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### IB Math

Lecture notes and guidelines used during times of my profession as a private tutor/lecturer.

## Contents1 Algebra2 Statistics3 Geometry and Trigonometry4 Calculus5 Matrix6 Vectors7 SyllabusAlgebra

### Sets of numbers

• Natural numbers(N) = {0, 1, 2, 3, 4, ...} excludes negative numbers
sum and product of two natural numbers is always a natural number
• Integers(Z) = {...,-3,-2,-1,0,1,2,3,...}
sum, product, and difference of two integers is always an integer
• Rational numbers(Q) = {p/q where p and q are integers and q≠0}
examples: -2, 0.1, 0.555, etc.
• Real numbers(R) = √2, π, ∛11...

### Approximation

• Rounding: approximation to a given degree of accuracy
\$2.065 to 2 decimal places -> \$2.07
• Significant figures: number of figures known with some degree of reliability
all non-zero digits and zeros between non-zero digits, and zeros placed after other digits to the right of the decimal point are significant
• Error = VA - VE
• Percentage error = VA - VE / VE x 100%

### Units

Scientific notation a x 10k

1=≤a<10

SI(Systeme International d'Unites) Units

• SI base units
• meter(m) for distance
• kilogram(kg) for mass
• second(s) for time
• ampere(A) for electric current
• kelvin(K) for temperature
• mole(mol) for amount of substance
• candela(cd) for intensity of light
• SI derived units

### Sequence and series

• Arithmetic sequence: a number sequence with common difference
common difference: constant increments between each term
Un = U1 + (n-1)d
• Arithmetic series: sum of an arithmetic sequence
Sn(sum of the first n terms) = n/2(u1 + un)
• Geometric sequence: a number sequence with common ratio
common ratio: quotient of current term divided by its previous term
Un = U1 x rn-1
• Geometric series
Sn = u1(rn-1)/r-1
• Simultaneous equations
1. method of substitution: express one of the equations for one variable in terms of the other, then substitute into the other equation
2. method of elimination: combine two or more equations so that one of the variables can be eliminated
• Quadratic equations ax2+bx+c = 0
1. factorizing:
• tip: find set of factors of c, and choose the set which gives sum = b ## Statistics

### Set

defined by brackets or intervals

• Intervals: defines the set of all the numbers inclusive
closed interval [-2, 3] all numbers from -2 to 3 inclusive
open interval (-2, 3) all numbers between but -2 and 3
unbounded interval (-∞,∞)
• Containment(∈) and shared properties
~ | ~ ~ such that ~
Neven = {2n∈N|n∈N} the set of numbers formed by doubling all the natural numbers
[a,b] = {x∈R: a≤x≤b}
• Subsets(⊂): a collection of elements from within a particular set
Z⊂R
• Empty set(∅): the only set that contains no elements at all
• Universal set(U): more general set which contains all
• Union(∪): combine all the elements from the original sets into a single set
• Intersection(∩): elements are common to both of the two sets
• Complement of a set S(¬S): containing elements which are not in S
S ∩ ¬S = ∅
• Compound relations: combination of intersections and/or unions
• Venn diagram: picturing operations between sets
• Sample space: a set consisting of all possible results of a trial or experiment or all events associated with a particular process. An individual outcome or result is also called an event.
• Propositions: mathematical logic dealing with basic statements
can be true, false, or indeterminate
• Compounds statements: all relations between two+ propositions
• Implication(p⇒q) if p, then q (p⇔q) q if and only p
• Conjunction (∧) = and
• Disjunction (∨) = or, possibly both
• Exclusive disjunction = or but not both
• Truth tables
• Contradiction p and ¬p cannot be true at the same time
• Tautology: proposition always true
• Implication: p⇒q
• Converse: q⇒p
• Inverse: ¬p⇒¬q
• Contrapositive: ¬q⇒¬p
• Containment 3∈Z, -3∉Z
• Subset Z⊂N A⊆Z
• Empty set {}
• Universal set U
• Union A ∪ B
• Intersection A ∩ B
• Complement Q(rational numbers) ∪ Q' = R(real numbers)

