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
 
   

 

AP and GP

A brainteaser about the sum of an A.P.

Part I

In a kingdom far, far away, there once lived a King and 10 wealthy jewelers, very well-known in their trade. Every year the 10 jewelers paid tax to the King, each gave the King a stack of 10 gold coins. The real coins weighed exactly 1 oz each. This year the King received report that one and only one stack contained 10"light" coins, each having exactly 0.1 oz of gold shaved off the edge. The King now ordered his personal adviser, Mr. Fischer, to IDENTIFY the crooked jeweller and the stack of light coins with JUST ONE SINGLE weighing on a scale.

It took Mr. Fischer, intelligent as he was, the whole afternoon to think of a perfect solution. Even then, he wouldn't reveal much to his aides. All he asked them to do was label the stacks of the coins 1, 2, 3, ..., 10. Take one coin from stack 1, two coins from stack 2, three coins from stack 3, four coins from stack 4, and so on up to stack 10.

"Weigh the coins you just collected from the stacks," he instructed his aides.

"54.3 oz was the reading, sir!" The aides said after the ONE and ONLY ONE weighing allowed by the King.

"Very well, the crooked jeweler was the who gave stack number ...!" Mr. Fischer whispered to himself.

"What did you say, sir?" The aides were eager to know.

"See how many coins you just weighed and you will know which stack was faulty!" Mr. Fischer wanted to test his aides.

 
Now which stack was the one from the crooked jeweller?

 

Part II

 

The King was so impressed with Mr. Fischer's many achievements (finding the crooked jeweller was just one) that he told Mr. Fischer,"Ask me for anything you want. Whatever you ask I will give you, up to half my kingdom!"

 

Mr. Fischer immediately took the King to the royal court whose marble floor was exactly an 8x8 chessboard. "Your majesty, if you so wish, please ask one of your servants to put one grain on the first square, two grains on the second square, four grains on the third square, eight grains on the fourth square and so on, doubling the amount of the grains when it comes to the next square until the last square of the chessboard is filled, then let your humble servant have all the grains placed on the chessboard!"

 

The King was again very pleased with Mr. Fischer. "How modest this Mr. Fischer is! How considerate! He could have asked a lot more!"

The King was about to grant Mr. Fischer what he asked for. Nonetheless, reason got the better of him. As a mere formality, the King summoned his book-keeper, Mr. Anderson, to calculate how much all this would cost him. When Mr. Anderson came back and showed the King his calculations, the King simply couldn't believe his eyes. Why?

 

Can you calculate how many grains there are on the chessboard?

(At that time, annual grain production of the whole world is roughly 10 16grains.)

 

What should the King do with this Mr. Fischer? Should he honour his word and let Mr. Fischer be his biggest creditor or should he take back his word and ...?

 

Part III

 
Using similar calculations, which of the following is the best deal for Tom's pocket money for the next 2 weeks?
(a) $ 7 every day
(b) $ 1 for the first day, $ 2 for the second day, $ 3 for the third day, ...
(c) 1 c for the first day, 2 c for the second day, 4 c for the third day, 8 c for the fourth day, ...
Afterthought
(1) How different are (a)(b)(c) if the period of 2 weeks is extended to a month?
(2) How will (a)(b)(c) look in a graph of total amount of pocket money vs time? Each of them represent a standard type of growth, can you name all of them?

 

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!

(A) Let's start with an easy one first.

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.

(3) A full explanation Does the answer agree with your intuition?

   

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

(1) A computer simulation for the Monty Hall problem - "Let's make a deal" applet
  just scroll down the page until you see 3 large doors #1, #2 and #3. Try the simulation repeatedly with your friends : one using the always STICK door strategy and the other using the always SWITCH door strategy. Do the statistics shown in small print below the 3 doors deviate significantly between the 2 strategies? Is there a clear winner and does that agree with your intuition?
   
(2) The winning strategy is ... with FULL explanation below
  There is a 1/3 chance that you'll hit the prize door, and a 2/3 chance that you'll miss the prize. If you do not switch, 1/3 is your probability to get the prize. However, if you missed (and this with the probability of 2/3) then the prize is behind one of the remaining two doors. Furthermore, of these two, the host will open the empty one, leaving the prize door closed. Therefore, if you miss and then switch, you are certain to get the prize! Summing up, if you do not switch your chance of winning is 1/3 whereas if you do switch your chance of winning is 2/3! More discussion
   
(3) History of the Monty Hall problem
 

This problem has aroused a heated debate for quite some time when it first appeared in 1991 and has never failed to vex and amaze people with its counter-intuitive solution ever since.

In 1991, Marylin Vos Savant** received the Monty Hall problem from Craig. F. Whitaker (Columbia, MD). She carefully explained the logic of the correct solution in a number of subsequent columns, but never completely convinced the doubters. Marylin's response caused an avalanche of correspondence, mostly from people who would not accept her solution (particularly advocates of "50-50" school). Eventually, she issued a call to Math teachers among her readers to organize experiments and send her the charts. Some readers with access to computers ran computer simulations. At long last, the truth was established and accepted. The matter continued at such length that it eventually became a notable news story in the New York Times and elsewhere.

**Marylin Vos Savant ran the popular "Ask Marylin" question-and-answer column of the U.S. Parade magazine. According to Parade, Marilyn vos Savant was listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ" with IQ score of 228.

 
   
 

 

STATISTICS

 

DATA

  • collection
  • organisation
  • presentation
  • analysis
 

Histogram - real data Interactive bar chart

 

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