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A-levels: Year 12 (Physics (Section 3- Mechanics and materials (Forces in…
A-levels: Year 12
Physics
Section 3- Mechanics and materials
Newton's laws of motion
Force and acceleration
Newton's first law of motion
An object will stay at rest or at a constant velocity unless acted upon by a force
Newton's second law of motion
The resultant force on a body is equal to its mass multiplied by its acceleration
Newton's third law of motion
For every force, there is an equal and opposite force
Weight
W = mg
Measured in Newtons
When an object is in equilibrium, the support force on it is equal and opposite to its weight
The gravitational field strength at the Earth's surface is 9.81Nkg^-1
F = ma
Where F1 > F2:
resultant force, F1-F2 = ma
F = Ma+ma = (M+m)a
Lift problems
When a = 0, T = mg (tension = weight)
When moving upwards and accelerating, a > 0
T = mg+ma > mg
When moving upwards and decelerating, then a <0
T = mg+ma < mg
When moving downwards and accelerating, a <0
T = mg+ma < mg
When moving downwards and decelerating ,a > 0
T = mg+ma > mg
Rocket thrust, T = mg+ma
Pulley problems
Mg-mg = (M+m)a
Slopes
mg sin θ - F0 = ma
Terminal speed
Drag force factors
The shape of the object
Its speed
The viscosity of th fluid, which is a measure of how easily the fluid flows past a surface
Motion of a powered vehicle
a = (F(engine) - F(resistive))/m
a = (mg-D)/m = g-D/m
On the road
Stopping distance = thinking distance +braking distance = ut+u²/2a
Thinking distace = speed X reaction time
Braking distance = u²/2a
Vehicle safety
Impact time t = 2s/(u+v)
Acceleration a = (v-u)/t
Impact force F = ma
Force and momentum
Momentum and impulse
Momentum
mass X velocity
The unit of momentum is kgms^-1
Momentum is a vector quantity
For an object of mass m moving at velocity v, its momentum p = mv
The resultant force is proportional to the change of momentum per second
If m is constant, then
F = mΔv/Δt = ma
Impulse
F = mv-mu/t
Ft = mv-mu
Change of momentum
Represented by area under a force-time graph
Impact forces
Force of impact F = change of momentum/ contact time = mv/t
Impact force = area under force-time graph
Rebound impacts
Velocity before = +u
Velocity after = -v
If there is no loss of speed on impact, the v = u, so
F = -2mu/t
Conservation of momentum
Newton's third law of motion
When two objects interact, they exert equal and opposite forces on each other
The principle of conservation of momentum
For a system of interacting objects, the total momentum remains constant, provided no external resultant force acts on the system
F1 = (mAvA-mAuA)/t
F1 = (mBvB-mBuB)/t
F2 = -F1
mBvB-mBuB = -(mAvA-mAuA)
mBvB+mAvA = mAuA+mBuB
Elastic and inelastic collisions
An elastic collision is one where there is no loss of kinetic energy
An inelastic collision occurs where the colliding objects have less kinetic energy after the collision than before the collision
Explosions
When two objects fly apart after being initially at rest, they recoil from each other in opposite directions with equal and opposite amounts of moments
The total initial momentum = 0
The total momentum immediately after the explosion
=momentum of A + momentum of B
=mAvA + mBvB
Using the principle of the conservation of momentum, mAvA + mBvB = 0
∴ mBvB = -mAvA
On the move
Speed and velocity
Displacement is distance in a given direction
Speed is defined as change of distance per unit time
Velocity is defined as change of displcaement per unit time
Scalar- magnitude but no direction
Vector- both a magnitude and a direction
Motion at constant speed
speed v = s/t
distance travelled s = vt
For an object moving at constant speed on a circle of radius r, its speed: v = 2πr/T
Motion at changing speed
average speed = s/t
v = Δs/Δt
Distance-time graphs
The speed of the object = distance travelled / time taken = gradient of the line
Acceleration
Change of velocity per unit time
Uniform acceleration
acceleration = change of velocity / time taken = (v-u)/t
v = u+at
Non-uniform acceleration
the gradient of the tangent to the curve at point P = height of gradient triangle / base of gradient triangle
Motion along a straight line at constant acceleration
The dynamics equations for constant acceleration
s = (u+v)t/2
s = ut+1/2at²
v = u+at
v² = u²+2as
The area under a velocity-time graph is the displacement
Free fall
Gravity affects everything equally
The acceleration caused by the force of gravity = 9.81ms^-2
Objects in free fall create a straight line along velocity-time graphs
Objects released at the same time reach the ground at the same time
Projectile motion
Priniples of projectiles
The vertical and horizontal components of an object's velocity are independent
The acceleration of the object is always equal to g and is always downwards because the force of gravity acts downwards
The horizontal velocity of the object is constant because the acceleration of the object does not have a horizontal component
Vertical component y = 1/2gt²
Horizontal component x = Ut
The effects of air resistance
Lower maximum height
Smaller horizontal distance
Vertical distances and velocities
y = Ut sin θ-1/2gt²
vy = U sin θ-gt
Work, energy and power
Forces in equilibrium
Vectors and scalars
A vector is any physical quantity that has a direction and a magnitude
A scalar is any physical quantity that is not directional
OB = OA + AB
Resultant forces can be found using a scale diagram- connecting the forces tip-to-tail
Addition of two perpendicular vectors
Two perpendicular vectors form a right-angle triangle
Resultant = root (F1² + F2²)
The angle between the resultant force and F1 is called theta (θ)
tan θ = F2/F1
Resolving a vector into two perpendicular components
Adjacent = resultant force X cos θ
Opposite = resultant force X sin θ
Balanced forces
Equilibrium of a point object
When two forces act on a point object, the object is in equilibrium only if the two forces are equal and opposite
When three forces are acting on an object, their combined effect is zero only if the resultant of any two forces is equal and opposite to the third force
The principle of moments
Turning effects
The moment of a force about any point is defined as the force X the perpendicular distance from the line of action to the force of the point
The moment of the force = F X d
The sum of the clockwise moments = the sum of the anticlockwise moments
W1D1 = W2D2
W1D1 = W2D2 + W3D3
Centre of mass
Th centre of mass of a body is the point through which a single force on the body has no turning effect
The centre of mass for a 2D object is the intersections of the lines of symmetry
The centre of mass always lies directly below the pivot
More on moments
S = W0 + W1 +W2...
