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IB Math

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


    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...


    • 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%


    Scientific notation a x 10k


    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
      2. Quadratic formula



    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
    • 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


    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
    • Contradiction P∧¬P

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


    • 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.

        76b133327e4c98581ddb42a557522731.png 2061fa9538c1c341e87863dcdba91233.png

    • 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

                cb45abbc2e224df42bfbbbc0b32c2e89.png  where 71a583026c8839368732382b1af3c818.png  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 = 
    • Gradient(slope)
    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

    V = Area of a base x height / 3



    • 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
    • Zero gradient
    points where f'(x) = 0
    • Local maxima and minima
    • Stationary point
    • Point of inflexion

    Quadratic functions

    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

    • Quadratic formula



    • 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).


    • 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)


    • Gradient:
    • 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


    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
    • Properties(p.439-441)
      • Multiplication rule
      • Division rule
      • Power rule
      • the nth root rule
    • Special bases
      • log base 10
      • natural log


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

    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

    Integration by parts

    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).


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

    • Matrix multiplication
    (row x column)

    • Identity matrix

    IX = XI = X

    • Zero matrix

    A+O = O+A = A

    For a function 

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


    • 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


    Higher Level - 
    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
    2/28 (Sat.) Integration by substitution
    - Integration by substitution
    - Curves defined implicity
    TBD Integration by parts
    - Inverse circular functions
    Higher Level - 
    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

    Standard Level 
    - 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
    1/9 (Fri.) Financial Math
    - Interest rate calculation
    - Time value of money
    1/10 (Sat.) Special topics of interest N/A

    Higher Level - 
    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
    12/27 (Sat.) Chain Rule
    - The chain rule
    - Circular function differentiation
    - Integration of products
    12/30 (Tue.) Integration by substitution
    - Integration by substitution
    - Curves defined implicity
    - Inverse circular functions

    Standard Level 
    - term2
    Date Topics Emphasis
    12/18 (Thu.) Functions
    - Quadratic functions
    - Exponential functions
    - Circular(Trigonometric) functions
    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

    Higher Level - term1
    Date Topics Remark
    12/12 (Fri.) Ice-breaking
    Numbers & Algebra
    Units 1-2
    12/13 (Sat.) Sets, logic and probability
    12/15 (Mon.) Functions
    12/16 (Tue.) Trigonometry
    12/17 (Wed.) Review