    how to generate sine, cosine and tangent curves; interactive exercise Definition Draw a circle with cenre (0,0) and radius 1. P is a point on the circle which makes an angle x with the positive x-axis. Idenitify the lengths sinx, cosx, tanx, cotx. Trig graphs the 4th applet the 3rd applet y=Asinx, y=sinx+d, y=sin(x-c), y=sinbx Revision the 3rd applet "recognise functions 3" the 4th applet "recognise graphs 3" Application Polynomials Synthetic Division - finding the quotient and remainder without using long division Application - a suprising application of polynomials in our daily life Proof by dissection - a visual proof of the expansion of the square of (a+b) a brain  Useful websites excellent thinkquest library created by pre-collegiate students

AP and GP

A brainteaser about the sum of an A.P.

Part I  LINEAR PROGRAMMING

 IMPORTANT POINTS TO NOTE IN LINEAR PROGRAMMING 1 Define clearly what x and y represent. Clue can usually be found from the last sentence of the question. 2 Set up appropriate constraints . Some favourite ones are x >= 0 , y >= 0 ( or > 0) x , y are integers (check if applicable) 3 State clearly what is to be maximised or minimised. To minimise P = \$ (3k x + k y) Remember to include k whenever we don't know the exact production cost of X and Y. 4 Select the best scale by examining the constraints. If x >= 0 , y >= 0 3x + y > 240 x + 2y < 300 then the x-axis must include at least up to ____ and the y-axis must include at least up to ____ . 5 Label the x,y-axes and write down the scale of the axes. Draw the corresponding line for each constraint, decide whether it should be a dotted line (if > ) or a solid line (if >= ). Label the line by an equation (e.g. 3x + y = 240). Depending on the inequality sign, add the appropriate arrows and update the region . 6 Shade the feasible region. If x, y are restricted to be integers, make sure you add "solution are points with integral coordinates in the shaded region." 7 To max \$ (k x + 3k y) or max \$ (2x + 6y - 200) . In both cases, we only need to draw x+ 3y = 0 and then parallel lines x +3 y = C. To draw x+3y=0, rewrite as y = -x/3 and plot (0,0) and (3,-1). Note that A,B,C,D are the only vertices for this feasible region. P ,Q, R are not vertices. Why ? Furthermore, if x, y must be integers and the vertex D does not have integral coordinates. Select another point closest to D but with integral coordinates. 8 Lastly, answer the question wisely. Find the number of cars and minibuses that minimise the running cost. The answer should be "5 cars and 8 minibuses" instead of "the minimum cost is \$ 10,000".  PROBABILITY

three interesting problems; Birthday problem, Monty Hall problem and one involving infinite series

The Birthday problem Surprise - Surprise !

A challenge to your intutive sense but quite simple in a mathematical sense!

At least how many people must be gathered so that we can be 100% certain that some of them share the same birthday?

(B) Now comes the serious part ...

How many people should there be in a class so that there is at least a 50% chance that some of them share the same birthday?

Should it be half of what we have in (A) above? How about one third? say 180 people? 120 people? Think about your own class ( around 40 in size ), do you know any 2 people having the same birthday? Ask another class, is it common to have 2 or more people having the same birthday?

Discussion

(1) A computer simulation

(2) How to approach the problem mathematically? (Assume there are only 365 different birthdays.)

 (a) Suppose there are 3 people in a class. Find the probability that (i) all have different probability (ii) at least some have same birthdays (b) Repeat (a) with 5 people in a class. How about n people? Guess how large should n be so that the probability that some of them have the same birthday is greater than 0.5. A probability problem involving an infinite series

2 monkeys throwing coconuts at each other !

The famous Monty Hall Problem *

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

If you were the contestant, which of the following would have a better chance to win the big prize?

Strategy 1 (stick): Stick with the original door

Strategy 2 (switch): Switch to the other door

or it doesn't matter since the two strategies have equal chance of winning the big prize

*This problem was named Monty Hall in honor of the long time host of the American television game show "Let's Make a Deal." During the show, contestants are shown three closed doors. One of the doors has a big prize behind it, and the other two have junk behind them. The contestants are asked to pick a door, which remains closed to them. Then the game show host, Monty, opens one of the other two doors and reveals the contents to the contestant. Monty always chooses a door with a gag gift behind it. The contestants are then given the option to stick with their original choice or to switch to the other unopened door.

Discussion  STATISTICS

DATA

• collection
• organisation
• presentation
• analysis

Normal Distribution (Gaussian distribution)

 Normal distributions are important because a lot of data in real life (e.g. social data) are distributed approximately normally, especially when it involves (1) a large population and (2) the data produced by a member of the population is independent of that produced by another member. Examples : Marks of a Mathematics test of 100,000 candidates Weights of the 5 kg packets of rice in a supermarket IQ score for a large group of people and many more Animation of falling balls by the University of Toronto The normal curve is bell-shaped and is given by the equation Unlike the functions sinx, cosx,… where the primitive functions can be easily found, the primitive function of f(x) cannot be expressed in a closed form (i.e. as a finite number of sum, difference, product and quotient of elementary functions ).The area under the bell curve cannot be found by integration as we know it. Instead, it will be looked up from the z (standard score) table: (1) traditional z table (2) interactive z table With the z table, many examples involving normal distributed data can be solved. More about normal distribution   