Two-support problems
Where x is in contact with the beam
SyD = Wdx
Sy = Wdx/D
Where y is in contact with the beam
SxD = Wdy
Sx = Wdy/D
If the centre of mass is closer to x than y, the
dx<dy and Sy<Sx
Couples
A pair of equal and opposite forces acting on a body, but not acting upon the same line
The total moment = Fx+F(d-x)=Fx+Fd-Fx=Fd
The total moment is the same, regardless of the point about which the moments are taken
Stability
If a body in stable equilibrium is released, it returns to its equilibrium position
When a body at stable equilibrium is at rest, its centre of mass is directly below the point of support
Tilting and toppling
Tilting
The clockwise moment of F about P = Fd
The anticlockwise moment of W about P = Wb/2
For tilting to occur Fd>Wb/2
Toppling
A tilted object will topple over if it is tilted too far
Toppling will occur if the centre of mass passes the pivot point
On a slope
The force parallel to the slope:
F = W sin θ
The force perpendicular to the slope:
Sx + Sy = W cos θ
Sx > Sy when x is lower than y
Equilibrium rules
The triangle of forces
For an object to be in equilibrium
Their vector sum F1 + F2 + F3 = 0
Any two of the forces give a resultant that is equal and opposite to the third force: F1 + F2 = -F3
F2sinθ3 = F3sinθ2
F1/sinθ1 = F2/sinθ2 = F3/sinθ3
The conditions for equilibrium of a body
The principle of moments must apply
The resultant force must be zero. If there are three forces, they must form a closed triangle
Matrerials
Section 4- Electricity
Electric current
DC circuits
Section 2- Waves and optics
Waves
Optics
Section 5- Skills in AS physics
About practical assessment
More on mathematical skills
Practical work in physics
Section 1- Particles and radiation
Quarks and leptons
Leptons at work
Lepton collisions
Neutrinos interact very little
Muons are very short lived
Electrons repel each other
Lepton-antilepton particles interact to produce hadrons
Electron-positron annihilations produce a quark and an antiquark
Neutrino types
Neutrinos travel almost as fast as light
They are produced in smaller numbers when particles in accelerators collide
Neutrinos and anti-neutrinos produced in beta decays are different to those produced by muon decays
Vμ is used for muon neutrinos
Ve is used for electron neutrinos
Lepton number
Must be conserved in interactions
Leptons have a lepton number of 1
Anti-leptons have a lepton number of -1
Hadrons have a lepton number of 0
Particle sorting
Large hadron collider:
the rest energy of the products = total energy before - the kinetic energy of the products
Matter and antimatter
Hadrons
Baryons
Mesons
Composed of quarks and antiquarks
Can interact through all four fundamental interactions, they interact through the strong interaction and through the elecromagnetic reaction if charged
Apart from the proton, which is stable, hadrons tend to decay through the weak interaction
Leptons
Interact through the weak interaction, the gravitational interaction, and the electromagnetic interaction if charged
The particle zoo
Mesons
Muon
Symbol μ
Negatively charged
Rest mass over 200 times the rest mass of the electron
Also known as the heavy electron
Pion
π meson
Can be positively charged, negatively charged, or neutral
Rest mass greater than a muon, but less than a proton
Kaon
K meson
Can be postively charged, neagatively charged, or neutral
Rest mass greater than a pion, but less than a proton
Particle consisting of a quark and an antiquark
Produced in twos through the strong interaction
Decay
A kaon can decay into pions, or a muon and an antineutrino, or an antimuon and a neutrino
A charged pion can decay into a muon and an antineutrino, or an antimuon and a neutrino, a π0 meson decays into high-energy photons
A muon decays into an electron and an antineutrino, an antimuon decays into a positron and a neutrino
Decays always obey the conservation rules for energy, momentum, and charge
Strange particles
Kaons are strange particles because they include pions as a product of their decay
Kaon decay took longer than expected
Created in twos
Quarks and antiquarks
Types of quarks
Quarks
Down
Charge -1/3
Strangeness 0
Baryon number 1/3
Strange
Charge -1/3
Up
Charge +2/3
Strangeness 0
Baryon number +1/3
Quantum phenomena
Matter and radiation
Inside the atom
The structure of the atom
Atoms are less than a millionth of a millimetre in diameter
Can only be seen using an electron microscope
We know, from Rutherford's alpha-scattering investigations that every atom contains
Electrons that surround the nucleus
A positively charged nucleus composed of protons and neutrons
Sub-atomic particles
Electrons
Negative charge (-1 relative to proton)
Held in the atom by the electrostatic force of attraction betewen them and the nucleus
Charge of -1.60 x 10 to the -19 coulombs
Weighs 1.67 x 10 to the -31 kg
Mass relative to proton is 0.0005 (1/2000)
Nucleons
Neutrons
No charge
Weighs 1.67 x 10 to the -27 kg
Mass relative to proton is 1
Nucleon- proton or neutron in the nucleus
Protons
Relative charge of 1
Charge of 1.60 x 10 to the -19 coulombs
Weighs 1.67 x 10 to the -27
Relative mass of 1
Nucleus
Contains almost all of the atoms mass
Is of the order 0.00001 times the diameter of a typical atom
An uncharged atom has an equal amount of protons and neutrons
Representing atomic structure
The letter(s) represent the chemical symbol
The number at the top is the nucleon number or the mass number
The bottom number is the protons number or the atomic number
Each type of atom is called a nuclide
Isotopes
Atoms of the same element with different numbers of neutrons
Isotopes are atoms with the same number of protons and different numbers of neutrons
Specific charge
Charge divided by mass
The electron has the highest specific charge of any particle
Measured in C kg to the -1
Stable and unstable nuclei
A stable isotope has nuclei that do not disintegrate
Strong nuclear force
Stable atoms have a force holding the nuclei together- this is called the
strong nuclear force
because it overcomes the electrostatic force of repulsion between the protons in the nucleus
Its range is no more than about 3-4 femtometres ( 1 fm = 10 to the -15 m), or the same as the diameter of a small nucleus
The range of electrostatic force has an infinite range, to which the force is negatively proportional
At seperations smaller than 0.5 fm, it is repulsive
Has the same effect between two protons as it does between a proton and neutron
Radioactive decay
Beta radiation
Consists of fast moving electrons
Represented by a β symbol
Result of a neutron changing into a proton in a neutron-rich environment, where a beta particle is created and emitted instantly
An antineutrino is also produced
No mass
No charge
Ghost particle
Results in a product with a higher atomic number
Can be stopped by a few millimetres of tin foil, or a few meters of air
Gamma radiation
Represented by the symbol γ
Electromagnetic radiation emitted by an unstable nucleus
No mass and no charge
Infinite range, and can only be partially stopped by thick sheets of lead
Emitted after an alpha or beta emmision
Alpha radiation
Alpha particles consist of two protons and two neutrons (a helium nucleus) and are represented with a α symbol
Atomic number of 2
Mass number of 4
Results in the product nucleus belonging to a different element
Stopped by a sheet of paper or a few centimetres of air
Neutrinos and antineutrinos
After beta emission, beta particles were found to have a maximum amount of kinetic energy
Either energy was not conserved in beta emisssions, or there was a ghost particle
The hypothesis of neutrinos and antineutrinos was unproven for over 20 years
Detected as a result of their interaction with cadmium nuclei in a large tank of water
Billions of neutrinos from the sun pass through our body every second without interacting
Photons
Electromagnetic waves
Light is just a small part of the spectrum of electromagnetic waves
In a vacuum, all electromagnetic waves travel at the speed of light, c, which is 3.00 X 10^8 ms^-1
The wavelength λ of electromagnetic radiation of frequency f in a vacuum is given by the equation: λ=c/f
Wavelengths of the sections of the electromagnetic spectrum
radio- >0.1m
microwave- 0.1m to 1mm
infrared- 1mm to 700nm
visible- 700nm to 400nm
ultraviolet- 400nm to 1nm
X-rays- 10nm to 0.001 nm
gamma rays- <1nm
An electromagnetic wave consists of an electric wave and a magnetic wave which travel together
