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...
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 nonzero digits and zeros between nonzero digits, and zeros placed after other digits to the right of the decimal point are significant
 Error = V_{A}  V_{E}
 Percentage error = V_{A}  V_{E} / V_{E} x 100%
Units
Scientific notation a x 10^{k}
 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 = U_{1} + (n1)d
 Arithmetic series: sum of an arithmetic sequence
 Sn(sum of the first n terms) = n/2(u_{1} + u_{n})
 Geometric sequence: a number sequence with common ratio
 common ratio: quotient of current term divided by its previous term
 Un = U_{1} x r^{n1}
 Sn = u_{1}(r^{n}1)/r1
 Simultaneous equations
 method of substitution: express one of the equations for one variable in terms of the other, then substitute into the other equation
 method of elimination: combine two or more equations so that one of the variables can be eliminated
 Quadratic equations ax^{2}+bx+c = 0
 factorizing:
 tip: find set of factors of c, and choose the set which gives sum = b
 Quadratic formula

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 ~
 N_{even} = {2n∈Nn∈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
 Estimate of unknown area = #points in unknown area/ total #points generated x known area
 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(AB) = P(A∩B)/P(B)
 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(AB) = 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 ⇒
 Tautology P∨¬P
 Contradiction P∧¬P
 Implication p⇒q
 Converse q⇒p
 Inverse ¬p⇒¬q
 Contrapositive ¬q⇒¬p
Statistics
 Discrete data counted
 Continuous data measured
 Frequency tables
 Midinterval 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
 Measures of dispersion
 Range
 Interquartile range = Q_{3}  Q_{1}
 Standard deviation
(yȳ) = S_{xy} (x¯x) / (S_{x})^{2}
 ℵ² test for independence
 ρvalue
 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.
 Permutation: order matters nPr = n! / (nr)!
 Combination: order does not matter nCr = n! / r! (nr)!
 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 SafetyFirst criterion
P(Ra < Rm) = [E(R_{P})R_{L}]/σ_{P}
 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
where plugin zvalue to get F(z) from ztable
C_{V} = 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
 timeseries data gathered from each time periods
 crosssectional 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
 Student's ttest
 used when sample size is small or variance unknown
used when H0: σ² = σ_{0}<super>2</super>
 Confidence interval (level of significance:probability of rejecting true H_{0})
 68% of observations fall in ±1σ
 95% of observations fall in ±1.96σ
 99% of observations fall in ±3σ
 Types of Bias
 datamining~ : repeatedly doing tests on same data sample
 sample selection~ : sample not really random
 survivorship~ : sampling only surviving firms
 lookahead~ : using information not available at the time to construct sample
 timeperiod~ : relationship exists only during the time period of sample data
Hypothesis Testing

