Wednesday, November 21, 2012

heres a cool trick i learnt regarding infinite decimal numbers - 
ie
3.711711711711711711.....


let S = 3.711711711...
S= 3 + 711/1000 + 711/1000^2 + 711/1000^3....

see the pattern - the "ratio" is 1:1000 each time, each element in the series.

therefore 1000.S = 3711 + 711/1000 + 711/1000^2 + 711/1000^3....
1000.S - S = 3711- 3    (the infinite recurring decimals subtract and are removed from the picture)


999 S = 3708

S = 3708/999 or 412/111

pretty nifty !

Saturday, November 3, 2012

second semesters exams just around the corner ! : O
this had been a really tough semester ! the work isn't too hard - but there is so much of it.

The math has been really fun.

Project/Team Management not so much since my team are a bunch of lazy ass teenagers...sheesh...

Statistics...its ok.


Here's a breakdown of the Maths we've covered.


* Proof of Pythagoras

* Trig ratios
* Natural Numbers, integers, Rational Numbers
* Proof root(2) is not a fraction
* Real Numbers as infinite decimals

* Intervals, closed and open notation

* Domain and range of a function
* Inverse trig functions
* The unit circle
* Elementary trig properties
* Periodicity of sin cos and tan

* Graphing of Functions

* Slope of a line between two points
* Average velocity for positions given in terms of time
* Instantaneous velocity defined as a limit.
* Derivatives, elementary calculus
* Polar coordinates
* Addition formulae for angles, double and half angle formulae

* Addition of two wave functions

* Modulus
* The Product Rule for differentiation of two functions
* The Chain Rule for Differentiation
* Trig identities and Simplifications
* Polynomials / RemainderTheorem
* Factorisation of Polynomials
* Completion of the square

* Leibniz Notation

* Implicit differentiation
* Differentiation of sin(x) and thus other trig ratios including powers and multiple angles
* Complex numbers, modulus of and argument of.
* Properties of modulus and complex conjugate of a complex number
* De Moivere's Theorem
* Square and Cube roots of complex numbers
* Relation to factoring polynomial functions


* Simple harmonic Motion

* Defn of natural log as the area under the graph of 1/x for x> 0
* graphing Exponential functions
* Decay / Half life
* Complex exponentials and their relation to trig functions
* Hyperbolic functions
* Derivatives of the inverse trig functions

* Local and Global maxima and minima, how to find them if they exist
* Vectors in R2 and R3 from both geometric POV and algebraic POV
* Length and dot product of vectors
* Angle between vectors, Orthogonality
* Parallel and perpendicular vectors
* Vector Cross Product
* Equations of lines and planes in R3
* Find a plane through 3 given non collinear points
* Row operations and simultaneous Equations

* Continuity on a point and on an interval.
* Max and min for continuous functions on closed bounding intervals
* Mean Value theorem, and consequence of a positive derivative 
* Rolle's Theorem as a special case
* Verbally posed max min problems
* Snell's Law
* Intermediate Value Theorem
* Approximate Solutions by repeated bi-sections
* Distance between two parallel lines.

* Indefinite integration - example of falling body under constant gravity
* Differential eqn's for Projectiles
* Geometric Sequence and its limit
* Summation
* Parametric curves in the plane
* Slope and equation of the tangent line
* Power Series
* Gregory Series
* Maclaurin Series
* Binomial Series
* Escape Velocity under the inverse square law of attraction

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