They vibrate at right angles to each other and to the direction in which they are travelling
They vibrate in phase with each other
Emitted by a charged particle when it loses energy
A fast-moving electron is stopped or slows down or changes direction
An electron in a shell of an atom moves to a different shell of lower energy
Emitted as short bursts of waves, each burst leaving the source in a different direction
Photons are packets of electromagnetic waves
Photon theory was established by Einstein in 1905, when he used his ideas to explain the photoelectric effect
Photon energy E = hf
h is Planck's constant
h = 6.63 X 10^-34 Js
Particles and antiparticles
Antimatter
When matter and antimatter meet, they destroy each other
Positrons
Positrons are used in PET (positron emitting tomography) scans
Antimatter electrons
Positron emission takes place when a proton changes into a neutron in a unstable nucleus with too many protons
Positively charged
Represented as a e+, except when beta radiation, where it is B+
Predicted in 1928 by English physicist Paul Dirac- his theory of antiparticles predicted that for every type of particle there is a corresponding antiparticle that:
Has exactly the same rest mass as the particle
Has exactly opposite charge to the particle if the particle has a charge
Annihilates the particle and itself if they meet, converting their total mass into photons
Pair production
Dirac predicted that a photn with sufficient energy passing near a nucleus or an electron can suddenly change into a particle-antiparticle pair, which would then seperate from each other
eVs
The energy of a particle/ antiparticle is often expressed in millions of electron volts
1 eV is defined as the energy transferred when an electron is moved through a potential difference of 1 volt
1 MeV = 1.60 X 10^-13 J
1 eV - 1.60 X 10^-19
E = mc²
Minimum energies
Photons produced in pair-annihilation: hfmin = E0
Photon needed in pair-production: hfmin = 2E0
Particle interactions
The electromagnetic force
Momentum
Mass X velocity
Transferred between objects if no other forces act on them
If two protons approach each other, they repel and move away and exchange virtual photons
Feynman described these photons as virtual because if we detect them, then we prevent the force working
The weak nuclear force
Responsible for beta decay
B- decay
n -> p + B- + ̅ν
Occurs in a neutron-rich environment
B+ decay
p -> n + B+ + v
Occurs in a proton-rich environment
Weaker than the strong nuclear force
Carried by W bosons
The W- boson decays into a B- particle and an antineutrino
The W+ boson decays into a B+ particle and a neutrino
Electron capture
Caused by a reaction between a proton and an inner shell electron
p + e- -> n + v
Force carriers
The W boson is the carrier of the weak nuclear force
The photon is the carrier of the electromagnetic force
The pion is the carrier of the strong nuclear force
The graviton (undiscovered) is the carrier of gravity
Further Maths
Pure
Matrices
Matrix multiplication
Determinants and inverses of 2X2 matrices
Addition, subtraction and scalar multiplication
Linear simultaneous equations
Hyperbolic functions
Hyperbolic identities
Solving harder hyperbolic equations
Defining hyperbolic functions
sinh x = e^x-e^-x/2
cosh x = e^x+e^-x/2
tanh x = sinh x/ cosh x = e^x-e^-x/e^x+e^-x
arsinh x = ln(x+√x²+1)
arcosh x = ln(x+√x²-1)
artanh x = 1/2 ln(1+x/1-x)
Let y = arcosh x
Then cosh y = x
e^y+e^-y/2 = x
e^y+e^-y = 2x
e^y+1/e^y = 2x
e^2y+1 = 2xe^y
e^2y-2xe^y+1 = 0
e^y = (2x±
Matrix transformations
Further ransformations in 2D
Invariant points and invaiant lines
Matrices as linear transformations
Transformations in 3D
Rational functions and inequalities
Functions in the form y = ax+b/cx+d
If y = ax+b/cx+d:
the intercepts are (0, b/d) and (-b/a, 0)
the asymptotes are x = -d/c and y = a/c
When solving inequalities algebraic fractions, multiply both sides by the square of the denominator
Functions in the form y = ax²+bx+c/dx²+ex+f
If y = ax²+bx+c/dx²+ex+f then:
when x = 0, y = c/f
when y = 0, x is a solution to ax²+bx+c = 0
vertical asymptotes are solutions to dx²+ex+f = 0
the horizontal asymptote is y = a/d
Cubic and quartic inequalities
To solve cubic or quartic inequalities, make one side zero and sketch the graph. Describe in terms of x the regions of the graph above or below the x-axis
Oblique asymptotes
When d = 0, the asymptote is a non-horizontal line
Futher applications of vectors
Cartesian equation of a line
Intersections of lines
Vector equation of a line
The scalar product
Further calculus
Volumes of revolution
Mean value of a function
Series
Using standard formulae
Method of differences
Sigma notation
Maclaurin series
The ellipse, hyperbola and parabola
Introducing the ellipse, hyperbola and parabola
The equation x²/a²+y²/b² = 1 represents an ellipse centred at the origin, with x-intercepts (±a, 0) and y-intercepts (0, ±b)
The equation x²/a²-y²/b² = 1 represents a hyperbola with x-intercepts (±a, 0) and asymptotes y = ±b/a x
Prove that the asympotes of a hyperbola with equation x²/a²-y²/b² = 1 are y = ±b/a x
The asymptotes pass through the origin, so their equations are y = ±mx
Consider a line of the form y = mx. Whether or not this line crosses the hyperboa depends on the value of m
x²/a²-(mx)²/b² = 1
b²x²-a²m²x² = a²b²
(b²-a²m²)x² = a²b²
Solutions only exist if b²-a²m² > 0
a²m² < b²
m² < b²/a²
-b/a < m < b/a
The equation xy = c² represents a hyperbola with vertices at (c, c) and (-c, -c) and coordinate axes as asymptotes
The equation y² = 4ax represents a parabola with its vertex at the origin, tanget to the y-axis
Solving problems with ellipses, hyperbolas and parabolas
The discriminant can be used to determine the number the number of intersections
Transformations of curves
Replacing x by (x, -p) and y by (y, -q) results in a translation of the curve by the vector (p q)
Replacing x by x/p and y by y/q results in a horizontal stretch with scale factor p and a vertical stretch with scale factor q
Replacing x with -x reflects a curve in the y-axis
Replacing y with -y reflects a curve in the x-axis
Replacing x with y and y with x reflect the curve in the line y =x
Replacing x with -y and y with -x reflects a curve in the ine y = -x
Roots of polynomials
Roots and coefficients
If p and q are the roots of the quadrtic ax²+bx+c = 0 then:
p+q = -b/a
pq = c/a
If a quadratic epression has roots p and q then:
(x-p)(x-q)=0
x²-(p+q)x+pq = 0
ax²-a(p-q)x+pq = 0
Hence: b = -a(p+q) and c = apq
So: p+q = -b/a and pq = c/a
Cubic and quartic equations
If p, q and r are the roots of the cubic ax^3+bx²+cx+d = 0 then:
p+q+r = -b/a
pq+qr+rp = c/a
pqr = -d/a
If p, q, r and s are the roots of the quartic ax^4+bx^3+c²+dx+e = 0 then:
p+q+r+s = -b/a
pq+pr+ps+qr+qs+rs = c/a
pqr+pqs+prs+qrs = -d/a
pqrs = ea
Finding an equation with given roots
You can use the realtionships between roots of polynomials and their coefficients to find unknown coefficients in an equation with given roots and vice versa
Complex solutions to polynomial equations
Complex solutions of real polynomials come in conjugate pairs
If f(z) = 0 then f(z*) = 0
(x-z)(x-z*) = x²-2Re(z)x+|z|²
Transforming equations
Given an equation in x that has a root x = p, if you make a substitution u = f(x), then the resulting equation in u has a root u = f(p)
Factorising polynomials
a²+b² = (a+bi)(a-bi)
Complex numbers
Definition and basic arithmetic of i
i² = -1
i = √-1
A complex number is one that can be written in the form x+iy
z is used to denote an unknown complex number and C is used for the set of all complex numbers
The original purpose of complex numbers was to solve quadratic equations with negative discriminants
If two complex numbers are equal, then their real parts are the same and their imaginary parts are the same
Division of complex conjugates
Complex conjugates differ only in the sign of the imaginary part
If z = x+iy, then the complex conjugate of z, z* = x-iy
To divide by a complex number, write as a fraction and multiply the numerator and the denominator by the conjugate of the number in the denominator
Prove that zz
is always real
Let z = x+iy where x and y are real
Then z
= x-iy
So: zz* = (x+iy)(x-iy)
= x²-ixy+ixy-i²y²
= x²-(-y²)
= x²+y²
which is real
Geometric representation
Real numbers can be represented on a number line
To represent complex numbers you can add another axis, perpendicular to the real number line, to show the imaginary part. This is called an Argand diagram
