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I know this question can be done by elementary means by partitioning various areas, but is it possible to evaluate it using calculus? A= \iint_D f(x_1,x_2) \left | \frac{\partial (x_1,x_2)}{\partial (y_1,y_2)} \right | dy_1 dy_2

Very cool question! I really like the second one. Very original. Will try tomorrow if I get time after school.

Using the Sums/Products and Products/Sums expression I presume?

Do you know any faster ways? Green's Theorem is the fastest that I know so far. For those confused, he used the parametric definition of the ellipse x=acos@ and y=bsin@.

Here is a neat result. Prove that for all natural numbers N, \int_{0}^{\frac{\pi}{2}} \cos^{2n}x dx = \frac{1}{2^{2n}} \binom{2n}{n}\frac{\pi}{2}

I like this question. The hardest part is finding a closed expression for the sum that determines the number of m-gons. I think a good Extension 2 student should be able to do so if they play around a bit with Complex Numbers. I will post a solution later if nobody else does.

No, I'm afraid you are mistaken. We define the Logarithmic function to be the inverse of its equivalent exponential function, but there is no guarantee that it immediately implies an exponential form immediately. For simplicity, I will use the functions f(x)=e^x and its equivalent inverse...

I'll probably come. Can you sign me up for MX1 and MX2? Interested in Q8s. I also want to meet RealiseNothing and barbernator.

I've heard rumours that the Year 12s this year are intending to lock all the toilets and toilet doors except for one bathroom.

Yes.

Can't do that because the numerator is a separate function class from the denominator.

Translation: "Two lines pass through P_0 and Q_0, and are parallel to v and w respectively. Find the shortest distance between the two lines". Recall that the shortest distance between a point P and another line containing point Q and also in the same direction as v, since there are...

Differentiate both sides of what Sy123 has with respect to x, then make d/dx (5^x) the subject.

Never heard of this technique before.

No, there is no asymptote at x=0. The function is well defined there, and is equal to 1.

No, the integral is clearly convergent in the interval [0,1] by the Comparison test, not to mention the fact that it has no singularities.

To really confirm for some classes of functions, say Polynomials, you could actually differentiate the function until it reduces to a constant and by observing whether that constant is positive, negative, or equal to 0, we can immediately observe its 'nature'. We have 3 cases essentially...

Hmm from where did you get this question? Are you sure you have typed it out correctly? I only ask because I've seen you make typos before.

It's the exact same thing, except they used the Weierstrass Substitution formula for the cosine function, instead of the tangent function, as deswa1 did above.

Prawnchip, from where did you get this question? For some reason, it's not working out nicely for me. The usual way would be to let u= \sqrt[3]{\alpha} which implies that \alpha = u^3 then find P(\alpha)=0 but the polynomial transforms to a polynomial of degree six, rather than preserving the...