• Sample space: a set consisting of all possible results of a trial or experiment, or all events associated with a particular process
• Event: an individual outcome or result

Probability

likelihood of an occurrence
P(¬S)+P(S) = 1
• Monte-Carlo method
Estimate of unknown area = #points in unknown area/ total #points generated x known area
• Tree diagrams
• Types of events
• Combined events P(A∪B) = P(A) + P(B) - P(A∩B)
• Mutually exclusive events P(A∪B) = P(A) + P(B)
• Independent events P(A∩B) = P(A) x P(B)
• Conditional probability P(A|B) = P(A∩B)/P(B)

• Heads and Tails
• Dice

• Venn diagram
• Tree diagram

• Combined events P(A∪B) = P(A) + P(B) - P(A∩B)
• Mutually exclusive events P(A∪B) = P(A) + P(B)
• Independent events P(A∪B) = P(A) x P(B)
• Conditional probability P(A|B) = P(A∩B)/P(B)

### Discrete Mathematics

• Proposition: basic statement dealing with a mathematical logic
• Conjunction ∧ (and)
• Disjunction ∨ (or)
• Exclusive disjunction (or but not both)
• Implication ⇒
• Equivalence ⇔

• Tautology P∨¬P

• Implication p⇒q
• Converse q⇒p
• Inverse ¬p⇒¬q
• Contrapositive ¬q⇒¬p

### Statistics

• Discrete data counted
• Continuous data measured

• Frequency tables
• Mid-interval values
• Upper & lower boundaries
• Frequency polygons only for continuous data
• Frequency histograms
• Stem and leaf diagrams

• Cumulative frequency
• Percentiles and quartiles
• Whisker plots
• Outliers

• Measures of central tendency
• Mode
• Median
• Mean
• Measures of dispersion
• Range
• Interquartile range = Q3 - Q1
• Standard deviation
• Correlation
• Linear regression

(y-ȳ) = Sxy (x-¯x) / (Sx)2

• ℵ² test for independence
• ρ-value
• Weighted average • Geometric mean • Harmonic mean • measurement scales(NOIR) - Nominal(no context; weakest), Ordinal(accoding to characteristics), Interval(special meaning to difference to numberical values), Ratio(scale amounts; strongest)
• Correlation: degree of linear dependence between the variables
• 1 in the case of an increasing linear relationship,
• −1 in the case of a decreasing linear relationship,
• some value in between in all other cases
• the closer the coefficient is to either −1 or 1, the stronger the correlation between the variables.  • Bayes formula • Permutation: order matters nPr = n! / (n-r)!
• Combination: order does not matter nCr = n! / r! (n-r)!
• Mean Absolute Deviation • Standard deviation • Chebyshev’s inequality : no more than 1/k2 of the values are more than k standard deviations away from the mean. • Sharpe ratio: measure of the excess return (or Risk Premium) per unit of risk in an investment asset • Roy's Safety-First criterion

P(Ra < Rm) = [E(RP)-RL]/σP

• Skewness
• measure of the asymmetry
• Positive skew: mode < median < mean
• negative skew: mode > median > mean • Kurtosis
• measure of the "peakedness"; can be either leptokurtic or platykurtic

### Probability Concepts

• Probability distribution: of all possible outcomes for a random variable
• discrete distribution: finite number of possible outcomes
• continuous distribution: infinite number of possible outcomes
• Probability function p(x): probability that a discrete random variable will take on the value x
• Probability density function f(x): probability a continuous random variable will take on a value within a range
• Cumulative distribution function F(x): probability a random variable will be less than or equal to a given value
• Binomial random variable: probability of exactly x successes in n trials
• Confidence interval: a range of values around an expected outcome within which we expect the actual outcome to occur some specified percent of the time
• 90% confidence interval = Χ ± 1.65σ
• 95% confidence interval = Χ ± 1.95σ
• 99% confidence interval = Χ ± 2.58σ
• Degrees of freedom
• sufficiently high df is approximately normal
• higher degrees of freedom, thiner tails
• Standard normal distribution: μ=0 σ=1
• z-value standardization where plug-in z-value to get F(z) from z-table