Complex numbers can be added or subtracted geometrically in the same way as you add vectors
Taking a complex conjugate results in a reflection in the real axis
Modulus and argument
Modulus
Distance from the origin
Represented by |z|
Argument
Angle relative to the positive real axis
Represented by arg z or θ
Modulus-argument form is often denoted with z = [r, θ]
Cartesian form- numbers represented as z = x+iy
Radians
The most commonly used unit of angle in advanced mathematics
360° = 2π radians
One radian ≈ 60°
Converting between modulus-argument and Cartesian form
To convert from modulus-argument and Cartesian form:
x = |z| cos θ
y = |z| sin θ
To convert from Cartesian to modulus-argument form:
|z| = √(x²+y²)
tan θ = y/x
A complex number z with modulus r and argument θ can be written as z = r(cos θ + i sin θ)
Locus in the complex plane
The distance between points representing complex numbers z1 and z2 on the Argand diagram is given by |z1-z2|
The equation |z-a|=r represents a circle with radius r and centre at the point a
The locus of a points satisfying arg(z-a) = θ is the half line starting from the point a and making angle θ with the positive x-axis
Operations in modulus-argument form
For complex numbers z and w,
|zw| = |z||w|
arg (zw) = arg z+arg w
sin(A+B) = sin A cos B+sin B cos A
cos(A+B) = cos A cos B-sin A sin B
Let z1 = r1(cos θ1+i sin θ1) and
z2 = r2(cos θ2+i sin θ2)
z1z2 = r1r2(cos θ1+i sin θ1)(cos θ2+i sin θ2)
= r1r2(cos θ1 cos θ2-sin θ1 sin θ2+i(sin θ1 cos θ2+sin θ2 cos θ1))
=r1r2(cos(θ1+θ2)+i sin(θ1+θ2))
So |z1z2| = r1r2
arg (z1z2) = θ1+θ2
For complex numbers z and w,
|z/w| = |z|/|w|
arg (z/w) = arg z - arg w
Proof by induction
Induction and matrices
Induction and divisibility
Induction and series
Indution and inequalities
The principle of induction
Polar coordinates
Changing between polar and Cartesian coordinates
Curves in polar coordinates
Some features of polar curves
Applied
Discrete
Critical path analysis 1
Linear programming and game theory 1
Constrained optimisation
To translate the problem into mathematical terms, you first produce a linear programming formulation
Identify the quantities you can vary. THese are the decision variables (or control variables)
Identify the limitations on the values of the decision variables. These are the constraints
Identify the quantity to be optimised. This is the objective function
Standard linear programming solution format:
Maximise or minimise x + y
Subject to x ≤ n1
y ≤ n2
ax + by ≤ n3
x ≥ 0, y ≥ 0
The feasible region is the set of (x, y) values that satisfy all of the constraints. It is the unshaded region on the graph
The objective line is a line joining all points (x, y) for which the objective function takes a specific value
Strategy
3 Identify the objective function
4 If the problem has two variables, solve graphically
2 Express the constraints as inequalities
5 Answer the questions
1 Identify the decision variables and label them x, y, z...
Zero-sum games
In a zero-sum game, the sum of the gains made by the players on each play is zero
A play-safe strategy gives the best guarenteed outcome regardless of what the other player does
A game has a stable solution if neither player can gain by changing from their play-safe strategy
The value of a game is the payoff to the row player if both players use their best strategy
A game has a stable solution if maximum of row minima = minimum of column maxima
In a payoff matrix, row i dominates row j if, for every column, the value in row i ≥ value in row j
Similarly, column i dominates column j if, for every row, the value in column i ≤ value in column j
Strategy
2 Check whether the matrix can be reduced by using dominance
3 Find the maximum of the row minima and the minimum of the row maxima to determine each player's play-safe strategy
1 If necessary, construct the matrix of payoffs for the row player
4 Decide whethere the game has a stable solution and, if so, find its value
5 Answer the question in context
Mixed-strategy games
If payoffs x1, x2, ..., xn occur with probabilities p1, p2, ...pn, the expectation or expected payoff E(x) is given by
E(x) = x1p1 + x2p2 + ... + xnpn = Σxipi
Mixed-strategy games can be solved graphically in most cases with linear programming
B's payoffs are the negative of A's payoffs
Optimal strategies occur where the expected payouts of playing each row with probability p are equal
B1:
A1B1P+(1-P)A2B1
B2:
A1B2P+(1-P)A2B2
B1 strategy = A2 strategy
Solve for P
Substitute P back into one of the previous equations to get your optimalexpected payoff
Value for A, v = -Value for B, -v
Strategy
1 If necessary, construct a payoff matrix
2 Use dominance to reduce the matrix as much as possible
3 Find the play-safe strategies and decide whether the game has a stable solution
4 For mixed-strategies and decide whether the game has a stable solution
5 Solve the resulting linear programming problem to find the optimal strategies and the value of the game
6 Answer the question in context
Graphs and networks 1
Graphs and networks
A graph consists of a number of points (vertices/ nodes) connected by a number of lines (edges/ arcs)
A graph is connected if you can travel from any vertex to any other vertex
A graph with no loops or multiple edges is a simple graph
The graph formed by using only some of the vertices and edges of a graph is a subgraph of the original graph
If you add extra vertices along the edges of a graph the result is called a sub-division of the graph
If a simple graph has an edge connecting all possible pairs of vertices, it is a complete graph
The complete grpah with n vertices is called Kn
In some graphs the vertices belong to two distinct sets, and each edge joins a vertex in one set to a vertex in the other- this is called a bipartite graph
The complete bipartite graph connecting m vertices to n vertices is called Km, n
The number of edges meeting at a vertex is called the degree or order of the vertex
Total numbers of degrees = 2 X number of edges
Abstract algebra
Binary operations
Sets
Collection of individual objects called elements
Finite or infinite
General sets
N represents the set of natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9}
Z represents the set of integers, positive or negative, including 0
Z+ is the set of positive integers
Z- is the set of all negative integers
Q represents all rational numbers, numbers that can be written in the form a/b, where a and b are real
R represents all real numbers, including those which are irrational and transcendental
C represents all numbers that have both an imaginary part and a real part
A function is a binary operation if it can be applied to any two elements of a set so that the result is also a member of the set
A binary operation,
*, is commutative if a
b = b*a for all a and b
A binary operation, ☆, is associative if (a☆b)☆c = a☆(b☆c)
The identitiy element, e, of a set under an operation ☆ is such that a☆e = e☆a = a for all values in the set
The identity element is unique
The inverse, a^-1, of an element a under an operation ☆ is such that a☆a^-1 = a^-1☆a = e
An element, a, is a self-inverse if a^-1 = a
Cayley tables
Named after the nineteenth-century mathematician Arthur Cayley
A table showing the result of a binary operation on all possible pairs of elements
Strategy
1 Look along the columns to find one the same as the initial column of elements; the element at the top of the column is the identity element, e. Alos check along the relevant row to make sure it equals the initial row of elements
2 Find all the instances of e in the table
3 Work down the rows. For each element, a, at the start of the row the element at the top of the column containing e is the inverse of a
Modular arithmetic
If a ≡ b (mod n) and c ≡ d (mod n) then:
a+c = b+d (mod n)
a-c = b-d (mod n)
ac = bd (mod n) as long as a, b, c, d, n are integers
Strategy
1 Draw the table for the set {0, 1, 2,..., (n-1)}
2 Calculate a*b for each pair of integers and write the answers (mod n) in the tables
3 Use the fact that a
e = e
a for the identity element e
The notation +n represents addition modulo n, where n is a positive integer
The notation Xn represents multiplication modulo n, where n is a positive integer
Maths
Pure
Algebra 1
Argument and proof
A proof is a logical argument for a mathematical statement. It shows that something must be either true or false.
