No.35

Graphing polar equations
Limacons 

, where a and b are NOT equal to 0

• If  the graph of the Limacon has an inner loop!
• If  the graph of the Limacon is "dimpled."
• If the graph of the Limacon is considered "convex."

The graphs of Limacons are generated as the angle increases from 0 to 2.

a.   Limacon with inner loop: 
Note that a = 2 and b = 3 and 

b.   Dimpled Limacon: 

Note that a = 3 and b = 2 and 

c.   Convex Limacon:  .   Note that the graph is not quite circular!

Note that a = 8 and b = 2 and 

No.34

The polar coordinate system is a two-dimensional coordinate system in which each
point P on a plane is determined by a distance r from a θ point O that is called the pole (or origin) and an angle from a θ direction. The point P is represented by the ordered pair (r; θ ) and r; θ are called polar coordinates.

No.33

Present Value

So $1,000 now is the same as $1,100 next year (at 10% interest).

We say the Present Value of $1,100 next year is $1,000

Because we could turn $1,000 into $1,100 (if we could earn 10% interest).

Now let us extend this idea further into the future ...

How to Calculate Future Payments

Let us stay with 10% Interest. That means that money grows by 10% every year, like this:

So:

• $1,100 next year is the same as $1,000 now.
• And $1,210 in 2 years is the same as $1,000 now.
• etc

In fact all those amounts are the same (considering when they occur and the 10% interest).

No.32

The Standard Deviation is a measure of how spread out numbers are.Its symbol is σ (the greek letter sigma). The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?". The Variance is defined as the average of the squared differences from the mean. 

Example:

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm. Find out the Mean, the Variance, and the Standard Deviation. Your first step is to find the Mean:

Mean =	600 + 470 + 170 + 430 + 300	=	1970	= 394
	5		5

No.31

No.30

The hyperbola is centered on a point (h, k), which is the "center" of the hyperbola. The point on each branch closest to the center is that branch's "vertex". The vertices are some fixed distance a from the center. The line going from one vertex, through the center, and ending at the other vertex is called the "transverse" axis. The "foci" of an hyperbola are "inside" each branch, and each focus is located some fixed distance c from the center.

No.29

An ellipse usually looks like a squashed circle. By placing an ellipse on an x-y graph (with its major axis on the x-axis and minor axis on the y-axis), the equation of the curve is:

x²/a² + y²/b² = 1

(similar to the equation of a hyperbola: x²/a² − y²/b² = 1, except for a "+" instead of a "−")

No.28

A parabola is a curve where any point is at an equal distancefrom:

• a fixed point (the focus), and
• a fixed straight line (the directrix)

• the axis of symmetry (goes through the focus, at right angles to the directrix)
• the vertex (where the parabola makes its sharpest turn) is halfway between the focus and directrix.

No.27

The current pi memory champion is a retired Japanese engineer Akira Haraguchi, who in 2006 recited 100,000 digits in front of witnesses in Tokyo. The event took nearly 17 hours. Previously, Mr. Haraguchi set three earlier records by reciting 54,000 digits in September 2004, 68,000 digits in December 2004, and 83,431 in July 2005. To date, however, the Guinness Book of World Records has not officially verified Haraguchi’s attempts.The current Guinness-record holder for memorized digits of pi is held by Lu Chao, a 24-year-old student from China, who recited 67,890 digits in 24 hours and four minutes.

No.26

Math is the greatest thing that God created. This song is basically about how math is closely related to our daily life.

No.25

Permutation: order matters. Example, 12 candidates running for one president, one vice president and one secretary. There are 12*11*10 ways. Because it asks specific orders.
Combination: order does not matter. Example, 12 candidates running for three positions. How many ways? In this case, since order does not matter, use combination. 12C3.

No.24

Mathematical induction:
Step1. Prove n=1 is true
             Left hand side equal to the first number in the sequence
             Right hand side equal to the number when you plug n=1in the other side of the equation
             If left is equal to right, then the statement is true
Step2. Assume n=1is true. Prove n+1is true
             Left hand side is the sequence when you remove n by n+1
             Right hand side is the sequence when you also plug n+1
             If they are equal, the statement is true. Hence,