• Covariance • Coefficient of Variation

CV = σ / μ

• Monte Carlo Simulation: to estimate distribution of derivatives prices or of Net Present Values
• Continuous compounding = ln(1+HPR)

### Sampling and Estimation

• Sampling: to make inferences about the parameters of a population
• time-series data- gathered from each time periods
• cross-sectional data- data from a single time period
• Stratified sampling: random picks within subgroups
• Central limit theorem
• sample mean for large sample sizes will be distributed normally
• as sample size increases, becomes more accurate in respect to population data
• holds for n > 30
• Test statistic: difference between population sample and hypothesized value
• Z-test • Standard error • Student's t-test
• used when sample size is small or variance unknown • Chi square test

used when H0: σ² = σ0<super>2</super>

• Confidence interval (level of significance:probability of rejecting true H0)
• 68% of observations fall in ±1σ
• 95% of observations fall in ±1.96σ
• 99% of observations fall in ±3σ
• Types of Bias
• data-mining~ : repeatedly doing tests on same data sample
• sample selection~ : sample not really random
• survivorship~ : sampling only surviving firms
• look-ahead~ : using information not available at the time to construct sample
• time-period~ : relationship exists only during the time period of sample data

### Hypothesis Testing

• H0: hypothesis set up to be nullified or refuted in order to support an alternate hypothesis.
• H1: alternative hypothesis
• Type I error(Significance level): rejecting a null hypothesis when it is actually true; decreases as confidence interval(tradeoff) increases
• Type II error(1 - Power of test): failing to reject a null hypothesis when the alternative hypothesis is the true; increases as confidence interval increases
• Volatility estimation
• unbiased~ : has an expected value equal to the true value of the population parameter
• consistent~ : more accurate the greater the sample size
• efficient~ : has the sampling distribution that is less than that of any other unbiased estimator
• Statistical significance omits transaction costs, taxes, risk factor from economical significance

## Geometry and Trigonometry

### Cartesian Geometry

Brief overview of basics

• Midpoint

• Pythagorean theorem
d =
m =
• Parallel lines
two lines are parallel if and only if they have the same gradient
• Perpendicular lines: negative reciprocal
negative reciprocal of m1 = 1/m1
• Collinear points: points which lie on the same straight line
• Axis intercepts
y-intercept intersection of a line with the y-axis when x=0
x-intercept intersection of a line with the x-axis when y=0
• Point of intersection
no point of intersection => different (likely parallel) lines
1 point of intersection => different gradients
infinite number of intersection points => coincident lines

### Right-angled Trigonometry

• Trigonometric ratios • Finding the sides and the angles of a right-angled triangle
• Isosceles triangle
• Any triangle
• Rectangle
• Rhombus
• Trapezium
• Parallelogram
• Circle
• The sine rule
• The cosine rule
• Area of a triangle

### Geometry of Solids

Solid Area of curved face Volume
Cone

V = Area of a base x height / 3
Sphere

Cylinder

## Calculus

• Gradient of a curve

• Finding the gradient of a curve at a general point P

• Derivative of polynomials
when y = x^n , where n is a real number,
when y = a * x^n, where n is a real number,
when  , where u and v are functions of x,
• Negative powers
• Second derivative ∂²y/∂x²
• Tangent equation
• Gradient of the tangent line
• Increasing and decreasing functions
points where f'(x) = 0
• Local maxima and minima
• Stationary point
• Point of inflexion

y = ax2 + bx + c
• parabola shape with axis of symmetry
• Completed square form F(x)=a(x-r)2 + s
axis of symmetry at r
check that f(r-h) = f(r+h)