Strategy
1 Decide which method of proof to use
2 Follow the steps of your chosen method
3 Write a clear conclusion that proves/ disproves the statement
Statement that can be assumed to be true are sometimes known as
axioms
Types of proof
Direct/ deductive proof
To use direct proof you:
Assume a statement, P, is true
Use P to show that another statement, Q, must be true
Disproof by counter example
You need to find just one example that does not fit the statement
Proof by exhaustion
You list all the possible cases and test each one to see if the result you want to prove is true
All cases must be true for proof by exhaustion to work
Index laws
ax²
2 is the power/ exponent/ indices
a is the coefficient
x is the base
Rules
Rule 1: Any number raised to the power zero is 1
x to the zero = 1
Rule 1 has an exception when x=0, as x 0 to the zero is undefined
Rule 2: Negative powers may be written as reciprocals
x to the -n = 1/x to the n
Rule 3: Any base raised to to the power of a unit fraction is a root
Laws
Law 2: To divide terms you subtract the indices
Law 3: To raise one term to another power you multiply the indices
Law 1: To multiply terms you ad the indices
To use the index laws, the bases of all the terms must be the same
Strategy
3 Simplify expressions as much as possible
4 Give your answer in an appropriate format that is relevant to the question
2 Apply the laws of indices correctly
1 Use the information in the question to write an expression or equation involving indices
Surds
A rational number is one that you can write in the form a/b, where a and b ≠ 0
Numbers that cannot be written in the form a/b are irrational numbers, if you express them as decimals, they have an infinite number of non-repeating decimal places
Roots of numbers can be irrrational
Irrational numbers involvig roots are called surds
Laws
√a X √b = √ab
(√a)/(√b) = √(a/b)
If the fraction is in the form k/√a, multiply numerator and denominator by √a
k/(a±√b), multiply the numerator and denominator by a±√b
k/(√a±√b), multiply numerator and denominator by √a±√b
Strategy
2 If possible, simplify surds. They should have the lowest possible number under the root sign
3 Rationalise any denominator containing surds
1 Use the information given to form an expression involving surds
Quadratic functions
A quadratic function can be written in the form ax² + bx + c, where a, b and c are constants and a ≠ 0
A quadratic equation can be written in the the form ax² + bx + c = 0
Curves in this form cross the y-axis when x = 0
Curves in this form are symmetrical about their vertex
A vertex is a turning point
For a > 0, the vertex is always a minimum point
For a<0, the vertex is always a maximum point
Curves in this form cross the x-axis whenever y = 0
You can factorise quadratics into the form (mx + p)(nx + q)
The solutions would be x=-p/m or x=-q/n
Completing the square:
ax² + bx + c = a(x + b/2a)² + q
q = c-b²/4a
When a=1 and q=0, the expression is known as a perfect square. Perfect squares only have one root, so they only touch the x-axis once
ax² + bx + c = 0:
x=(-b±√(b²-4ac))/2a
The discriminant (Δ) is b²-4ac
Δ>0: two roots
Δ=0: one root, repeated once
Δ<0: no roots
Strategy
2 Sketch the curve using appropriate axes and scale
3 Mark any relevant points in the context of the question
1 Factorise the equation
Simultaneous equations
Methods of solving simultaneous equations
By eliminating one of the variables
By substituting an expression for one of the variables from one equation into the other
Graphically
A straight line can intersect a quadratic curve at either one, two or zero points, which can be determined using the discriminant
Strategy
2 Use either elimination or substitution to solve your equations
3 Check your solution and interpret it in the context of the question
1 Use the information in the question to create the equations
Lines and circles
A straight line can be written in the form
y-y1=m(x-x1)
The gradient of a straight line through two points
(x1, y1) and (x2, y2) is m=(y2-y1)/(x2-x1)
The distance between two points (x1, y1) and (x2, y2) is
√((x1-x2)²+(y1-y2)²)
The coordinates of the midpoints of the line joining (X1, Y2) and (X2, Y2) are ([X1+X2]/2, [Y1+Y2]/2)
If m1 = m2, the two lines are parallel
If m1 X m2 = -1, the two lines are perpendicular
The equation of a circle, centre (a, b) and radius r, is
(x-a)²+(y-b)² = r²
For a circle centred at the origin, a=o and b=0, so the equation of the circle is x² + y² = r²
If a triangle passes through the centre of the circle, and all three corners touch the circumference of the circle, then the triangle is right-angled
The perpendicular line from the centre of the circle to a chord bisects the chord
Any tangent to a circle is perpendicular to the radius at the point of contact
Strategy
2 Apply any relevant rules and theorems. Draw a sketch if it helps
3 Show your working and give your answer in the correct form
1 Choose the appropriate formulae
Inequalities
Inequalities can be represented on a number line
An empty circle represents < or >
A black (shaded) circle represents ≤ or ≥
When you multiply or divide an inequality by a negative number, you reverse the inequality sign
A quadratic inequality has a range of values instead of up to two specific values
Strategy
2 Solve the inequalities and, if requested, show them on a suitable diagram
3 Write a clear conclusion that answers the question
1 Use the information in the question to write the inequalities
Polynomials and the binomial theorem
The binomial theorem
Pascal's triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
A binomial expression has two terms
The r th coeffiicient in the n th row is nCr ≡ n!/(n-r)!r!
n! stands for the product of all integers from 1 to n.