 H_{0}: hypothesis set up to be nullified or refuted in order to support an alternate hypothesis.
 H_{1}: 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
Quadratic functions
 y = ax^{2} + bx + c
 parabola shape with axis of symmetry
 Completed square form F(x)=a(xr)^{2} + s
 axis of symmetry at r
 check that f(rh) = f(r+h)
Factorising techniques
 3 quick special cases
 No constant term x(xr)
 Difference of two squares (xr)(x+r)
 Sum of two squares (xr)(xr)
 never factorize
 General case hitandmiss process
 Find factors for a and c
 Test for the case matching b
 Transformation into quadratic equation
 Fatorising a polynomial
 Factor theorem For polynomial p(x), if p(t) = 0, then (xt) is a factor of p(x).
 Extended factor theorem If p(t/s) = 0, then (sxt) is a factor of p(x).
 Method of equating coefficients If ax^{2}+bx+c = lx^{2}+mx+n, then a=l, b=m, c=n.
 If a polynomial p(x) is divided by a linear polynomial sxt, the remainder is p(t/s).
Exponentiation
 Power functions f(x) = x^{a}
 Exponential functions f(x) = a^{x}
Investigating shapes of graphs
 f(x) = ka^{λx} + 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
 Find an expression for f'(x)
 List the values of x for which f'(x) = 0
 Find the sign of f'(x) in intervals to the left and to the right of that value
 If these signs are  and + respectively, => minimum point. If + and , => maximum point
 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
 Find f'(x)
 List the values of x for which f'(x)=0
 Find f"(x)
 For each x with f'(x)=0, find the sign of f"(x). If the sign is + => minimum point; if  => maximum
 For each value of x which gives a minimum or maximum, calculate y = f(x)
Differentiation
 Gradient:
 Differentiation: process of finding the gradient formula for a curve
 Derivative: the gradient formula for a curve
 If f(x) = x^{n}, where n is a positive integer, then f'(x) = nx^{n1}
 The derivative of af(x) + bg(x), where a and b are constants, is af'(x) + bg'(x)
 proof f(x)=ax^{2}+bx+c
Integration
 reverse process of differentiation
 Finding a function from its derivative
 ex.> f'(x) = 2x + x^{2}
 Indefinite integral ∫f(x) dx
 Arbitrary constant
 ex.> f'(x) = 6x^{2}  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/s^{2}, 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(30t) m/s^{2}, 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.
 to find the area under the graph f(x) from x=a to x=b:
 Find the simplest integral of f(x), call it I(x)
 Work out I(a) and I(b)
 Area = I(b)  I(a)
 ex. 29.1.2 Find the area between the graph of y=2x  x^{2} and the xaxis.
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/t^{2}. Find how far the aircraft travels in this time.
 ex.> Find the area between y=x^{2} 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)=x^{3}  x^{2}  6x + 8 and g(x) = x^{3} + 2x^{2}  1, intersect at two points, and find the area of the region enclosed between the two curves.
 ex.> 29.7.3 y=3 + 2x  x^{2} Calculate the areas enclosed
Exponentials & Logarithms
 Exponential function f(x) = b^{x}
 where b is a positive real number and base b≠1
 if x is a real number and y is a positive real number
 b^{x} = y is equivalent to x = log_{b}y
 examples
 Properties(p.439441)
 Multiplication rule
 Division rule
 Power rule
 the nth root rule
Chain Rule
Circular functions
Integrating products
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 liftoff. 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 T^{1.2}
 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
 If x=s(u), then
 where g(u) = f(s(u)), and p = s^{1}(a), q = s^{1}(b)
 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
 (row x column)
IX = XI = X
A+O = O+A = A
For a function
 AC = CA = I
 AA^{1} = A^{1}A = I
Vectors
 Displacements in the plane in 3D
 Scalar product
 Perpendicular vectors v . w = 0
 Parallel vectors v . w = vw
Syllabus
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.6466)
Review exercise 4(Pg.248250)
Review exercise 6(Pg.358360) 
2/14 (Sat.) 
Calculus overview
 Circular/Trigonometric functions
Exponentials & Logarithms 
Review exercise 8(Pg.461464) 
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
Wrapup 
N/A 
Higher Level 
term3 
Date 
Topics 
Assignment 
1/2 (Fri.) 
Integration by Substitution
 Differential equations
 Curves defined implicitly
 Inverse circular functions 
Review exercise 8(Pg.381384) 
1/3 (Sat.) 
Probability Distribution
 Discrete probability distribution
 Central limit theorem
 tdistribution 
Review exercise 9(Pg.510514) 
1/9 (Fri.) 
Hypothesis Testing
 Significance test
 Chisquare test
 Linear regression 
N/A 
Standard Level
 term3 
Date 
Topics 
Assignment 
1/2 (Fri.) 
Calculus Extension
 Product & quotient rules
 Circular function
 The chain rule
 Integration 
Review Exercise 4(Pg.209214) 
1/3 (Sat.) 
Vectors and Matrices
 Scalar products
 Sets of linear equations
 Lines in 3dimensions
 Inverse matrices
 Determinants 
Review Exercise 9(Pg.422425) 
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 
Higher Level 
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.6466)
Review exercise 4(Pg.248250)
Review exercise 6(Pg.358360) 
12/22 (Mon.) 
Differentiation
 Introduction to fundamentals of differentiation
 Tangents and normals
 Area calculation
Exponentials and logarithms 
Review exercise 8(Pg.461464) 
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
Wrapup 
N/A 
Standard Level
 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
 Rightangled 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
Wrapup 
Higher Level  term1
Date 
Topics 
Remark 
12/12 (Fri.) 
Icebreaking
Diagnostics
Numbers & Algebra 
Units 12 
12/13 (Sat.) 
Sets, logic and probability 

12/15 (Mon.) 
Functions 

12/16 (Tue.) 
Trigonometry 

12/17 (Wed.) 
Review 

