High School Mathematics

Revision Lecture 1: Sep 7 2010

Table of ContentsGeneral Mathematics• Power and Logarithms• Numbers and Polynomials• Simple Identity and Quadratic Equations• Number sequences, AP, GP, summation

Additional Mathematics• Mathematical Induction• Limits and Derivatives• Integration

Skip Linear Algebra, Probabilities, Geometry, Complex numbers.

Power

a picture of 2x

(source: wikipedia – logarithm)

Definition: given integers a and n,

Identities:

What is a0?

Logarithm

Power: given x, compute y

Logarithm: given y, compute x

Logarithm is the “inverse” operation of power.

Given y, find x such that ax = y, the answer is called logay

a picture of 2x

Logarithm

logb(xp) = p logb(x).

NumbersIntegers Z

Rational number Q - a/b for a,b integers

Real number R

Irrational number - real number but not rational number

Complex number – won’t discuss

PolynomialsPolynomial is an expression of one variable and numbers, using only addition, subtraction, multiplication, and non-negative integer exponents.

For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not

Polynomial addition:

Polynomial multiplication:

Polynomial division:

Polynomial EquationsGiven compute x.

(wikipedia – quadratic equation)Exercise: solve

Number SequencesArithmetic sequence: the difference between two consecutive terms is the same

e.g. 2,5,8,11,14,17… 3,10,17,24,31,… 5,2,-1,-4,-7,-10,…

Geometric sequence: the ratio between two consecutive terms is the same

e.g. 1,2,4,8,16,32,64… 3,-9,27,-81,243,… 4,2,1,1/2,1/4,1/8,…

e.g.

Sum of Number Sequences

What is the sum of an arithmetic sequence: a+(a+d)+(a+2d)+…+(a+(n-1)d)?

The answer is

What is the sum of a geometric sequence: a+ar+ar2+…+arn-1?

The answer is

We will prove these in a later lecture. The proofs can also be found here:

http://en.wikipedia.org/wiki/Arithmetic_progression

http://en.wikipedia.org/wiki/Geometric_progression

Fact: If m is odd and n is odd, then nm is odd.

Statement: for an odd number m, mk is odd for all non-negative integer k.

• m is odd by definition.• m2 is odd by the fact.• m3 is odd because m2 is odd and the fact.• mi+1 is odd because mi is odd and the fact.• So mi is odd for all i.

Idea of induction.

Mathematical Induction

Mathematical InductionProve

Basis: Show that the statement holds for n = 0.

Inductive step: Show that if statement is true for k, then it is also true for k+1.

0+1+2+ … +k+(k+1)

= (0+1+2+ … +k)+k+1

= k(k+1)/2 + k+1

= (k+1)(k/2+1)

= (k+1)(k+2)/2

assumption (induction hypothesis)

That is, assuming 0+1+…k=k(k+1)/2, want to prove 0+1+…(k+1)=(k+1)(k+2)/2.

Objective: Prove a statement is true for any non-negative integer

This is to prove

The idea of induction is to first prove P(0) unconditionally,

then use P(0) to prove P(1)

then use P(1) to prove P(2)

and repeat this to infinity…

Mathematical Induction

Don’t worry, we are going to study mathematical induction in details.

Limits and DerivativesGiven a function, want to find the “slope” of a given point.

(Recall that the slope is defined as (y2-y1)(x2-x1).)

Informally, we’d like to determine how y changes as x changes.

f(x)=x2

Say we’d like to compute the slope of x2 at point p1.

Take p2 closer, the answer is better.

So the answer is

Limits and Derivatives

f(x)=x2

We say dx2/dx = 2x

That is, when we increase x by a small number c,then x2 increases by 2c.

In principle, we can do differentiation in this way for every function f.

In practice, we have a table and a set of rules for convenience of computation.

See http://www.cse.cuhk.edu.hk/~chi/csc2110/notes/calculus.pdf

IntegrationThere are two definitions of integration(1) It is the “reverse” operation of differentiation (i.e. undo the differentiation).(2) It computes the area of a function

For (1), there is not much to say, mostly look at the rules and do the calculations.See http://www.cse.cuhk.edu.hk/~chi/csc2110/notes/calculus.pdf

(source: wikipedia – integral)

For (2), we mean given a starting pointand an ending point, compute the area“under” the curve.

As you see in the figure, one can approximately compute the area using smaller and smaller rectangles.

IntegrationWhat is the connection between (1) and (2)?

The answer is: where dF(x)/dx = f(x)

To see why it makes sense, consider the following area that represents your income

f(x)=x

F(x)

To compute the area, one can

plot another graph to do so.

Call this function F(x).the total income after b years is simply F(b).

Integration

f(x)=x

To compute the area, one can

plot another graph to do so.

What is the relation between f(x) and F(x)?

F(x)

The slope of every point

in F(x) is equal to f(x)

e.g. slope at x=5

is equal to 5

e.g. slope at x=4

is equal to 4

Therefore, dF(x)/dx = f(x)

This is why we first compute the “reverse differentiation” F(x)

and then compute the area from under f from a to b by F(b)-F(a).

SummaryYou should remember what you learnt in general mathematics,

especially power and logarithm, polynomial and number sequences.

Mathematical induction will be covered in details in this course.

Differentiation and integration will not be essential in this course,

but they are important in other courses (e.g. numerical analysis,

probability, etc). Tutor will tell you more about how to do calculations.

We just try to give you a very brief idea of what they are. They are not

very difficult once you understood it. In additional mathematics it took

many months to do calculations for more complicated functions, but

these can be picked up gradually.