### Factorising techniques

• 3 quick special cases
1. No constant term x(x-r)
2. Difference of two squares (x-r)(x+r)
3. Sum of two squares (x-r)(x-r)
never factorize
• General case hit-and-miss process
1. Find factors for a and c
2. Test for the case matching b

• Transformation into quadratic equation
• Point of intersection
• Fatorising a polynomial
1. Factor theorem For polynomial p(x), if p(t) = 0, then (x-t) is a factor of p(x).
2. Extended factor theorem If p(t/s) = 0, then (sx-t) is a factor of p(x).
3. Method of equating coefficients If ax2+bx+c = lx2+mx+n, then a=l, b=m, c=n.
• The remainder theorem
If a polynomial p(x) is divided by a linear polynomial sx-t, the remainder is p(t/s).

### Exponentiation

• Power functions f(x) = xa
• Negative integer powers
• Fractional indices
• Exponential functions f(x) = ax

### Investigating shapes of graphs

• Exponential functions
f(x) = kaλx + C
• Stationary points: points of a graph at which the gradient is 0
• Local maxima and minima
• f'>0 increasing function
• f'<0 decreasing function
1. Find an expression for f'(x)
2. List the values of x for which f'(x) = 0
3. Find the sign of f'(x) in intervals to the left and to the right of that value
4. If these signs are - and + respectively, => minimum point. If + and -, => maximum point
5. For each value of x which gives a minimum or maximum, calculate y = f(x)
• Application to roots of equations

### Second derivatives

• examines the way in which the graph is bending
• f" > 0 => concave up
• f" < 0 => concave down
• f" = 0 ?<=>? Point of inflexion f"
• Local maxima and minima
1. Find f'(x)
2. List the values of x for which f'(x)=0
3. Find f"(x)
4. For each x with f'(x)=0, find the sign of f"(x). If the sign is + => minimum point; if - => maximum
5. For each value of x which gives a minimum or maximum, calculate y = f(x)

### Differentiation

• Differentiation: process of finding the gradient formula for a curve
• Derivative: the gradient formula for a curve • Differentiation rule
If f(x) = xn, where n is a positive integer, then f'(x) = nxn-1
The derivative of af(x) + bg(x), where a and b are constants, is af'(x) + bg'(x)
proof f(x)=ax2+bx+c

### Integration

reverse process of differentiation
• Finding a function from its derivative
ex.> f'(x) = 2x + x2
• Indefinite integral ∫f(x) dx
• Arbitrary constant
ex.> f'(x) = 6x2 - 5x; f(x) passes through (2,3)
• Application to kinematics
ex.> 28.3.1 For the first few seconds of a race a horse's acceleration, a m/s2, is modelled by the equation a = 6 - 1.2t, where t is the time in seconds from a standing start. Find an expression for the distance it covers in the first t seconds. Hence find the horse's acceleration 5 seconds after the start, how fast it is then moving and how far it has run.
ex.> 28.3.2 A train is traveling on a straight track at 48 m/s when the driver sees an amber light ahead. He applies the brakes for a period of 30 seconds, producing a deceleration of 1/125 * t(30-t) m/s2, where t is the time in seconds after the brakes are applied. Find how fast the train is moving after 30 seconds, and how far it has traveled in that time.
• The area under a graph
to find the area under the graph f(x) from x=a to x=b:
1. Find the simplest integral of f(x), call it I(x)
2. Work out I(a) and I(b)
3. Area = I(b) - I(a)
ex. 29.1.2 Find the area between the graph of y=2x - x2 and the x-axis.
• Definite integrals

ex.> 29.3.1 An aircraft accelerates along the runway as it takes off. After t seconds it has traveled s meters and has a speed of v m/s. For 10<t<20 the motion is modeled by the equation v = 100 - 4000/t2. Find how far the aircraft travels in this time.