You read it as n factorial
(a+b)^n ≡ a^n+nC1a^(n-1)+nC2a^(n-2)b²+...+nCra^(n-r)b^r+...+b^n
(1+x)^n ≡ 1+nx+(n(n-1)/2!)x²+(n(n-1)(n-2)/3!)x^3+...+x^n
Strategy
2 Use Pascal's triangle or the binomial theorem to find the required terms of the binomial expansion
3 Use your expansion to answer the question in context
1 Create an expression in the form (1+x)^n or (a+b)^n
Algebraic division
The factor theorem states that if f(a) = 0, (x-a) is a factor of f(x)
Strategy
3 Factorise the quadratic quotient to fully factorise the polynomial
2 Divide the polynomial by the factor to get a quadratic quotient
1 Apply the factor theorem as necessary to find your first factor
Expanding and factorising
Polynomials are algebraic expressions that can have constants, variables, coefficients and powers, all combined using addition, subtraction, multiplication and division
The highest power in a polynomial is called its degree
A statement that is true for all values of the variables is called an identity
You write an identity using the symbol ≡
Strategy
1 Look for obvious common factors and factorise them out
2 Write an identity and expand to compare coefficients
3 Write your solution clearly and use suitable units where appropriate
Curve sketching
If a<0 the transformation y = f(ax) reflects the curve in the y-axis. If -1<a<1, the curve gets wider
If a<0 the transformation
y = f(x)+a translates the curve downwards
If a<0 the transformation
y = af(x) reflects the curve in the x-axis
If a<0 the transformation
y = f(x+a) moves the curve to the right
A line, l, is an asymptote to a curve, C, if, along some unbounded section of the curve, the distance between C and l approaches zero
Strategy
3 Identify any x and y-intercepts and any asymptotes
4 Apply any suitable transformations
2 Identify the standard shape of the curve and identify any symmetry
5 Show all relevant information on your sketch
1 Define the function using any variables supplied in the question
Trigonometry
Sine, cosine and tangent
sin²θ+cos²θ ≡ 1
tanθ ≡ (a/c)/(b/c) ≡ sinθ/cosθ
Strategy
2 Draw either a quadrant diagram or a trigonometric graph to show the information
3 Use your knowledge of graphs, quadrant graphs, symmetry and transformations to help you answer a question
1 Use trigonometric identities to simplify expressions
The sine and cosine rules
The cosine rule states that, for triangle ABC,
a² = b²+c²-2bc cos A
Area of triangle ABC = 1/2 ab sin C
The sine rule states that for triangle ABC
a/sin A = b/sin B = c/sin C
or
sin A/a = sin B/b = sin C/c
Strategy
1 Draw a large diagram to show the information you have and what you need to work out
2 Decide which rule of combination of rules you need to use
3 Calculate missing values and add them to your diagram as you solve the problem
Differentiation
Differentiation from the first principles
The gradient of PQ is given by
mPQ = yQ-yP/xQ-xP = f(x+h)-f(x)/(x+h)-x = f(x+h)-f(x)/h
f '(x) = lim(h-->0) f(x+h)-f(x)/h
For a function af(x), where a is a constant, the derived function is given by af '(x)
For a function f(x) = a, where a is a constant, the derived function f '(x) is zero
Strategy
1 Substitute your function into the formula f '(x) = lim(h-->0) f(x+h)-f(x)/h
2 Expand and simplify the expression
3 Let h tend towards 0 and write down the limit of the expression, f '(x)
4 Find the value of the gradient at a point (a, b) on the curve by evaluating f '(a)
Differentiating ax^n and Leibniz notation
If f(x) = ax^n then f '(x) = nax^n-1
If h(x) = f(x)+g(x) then h '(x) = f '(x)+g '(x)
If y = f(x) then dy/dx = f '(x)
Strategy
1 Use the laws of algebra to make your expression the sum of terms of the form ax^n
2 Apply f(x) = ax^n ==> f '(x) = nax^n-1 to each term to find the derivative
3 Substitute any numbers required and answer the question
Rates of change
The rate of change of y with respect to x can be written dy/dx
v = dr/dt
The rate of change of velocity is called acceleration, a = dv/dt
Second derivative
y = f(x) ==> dy/dx = f '(x) ==> d²y/dx² = f ''(x)
Acceleration is a derivative of a derivative
Gradients and changing fuctions
The function is increasing (dy/dx > 0)
The function is decreasing (dy/dx < 0)
The function is stationary (dy/dx = 0)
Strategy
2 Differentiate to get a formula for the relevant rate of change
3 Evaluate under the given conditions
4 Apply this to the initial question, being mindful of the context and units
1 Read and understand the context, identifying any function or relationship
Tangents and normals
When lines m1 and m2 are perpendicular to each other, m1 X m2 = -1
m1 = -1/m2 for perpendicular lines
The tangent to the curve y = f(x) which touches the curve at the point (x, f(x)), has the same gradient as the curve at that point, giving mT = f '(x), is perpendicular to the tangent at that point, giving mN = -1/mT = -1/f '(x)
Strategy 1
To work out where a tangent or normal meets
1 Differentiate the function for the curve
2 Equate this to the gradient of the tangent or normal
3 Rearrange and solve for x
4 Substitute x in the function and solve for y
Strategy 2
To work out the area bound between a tangent, a normal and the x-axis/ y-axis
1 Work out the equation of the and, from it, the equation of the normal
2 Work out where each line crosses the required axis. Lines cut the x-axis when y = 0, and the y-axis when x = 0
3 Sketch the situation if required
4 Use A = 1/2 X base X height, where the base is the length between the intercepts on the x-axis or y-axis and height is the y-coordinate or x-coordinate respectively
Turning points
At a turning point, the gradient of the tangent is zero. Therefore, you can work out the coordinate of the turning point by equating the derivative to zero
A turning point is a stationary point, but a stationary point is not necessarily a turning point
At a maximum turning point, d²y/dx² < 0
At a minimum turning point, d²y/dx² > 0
Strategy 1
To identify the main features of a curve
1 Work out where it crosses the axes (x = 0 and y = 0)
2 Consider the behaviour of the curve as x tends to infinity and identify any asymptotes
3 Work out the coordinates of the turning points (dy/dx = 0) and determine their nature
4 Use the information you have found to sketch the function
Strategy 2
To optimise a given situation
3 Let the derivative be zero and find the value of x that optimises y
4 Examine the nature of the turning points to decide if it is a maximum or minimum
5 Put your turning point in the context of the question
1 Express the dependent variable as a function of the independent variable
2 Differentiate y with respect to x
Integration
Integration is the reverse process of differentiation
Integrating x^n with respect to x is written as
∫x^n dx = x^(n+1)/n+1 + c, n ≠ -1
∫af(x)dx = a∫f(x)dx where a is a constant
∫v(t)dt = r(t)+c
∫a(t)dt = v(t)+c
Fundamental theorem of calculus
∫f(x)dx = F(b)-F(a) where d/dx(F(x)) = f(x)
Strategy
1 Identify the variables and express the problem as a mathematical equation
2 Integrate
3 Use initial conditions to work out the constant of integration
4 Substitute c into the equation and answer the equation
Area under a curve
To find the area under a curve, you perform a series of calculations using a definite integral
A definite integral is denoted by ∫f(x)dx
b is called the upper limit, and a the lower limit
The area under a curve between the x-axis, x = a, x = b and y = f(x), is given by A = ∫f(x)dx = F(b)-F(a)
Strategy
1 Make a sketch of the function, if there isn't one provided
2 Udentify the area that has to be calculated
3 Write down the definite integral associated with the area
4 Evaluate the definite integral and remember that area is always positive
Exponentials and logarithms
The laws of logarithms
x = a^n
n = log a(x)
n is the log of x to base a
When n = 1
a^1 = a
log a(a) = 1
When n = o
a^0 = 1
log a(1) = 0
When n = -1
a^-1 = 1/a
log a(1/a) = -1
Three laws of logarithms
log a(xy) = log a(x)+log a(y)
log a(x/y) = log a(x)-log a(y)
log a(x^k) = k log a(x)
You can write log 10(x) as simply log (x)
Strategy
1 Convert between index notation and logarithmic notation
2 Apply the laws of logarithms if necessary and any results for special cases
3 Manipulate and solve the equation. Check your solution by substituting back into the original equation
Exponential functions
The general equation of an exponential function is y = a^x where a is a positive constant
The graph y = e^x has a gradient of e^x at any point (x, y)
The inverse of y = a^x is the logarithmic function, y = log a(x)
The inverse of y = e^x is log e(x) which can be written as y = ln (x)
ln (x) is called the natural (or Naperian) logarithm
Strategy
1 Draw or sketch a graph if it is helpful
2 Use what you know about the fradients of y =e^x and y = e^kx
3 Use the realtionship between an exponential function and its inverse
Exponential processes
An equation of the form y = Ae^kt gives an exponential model where A and k are constants
Strategy
1 Calculate data using the model
2 Consider sketching or using a graphical calculator to graph the model
3 Use your knowledge of exponential functions and logarithms to find rates of the change and solve equations
4 Compare actual data with your model an, where necessary, comment on any limitations
Curve fitting
y = ax^n becomes Y = nX + c, where Y = log (y), X = log (x)
y = kb^x becomes Y = mx+c, where Y = log (y), m = log (b) and c =log (k)
Strategy
1 Transform the non-linear functions y = ax^n and y =kb^x to linear functions using logarithms
2 Use the transformed data to draw a straight-line graph, using a line of best fit when necessary
3 Use your graph to calculate the constant the constants and work out the realtionship between x and y
Applied
Mechanics
Vectors
Definitions and properties
A scalar quantity has magnitude only. A vector quantity has both magnitude and direction.
