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You write "without loss of generality..." (or "wlog" if you want to be really terse) at the beginning of the proof. It means that whichever of the two positive constants is the larger, we shall relabel it a, and the other b.

Sounds perfect to me. The whole idea of the tangent is that it is the limit of a series of secants.

Re: Distinction Courses: Interest and discussion on the proposed cancellation of cour And how much public funds are squandered on a very, very small number of students in the AIS, I wonder? Alot more than $3000 each. But this is Australia, where elitism in sport (which ultimately benefits...

I think the middle terms form a GP with a= 450 x 1.03^(n-1) and the common ratio, r = 1.02/1.03.

I always knew that thing as Euclid's proof, rather than infinite descent. It is pretty much the same thing though. I suppose you could also do another proof where you invoke the Fundamental Theorem of Arithmetic (a.k.a. Prime Factorisation Theorem), but it would basically be the same as...

I think the question is correct - if a polynomial of order 2 has 3 distinct zeros then it must be the zero polynomial, hence all coefficients are zero.

One of the weirder suggestions that i have read for extension activities for gifted students is to get them to write essays on the topic "In what ways are transcendental numbers like running water?" (This was supposed to be for year 8 kids, or something like that.)

There are many rational numbers that do not have terminating decimal expansions.

When figuring out significant figure problems, zeros count when they are between non-zero numerals. One of the best ways to answer sig fig questions is to convert your answer into scientific notation.

I doubt you would be asked to use Fourier series to prove the irrationality of pi. I think the irrationality proof for e that you are talking about used the Taylor series expansion for the exponential function. When doing calculus, you should always assume that angles are measured in radians.

To get 100K at uni you would have to be an associate professor or higher. That won't happen until 15 or 20 years of work as a lecturer. The pay for a level A academic (i.e., starting pay) is almost as bad as a teacher's, but without any job stability - most of them are on post-doctoral...

Yeah, teaching at UNSW is real bad. And not about to get better in a hurry, either. The other half of the double degree might be OK, though. You could also consider UTS - I don't know much about their teaching degree. And MacQuarie is supposed to be particularly good at special ed, if that's...

UNSW never does any more practicum than the bare minimum required by the NSW Institute of Teachers. And they do it really late in the course in one big block at the end of 4th year. Have you considered some of the other unis as well? I think ACU and Uni of Wollongong have a good reputation...

If that is the case, then you need to get a new on-line tutor. Preferably one who actually knows some mathematics.

tan x = sin x / cos x, so if tan x - 2 sin x = 0 then sin x/ cos x - 2 sin x = 0 sin x(1/cos x - 2) = 0, so either sin x = 0 or cos x = 1/2. So x = 0, pi, 2pi or x = pi/3, 5pi/3.

I can see your mistake - you should have seventeen terms to sum, not sixteen, because the mother made the first deposit on the 0th birthday, not the first birthday. In this case, the sum is 8799.061333, not 7695.590476 (according to my computer).

How did you answer the third part? I calculated the probability of A winning within 4 turns by adding the probability that she wins on the first turn + probability she wins the second (obviously, this means that nobody wins the first round) + probability she wins the third turn. Now if you do...

Try Wolfram MathWorld (probably better for definitions) http://mathworld.wolfram.com/ or wikipedia (seems to have more stuff on characteristics) http://en.wikipedia.org/wiki/Rhombus

A monic polynomial has a leading coefficient of 1, not -1. You may wonder why this is the case, but it is defined that way to make some of the theorems simpler. For example, when you define the GCD (greatest common divisor) of 2 polynomials, you want to get a unique answer, so the GCD of two...

Well, e^(-x^2) is an even function, so if one of the corners of the rectangle was at the point (a,0), then there must be another corner at (-a,0), otherwise the shape won't be a rectangle. In this case, the area of the rectangle is 2ae^(-a^2) (assuming a>0).