• Properties
ex.> Find the area between y=x2 and the chord joining the points (-1,1) and (2,4) on the curve
ex.> 29.7.2 Show that the graphs of y=f(x) and y=g(x), where f(x)=x3 - x2 - 6x + 8 and g(x) = x3 + 2x2 - 1, intersect at two points, and find the area of the region enclosed between the two curves.
ex.> 29.7.3 y=3 + 2x - x2 Calculate the areas enclosed

### Exponentials & Logarithms

• Exponential function f(x) = bx
where b is a positive real number and base b≠1
• Logarithm to base b of y
if x is a real number and y is a positive real number
bx = y is equivalent to x = logby
examples
• Properties(p.439-441)
• Multiplication rule
• Division rule
• Power rule
• the nth root rule
• Special bases
• log base 10
• natural log

### Extension

consider a function in the following format:
• Differentiation
• Integration
• Sum rule
if y = u+v
• Product rule
if y = uv
• Quotient rule
if

### Exponentials and Logarithms

• Exponential function ,
• e = 2.718271828...
for which f'(0) = 1
• for any positive base b,
• Natural logarithm
• Natural log differentiation
• Reciprocal integral
• Irrational indices

### Volumes of revolution

• Rotation about the x-axis
• Rotation about the y-axis
• Rotating regions between curves

### Chain Rule

Circular functions

• Differentiation
• Applications

### Integrating products

• Integration by parts
• Arbitrary constant

### Differential Equations

the rate of change of one variable with respect to the other
• A rodent has mass 30 grams at birth. It reaches maturity in 3 months. The rate of growth is modelled by the differential equation , where m grams is the mass of the rodent t months after birth. Find the mass of the rodent when fully grown.
• A botanist makes a hypothesis that the rate of growth of hothouse plants is proportional to the amount of daylight they receive. If t is the time in years after the shortest day of the year, the length of effective daylight is given by the formula 12 - 4 cos 2π t hours. On the shortest day the height of one plant is measured to be 123.0 cm; 55 days later the height is 128.0 cm. What will its height be on the longest day off the year six months later?
• Gardeners are concerned about the spread of a species of beetle. All the specimens detected so far lie within a circular region of radius 25 kilometers, and it is suggested that the increase of the radius r kilometers might be modelled by a differential equation , where t denotes the time in months. What does this model predict for the radius of the region colonised by the beetle after t months?
• A hot air balloon can reach a maximum height of 1.25 km, and the rate at which it gains height decreases as it climbs, according to the formula , where h is the height in km and t is the time in hours after lift-off. How long does the balloon take to reach a height of 1 kim?
• When a ball is dropped from the roof of a tall building, the greatest speed that it can reach(terminal speed) is u. One model for its speed v when it has fallen a distance x is given by the differential equation , where c is a positive constant. Find an expression for v in terms of x.
• A steel ball is heated to a temperature of 700 degrees Celsius and dropped into a drum of powdered ice. The temperature falls to 500 degrees in 30 seconds. Two models are suggested for the temperature, T degrees, after t seconds:
• the rate of cooling is proportional to T
• the rate of cooling is proportional to T1.2
It is found that it takes a further 3 minutes for the temperature to fall from 500 to 100 degrees. Which model fits this information better?

### Integration by substitution

• Direct substitution
• Definite integrals
If x=s(u), then
where g(u) = f(s(u)), and p = s-1(a), q = s-1(b)
• Reverse substitution
If u=r(x), and if g(r(x)) = f(x), then   is equal to , with u replaced by r(x).