Vectors are equal if they have the same magnitude and direction
ka is a vector parallel to a and with magnitude k|a|
a is parallel to b only if a = kb
The vector sum of two or more vectors is called their resultant
A vector with a magnitude of 1 is called a unit vector
a and -a have the same magnitude but opposite directions
Subtracting a vector is the same as adding its negative
Strategy
2 Look for parallel, collinear and equal vectors
3 Break down vectors into a route using vectors you already know
1 Sketch a diagram using directional line segments, to show all the information given in the question
Components of a vector
Vectors i and j are unit vectors in the x and y directions
x = r cosθ and y =r sinθ
OP = r cosθi + r sinθ
r = √(x²+y²) and tan
(ai+bj)+(ci+dj) = (a+c)i+(b+d)j
(ai+bj)-(ci+dj) = (a-c)i+(b-d)j
k(ai+bj) = kai+kbj where k is a scalar
Two vectors are equal if and only if both their i and their j components are equal
If points A and B have position vectors a and b then
vector Ab = b-a
distance Ab = |b-a|
Strategy
1 Draw a diagram if appropriate
2 Convert to components if they're not already in component form
3 When solving vector equations remember that you can equate x and y components separately
Units and kinematics
Standard units and basic dimensions
All quantities in mechanics are defined in terms of three fundamental quantities or dimensions: mass, length and time
Force = mass X acceleration. The SI unit is the newton
Strategy
1 Convert units if they're inconsistent and perform any neccessary calculations
2 Check that dimensions have been conserved and that your final answer is in the correct units
Motion in a straight line- definitions and graphs
Position is a vector: the distance and direction from the origin O
Displacement is a vector: the change of position
Distance is a scalar: the magnitude of displacement
Velocity is a vector: the rate of change of displacement
Speed is a scalar: the magnitude of velocity
Average velocity = resultant displacement/ total time
Averge speed = total distance/ total time
The gradient of a displacement-time graph is the velocity
Acceleration is the rate of change of velocity. The gradient of a velocity-time graph is the acceleration
Acceleration = change in velocity/ time = (v-u)/t
The area between the v-t graph and the t-axis is the displacement
Strategy
1 Be clear whether you are being asked for displacement or distance, and velocity or speed
2 Use gradient to calculate velocity from an s-t graph, and acceleration from a v-t graph
3 Use area under a v-t graph to calculate displacement. Keep in mind that area below the t-axis is negative displacement
Eqations of motion for constant acceleration
s = displacement
u = initial velocity
v = final velocity
a = acceleration
t = time
v = u+at
s = 1/2 (u+v)t
s = vt-1/2 at²
v² = u²+2as
Strategy
1 Use the information in the question to list the known values and the variable you need to find. Be careful to distinguish between displacement and distance, and between velocity and speed
2 Choose the correct equation to use
3 Apply the equation to find the numerical value and use it to answer the original question
Motion with varable acceleration
Gradient of s-t graph = velocity at that instant
Gradient of v-t graph = acceleration at that instant
v = ds/dt
a = dv/dt = d²s/dt²
s = ∫v dt and v = ∫a dt
Strategy
1 Identify what dimensions you're dealing with and differentiate or integrate as appropriate
2 Include the constant of integration and calculate its value
3 Use the result of your differentiation or integration to answer the original question
Forces and Newton's laws
Forces
An object that is at rest or moving with constant velocity is in equilibrium
When you resolve in any direction for an object that is in equilibrium, the overall force in the direction will be zero
The frictional force always acts in a direction which opposes motion. There is no friction on smooth surfaces. Friction appears when rough surfaces try to move relative to each other
If forces F1, F2,..., Fn act on an object then the resultant force is R = F1 + F2 + ...Fn
Strategy
1 Resolve in two perpendicular directions (always in the direction of one of the forces) to find the sum of the components of all the forces in these two directions. Label the components P and Q
2 Draw a right-angled triangle with P and Q as the two shorter sides
3 To calculate the resultant R = √(P²+Q²) is the magnitude and ⍺ gives the direction. If the resultant R is known, use P = Rcos ⍺ and Q = Rsin ⍺ to find the components
Statistics
Collecting, representing and interpreting data
Single-variable data
The five-number summary gives the minimum value, lower quartile, median, upper quartile and maximum value
Box-and-whisker plots are useful for comparing sets of data
The distribution of data is how often each outcome occurs. Each outcome occurs with a given frequency
A continuity correction involves altering the endpoints of an interval of rounded data to include values which would fall in the interval when rounded
You can use a histogram to display continuous data
Strategy 1
2 Cosider whether you need to be able display relative or absolute frequencies
3 Consider whether you are more interested in displaying the distribution or the summary statistics
1 Consider whether you need to be able to display all of the values, including outliers
Choosing appropriate diagrams to represent data
Box plot
Pros
Highlights outliers
Makes it easy to compare data sets
Cons
Data is grouped into only four categories so some detailed analysis is not possible
Histogram
Pros
Clearly shows shape of distribution
Cons
Doesn't always highlight outliers
It is possible but not easy to estimate Q1, Q2 and Q3
Cumulative frequency curve
Pros
Makes it easy to find the number summary five
Cons
Doesn't always highlight outliers
If interval boundaries are not shown the degree of detail is not clear
Strategy 2
1 Consider what is being represented and whether your data is discrete of continuous
2 If necessary, identify any outliers or missing/incorrect data, and consider the effects of removing them
3 Read what is being asked for in the question and use the diagram to answer it
Sampling
Population- set of things in which you are interested
Sample- subset of the population
Paremeter- a number that describes the entire population
Statistic- a number taken from a single sample
Statistics can be used to estimate the parameter
Sampling methods
Methods if you are able to list every member of the population
Simple random sampling- every member of th population is equally likely to be chosen. For example, allocate each member of the population a number. Then use random numbers to choose a sample of the desired size
Systematic sampling- find a sample of size n from a population of size N by taking one member from the first k members of the population at random and then selecting every kth member after that, where k = N/n
Stratified sampling- when you know you want distinct groups to be represented in your sample, split the population into these distinct groups and then sample within each group in proportion to its size
Methods if you cannot list every member of the population
Opportunity sampling- Taking samples from members of the population you have access to until you have a sample of the desired size
Quota sampling- when you know you want distinct groups to be represented in your sample, decide how many members of each group you wish to sample in advance and use opportunity sampling until you have a large enough sample for each group
Cluster sampling- split the population into clusters that you expect to be similar to each other, then take a sample from each of these clusters
Strategy
1 Consider whether or not you can list every member of a population
2 Identify any sources of bias and any difficulties you might face in taking certain samples
3 Compare the different sampling methods you have available and choose the one that best suits your needs and limitations
A sampling method is biased if it is not representative of the population
Central tendency and spread
Continuous data- always numeric, it can take any value between two points on a number line
Raw data is produced in statistical investigations. It is useful to reduce this data to some key values, called summary statistics, which can be characterised as measures of central tendency and measures of dispersion.
