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Re: Discrete Maths Sem 2 2016 There are 7C3=35 elective combinations possible. There are 200 students. 200/35 > 5 => At least one elective combination must be taken by more than 5 students. (I.e at least 6 students.) Not much inspiration to give, just identify the pigeonholes based on what...

I don't have the time to carefully write out a full solution right now, which part of my outlined process are you struggling with?

Integrand's explanation above is the general procedure. There are often shortcuts you can take by noting the structure of the inequalities and the geometry of the domains. a) Since the first domain is a rectangle and the function is a sum a(x)+b(y), we can find maxima/minima by...

You should make a habit of checking dimensions to check that your answer makes sense. An intercept of plane is a real number (the ordinate at which the plane cuts the axis), the thing you wrote is a vector. This is how you would do the question posted...

Re: First Year Uni Calculus Marathon Originally I did want that (and to achieve this you can prove that the first thing is monotonic in n and bounded, which establishes convergence), but there is nothing logically wrong with skipping this step though so I cannot fault a proof that does not...

Re: First Year Uni Calculus Marathon Yep :), almost all of it is correct and exactly what I was looking for. Bounding that first expression above by 3^n does not achieve much since 3^n is unbounded, but the convergence of the first expression follows from the application of the monotone...

Re: First Year Uni Calculus Marathon This is actually a bit circular. How are you defining e^x and log(x) if not through one of these limits? In which case you need to prove existence of the limits / differentiability etc before you can invoke something like L'Hopital. To avoid such...

Re: First Year Uni Calculus Marathon ^ Bunch of unanswered questions of varying difficulty.

Re: First Year Uni Calculus Marathon Pretty much just the definition of Riemann integrability (See https://en.wikipedia.org/wiki/Riemann_integral) and basic properties of limits. It is not a difficult question in terms of technicality.

Re: First Year Uni Calculus Marathon Sure, that is one way to do it. (In exactly the same way the discrete weighted Jensen's proves discrete weighted AM-GM). I would expect in such an answer that a student would prove the continuous form of Jensen's though, as the main emphasis of the exercise...

Re: First Year Uni Calculus Marathon It is good practice to try to prove (weighted) AM-GM in this setting. Suppose f,w are non-negative functions on [0,1] that are continuous (or more generally just Riemann-integrable) and suppose that w has integral 1 on this interval. Prove that: \int_0^1...

Not that the above is not quite proving the full 0/0 case of L'Hopital, it is proving the weaker assertion that if f,g are diffble on an interval containing c and g'(c) is nonzero then f(x)/g(x)-> f'(c)/g'(c). The usual formulation of L'Hopital's is that if f,g are diffble on a interval except...

First one: 2 tan(t)/t -> 1 as t -> 0. This is a direct application of that. Second one: 0 e^x grows much faster than any polynomial as x gets large, so the denominator "wins". To prove this using L'Hopital, just differentiate numerator and denominator twice so your numerator becomes a constant...

Re: HSC 2016 4U Marathon - Advanced Level Good stuff :). (Although the denominator should be d+1 in your final answer to 4. Similarly, you have lost a constant factor on your way to the final expression in 3, although of course this does not affect the proof.) Ps, the expression you obtained...

Re: HSC 2016 4U Marathon It is a true statement, but then that sweeps the issue under the rug, as it is not obvious to someone who has not studied some analysis that a) the real exponential would have an analytic extension or b) there is a unique such extension. In other words, there is...

Re: HSC 2016 4U Marathon For this question to be well-posed, you very much need to specify what you mean by the quantity "e^(ix)". I.e. what is your definition of the quantity you are asserting to be equal to cis(x)? You either need to define what it means to raise a real number to a complex...

You could ignore most of the text in the question. All that matters is the recurrence: f_0(x)=1\\ \\ f_{n+1}(x):=\int_0^x f_{n}(t)^3+f_n(t)^2+f_n(t)\, dt The question is literally just to compute f_1,f_2,f_3 which amounts to expanding large polynomials and integrating. f_3 is something like a...

This is just quite tedious, not challenging.

Hsc physics is closer to hsc english than it is to any math course these days.

Re: HSC 2016 4U Marathon I know, but the question asks you to find N&C conditions on coefficients for: a) single root on unit circle b) both roots in unit disk. You found this condition by assuming that the roots are of the form r,conj(r), and that the coefficients of the polynomial were...