## Matrix

• elements: individual numbers in the matrix
• order p x q
• rows p
• columns q
• Scalar multiplication

• Matrix multiplication
(row x column)

• Identity matrix

IX = XI = X

• Zero matrix

A+O = O+A = A

For a function

• Determinant
• Inverse matrix
AC = CA = I
AA-1 = A-1A = I

## Vectors

• Displacements in the plane in 3-D
• Components of a vector
• Scalar product
• Perpendicular vectors v . w = 0
• Parallel vectors v . w = |v||w|
• Magnitude
• Unit vectors
• Position vectors

## Syllabus

 Saturni Februarii session Date Topics Assignment 2/7 (Sat.) Calculus overview - Graph transformation - Quadratic functions - Factorising techniques - Exponentiation - Investigating shapes of graphs - Second derivatives Review exercise 1(Pg.64-66) Review exercise 4(Pg.248-250) Review exercise 6(Pg.358-360) 2/14 (Sat.) Calculus overview - Circular/Trigonometric functions -Exponentials & Logarithms Review exercise 8(Pg.461-464) 2/21 (Sat.) Chain Rule - The chain rule - Circular function differentiation - Integration of products TBD 2/28 (Sat.) Integration by substitution - Integration by substitution - Curves defined implicity N/A TBD Integration by parts - Inverse circular functions -Wrap-up N/A
 term3 Date Topics Assignment 1/2 (Fri.) Integration by Substitution - Differential equations - Curves defined implicitly - Inverse circular functions Review exercise 8(Pg.381-384) 1/3 (Sat.) Probability Distribution - Discrete probability distribution - Central limit theorem - t-distribution Review exercise 9(Pg.510-514) 1/9 (Fri.) Hypothesis Testing - Significance test - Chi-square test - Linear regression N/A

 - term3 Date Topics Assignment 1/2 (Fri.) Calculus Extension - Product & quotient rules - Circular function - The chain rule - Integration Review Exercise 4(Pg.209-214) 1/3 (Sat.) Vectors and Matrices - Scalar products - Sets of linear equations - Lines in 3-dimensions - Inverse matrices - Determinants Review Exercise 9(Pg.422-425) 1/7 (Wed.) Probability Distribution - Probability concepts - Discrete probability distribution - Central limit theorem - Correlation Handout 1/9 (Fri.) Financial Math - Interest rate calculation - Time value of money TBD 1/10 (Sat.) Special topics of interest N/A

 term2 Date Topics Assignment 12/19 (Fri.) Visualization - Graph transformation - Quadratic functions - Factorising techniques - Exponentiation - Investigating shapes of graphs - Second derivatives Review exercise 1(Pg.64-66) Review exercise 4(Pg.248-250) Review exercise 6(Pg.358-360) 12/22 (Mon.) Differentiation - Introduction to fundamentals of differentiation - Tangents and normals - Area calculation -Exponentials and logarithms Review exercise 8(Pg.461-464) 12/26 (Fri.) Calculus Extension - Differentiation of exponentials and logarithms - Differentiation of products and quotients - Volumes of revolution TBD 12/27 (Sat.) Chain Rule - The chain rule - Circular function differentiation - Integration of products TBD 12/30 (Tue.) Integration by substitution - Integration by substitution - Curves defined implicity - Inverse circular functions Wrap-up N/A

 - term2 Date Topics Emphasis 12/18 (Thu.) Functions - Quadratic functions - Exponential functions - Circular(Trigonometric) functions Visualization Plotting a function Application problems 12/23 (Tue.) Introduction to Differential Calculus - Rational functions - Logarithmic functions - Gradient - Differentiation Understanding concept of gradients Application of differentiation in other fields 12/24 (Wed.) Geometry and Trigonometry - Right-angled Trigonometry - Geometry of solids Calculating areas & volumes 12/27 (Sat.) Statistics - Review of probability concepts - Sampling - Interpretation of data - Hypothesis testing Linear regression analysis Significance test 12/29 (Mon.) Matrices and etc. - Mathematical induction - Inverse matrices - Determinants Matrix concept and transformation Logical reasoning Wrap-up

 Date Topics Remark 12/12 (Fri.) Ice-breaking Diagnostics  Numbers & Algebra Units 1-2 12/13 (Sat.) Sets, logic and probability 12/15 (Mon.) Functions 12/16 (Tue.) Trigonometry 12/17 (Wed.) Review