Measures of central tendency
Median
The median of a set of data is the middle value of data listed in order of size
Mean
To work out the mean x̄ of a set of n observations, calculate their sum and divide the result by n x̄ = Σx/n
The mean of a set of data given in the form of a frequency distribution is given by x̄ = Σfx/Σf
Mode
The mode of a set of data is the value or category that occurs most often or has the largest frequency. For grouped data, the modal interview or modal group is normally given
Measures of spread
You should be familiar with four measures of spread
Interquartile range
Q1: (n+1)/4
Q3: 3(n+1)/4
The interquartile range (IQR) = Q3 - Q1
Variance- measures how spread-out the values are from the mean
The population variance of n observations with mean x̄ is defined as
σ² = Σ(x-x̄)²/n = Σx²/n - x̄²
Range
The range of a set of data is the largest value minus the smallest value
Standard deviation- the square roof of the variance √
The population standard deviation of n values with mean x̄ is defined as
σ = √(Σ(x-x̄)²/n) = √(Σx²/n - x̄²)
An unbiased estimate of the population varience using a sample of n observations with sample mean x̄ is given by the sample variance, s². The divisor of s² is n-1
s² = Σ(x-x̄)²/n-1 = Σx²/n-1 - (Σx)²/n(n-1)
The sample standard deviation of a set of n values is defined as
s = √(Σ(x-x̄)²/n-1) = √(Σx²/n-1 - (Σx)²/n(n-1))
Discrete- any one of a finite set of categories (non-numeric) or values (numeric), but nothing in between those values
Strategy
1 Identify the summary statistics appropriate to the problem
2 Calculate values of the required statistics, using a calculator where appropriate
3 Use the statistics to describe key features of the data set and make comparisons
4 If not already done, identify any outliers and remove them, then see how this affects the calculations
Outliers are values that lie significantly outiside the typical set of values of the variable
Data properties
Measures of central tendency
Mode
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Median
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Mean
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Measures of spread
Range
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IQR
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Standard deviation
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Bivariate data
Correlation
There are three types of correlation
Negative correlation- one variable increases as the other decreases
Zero correlation- the variables are uncorrelated
Positive correlation- the variables increase together
Variables that are statistically related are described as correlated.
The correlation coefficient, known as r, can have a value between -1 and +1 inclusively
For no correlation, r = 0
For perfect negative correlation, r = -1
For perfect positive correlations, r = 1
When a change in one variable does affect the other, they have a causal connection. Correlation without a causal connection is known as spurious correlation
Strategy
1 Draw a scatter diagram to identify any correlation between two variables
2 Identify data points that don't fit the general pattern shown by the data
3 Use correlation in the scatter diagram to determine the value of missing data points
Probability and discrete random variables
Probability
A random experiment is any repeatable action with a collection of clearly defined outcomes that cannot be predicted with certainity
The sample space for an experiment is the collection of all possible outcomes of the experiment
The total probability associated with a sample space is 1
The probability of an event happening is written 'P(event)'
If A and B are mutually exclusive, P(A or B) = P(A) + P(B)
P(A') = 1-P(A)
Complementary events are mutually exclsive and exhaustive- no other outcome exists
For a sample space with N equally likely outcomes, the probability of any one occuring is 1/N
If event A occurs in n(A) of the equally probable outcomes, the probability of A is given by P(A) = n(A)/N
Two events, A and B, are independent if the fact that A has occured does not affect the probability of B occuring
If A and B are independent, P(A and B) = P(A) X P(B)
P(A or B) = P(AuB)
P(A and B) = P(AnB)
The u and n symbols are called the union and intersection symbols respectively
Strategy
3 For unknown probabilities, consider using the 'probabilities total 1' result
1 Identify mutually exclusive events and use the addition rule
2 Identify independent events and use the multiplication rule
Binomial distribution
Conditions for a binomial probability
Two pssible outcomes in each trial
Fixed number of trials
Independent trials
Identical trials (p is the same for each trail)
P(X=x) = nCx p^X(1-p)^(n-x)
where n is the number of trials and p is the probability of success in any given trial
Written as X~B(n,p)
If X can only take integer values, P(X<x) = P(X≤x-1) and P(X>x) = 1 - P(X≤x)
Strategy
1 Check the conditions for a binomial distribution are met. List any assumptions
2 Identify the random variable and corresponding values of n and p
3 Calculate probablities using the addition and multiplication rules if necessary
Hypothesis testing 1
Formulating a test
The null hypothesis, H0, is a statistical statement representing your basic assumption
The alternative hypothesis, H1, is a statement that contradicts the null hypothesis
The null hypothesis always includesthe equality sign
One tailed-test: this only tests either below the value stated in H0 or only above the value stated in H0
Two-tailed tests: this tests both below and above the value stated in H0
Testing hypotheses
Let X be the random variable representing the number of things in the chosen sample who meet the target
X is called the test statistic and determines whether you accept or reject the null hypothesis
Assuming H0 is true, X is binomially distributed: X~B(n, p)
The critical region is the set of values that leads you to reject the null hypothesis. The acceptance region is the set of values that leads you to accept the null hypothesis
The critical value lies on the border of the critical region. It depecds on the significance level of the test. The critical region includes the critical value and all values that are more extreme than that
Every hypothesis test has a significance level. This is equal to the probability of incorrectly rejecting the null hypothesis
As you lower the significance level, you need more evidence to reject the null hypothesis and you lower the chance of making an incorrect conclusion
Strategy
1 Identify the critical region and draw a diagram if it helps
2 If the value from the sample lies in the critical region then you reject the null hypothesis. If the value does not lie in the criticaal region the you accept the null hypothesis
3 End with a conclusion that relates back to the situation described in the question
The critical region
If H0 is assumed to be true, for discrete random variables, a value lies within the critical region if the probability of X being equal to or more extreme than that value is equal to or less than the significance level
The p-value is the probability that x is equal to or more extreme than an observed value. If the p-value is greater than the significance level, you accept H0. If the p-value is less than or equal to the significance level, you reject H0
Strategy
1 Define X, state its distribution and write down H0 and H1
2 Assume that H0 is true and either find the critical region or calculate the p-value
3 Decide whether to accept or refect H0 and iterpret the result in the context of the question