l0

f (x)cos nπx

l dx for n = 0 1 2


f (x) =infinsumn=1

bn sin nπx

l

where

bn = 2

l int l

0

f (x)sin nπx

l dx for n = 1 2 3

19



11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





11 Vector Calculus

Gradient


nablaφ = partφ

partxi +

part φ

party j +

part φ

partzk

Divergence


by


partx +

part F 2party

+ part F 3

partz

Curl


nabla times F =

i j k

part

partx

part

party

part

partz

F 1 F 2 F 3

= (partF 3party

minus part F 2partz


minus part F 1partz

) j + (partF 2partx

minus part F 1party

)k



df

du =

nablaf middot u

Surface Normals


nablaF = partF

partxi +

part F

party j +

part F

partz k

20



Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





Line Integral int C

F middot dr =

int C

f dx + g dy + h dz


F middot n dS =

int int D

F 1n1 + F 2n2 + F 3n3 dudv

int int S

φ(xy z) dS =

int int D

φ(xy f (x y)) dxdy

| n middot k| =

int int D

φ(xy f (x y))

860698 983080partf

partx

9830812

+

983080partf

party

9830812

+ 1 dxdy



partx minus part P

party 983081 dxdy =

C P dx + Q dy


plane


F middot dS =

int int S

F middot n dS =

int int int V

nabla middot F dV





C

F middot dr


(Note If






21



12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





12 Statistics


bull P (A

cupB) = P (A) + P (B)

minusP (A

capB)


1048616n

x




xeminusλ

micro = λ σ2 = λ


int xminusinfin

f (t)dt

bull d

dx(F (x)) = f (x)


int infinminusinfin

xf (x) dx


int infinminusinfin



σradic

2πeminus

1

2(xminusmicroσ )

2


X

minusmicro

σ


1


0 otherwise


n

nsumi=1

xi


n minus 1

nsumi=1

(xi minus x)2 =n

n

sumi=1

x2i minus 1048616

n

sumi=1

xi10486172

n(n minus 1)

22



Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





Test Statistics


σradic

n


x minus micro




s p991770

1n1

+ 1n2



n1 + n2 minus 2


σ2 follows a χ2



radic p(1

minus p)n

where ˆ p = X

n



n





χ2nminus1α2

(n minus 1)s2

χ2nminus11minusα2


860698 1

n1+

1

n2


860698 1

n1+

1

n2



1057306 1 +

1

n


1057306 1 +

1

n

23



z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





z 00 01 02 03 04 05 06 07 08 09

00 5000 5040 5080 5120 5160 5199 5239 5279 5319 5359

01 5398 5438 5478 5517 5557 5596 5636 5675 5714 5753

02 5793 5832 5871 5910 5948 5987 6026 6064 6103 6141

03 6179 6217 6255 6293 6331 6368 6406 6443 6480 6517

04 6554 6591 6628 6664 6700 6736 6772 6808 6844 6879

05 6915 6950 6985 7019 7054 7088 7123 7157 7190 7224

06 7257 7291 7324 7357 7389 7422 7454 7486 7517 7549

07 7580 7611 7642 7673 7704 7734 7764 7794 7823 7852

08 7881 7910 7939 7967 7995 8023 8051 8078 8106 8133

09 8159 8186 8212 8238 8264 8289 8315 8340 8365 838910 8413 8438 8461 8485 8508 8531 8554 8577 8599 8621

11 8643 8665 8686 8708 8729 8749 8770 8790 8810 8830

12 8849 8869 8888 8907 8925 8944 8962 8980 8997 9015

13 9032 9049 9066 9082 9099 9115 9131 9147 9162 9177

14 9192 9207 9222 9236 9251 9265 9279 9292 9306 9319

15 9332 9345 9357 9370 9382 9394 9406 9418 9429 9441

16 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545

17 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633

18 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706

19 9713 9719 9726 9732 9738 9744 9750 9756 9761 9767

20 9772 9778 9783 9788 9793 9798 9803 9808 9812 9817

21 9821 9826 9830 9834 9838 9842 9846 9850 9854 9857

22 9861 9864 9868 9871 9875 9878 9881 9884 9887 9890

23 9893 9896 9898 9901 9904 9906 9909 9911 9913 9916

24 9918 9920 9922 9925 9927 9929 9931 9932 9934 9936

25 9938 9940 9941 9943 9945 9946 9948 9949 9951 9952

26 9953 9955 9956 9957 9959 9960 9961 9962 9963 9964

27 9965 9966 9967 9968 9969 9970 9971 9972 9973 9974

28 9974 9975 9976 9977 9977 9978 9979 9979 9980 9981

29 9981 9982 9983 9983 9984 9984 9985 9985 9986 9986

30 9987 9987 9987 9988 9988 9989 9989 9989 9990 9990



24



df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





df 90 95 975 99 995 999

1 3078 6314 12706 31821 63657 318309

2 1886 2920 4303 6965 9925 22327

3 1638 2353 3183 4541 5841 10215

4 1533 2132 2777 3747 4604 7173

5 1476 2015 2571 3365 4032 5893

6 1440 1943 2447 3143 3708 5208

7 1415 1895 2365 2998 3500 4785

8 1397 1860 2306 2897 3355 4501

9 1383 1833 2262 2822 3250 4297

10 1372 1812 2228 2764 3169 4144

11 1363 1796 2201 2718 3106 4025

12 1356 1782 2179 2681 3055 3930

13 1350 1771 2160 2650 3012 3852

14 1345 1761 2145 2625 2977 3787

15 1341 1753 2132 2603 2947 3733

16 1337 1746 2120 2584 2921 3686

17 1333 1740 2110 2567 2898 3646

18 1330 1734 2101 2552 2879 3611

19 1328 1729 2093 2540 2861 3580

20 1325 1725 2086 2528 2845 3552

21 1323 1721 2080 2518 2831 352722 1321 1717 2074 2508 2819 3505

23 1319 1714 2069 2500 2807 3485

24 1318 1711 2064 2492 2797 3467

25 1316 1708 2060 2485 2788 3450

26 1315 1706 2056 2479 2779 3435

27 1314 1703 2052 2473 2771 3421

28 1313 1701 2048 2467 2763 3408

29 1311 1699 2045 2462 2756 3396

30 1310 1697 2042 2457 2750 3385

40 1303 1684 2021 2423 2705 3307

50 1299 1676 2009 2403 2678 3262

60 1296 1671 2000 2390 2660 3232

80 1292 1664 1990 2374 2639 3195

100 1290 1660 1984 2364 2626 3174

200 1286 1653 1972 2345 2601 3132

infin 1282 1645 1960 2326 2576 3090


25



References


2011



2005

26





References


2011



2005

26





Documents

Fomula Maths