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Re: HSC 2016 4U Marathon You are ignoring the possibility that both roots of the quadratic are real and do not coincide, in which case they are not complex conjugates of each other. (Also, you are allowing for the discriminant to equal 0 in the first part but not the second, is there a reason...

Re: MATH1231/1241/1251 SOS Thread Here is a fun question to think about for those who like algebra. (Not an ideal thread to post this in, but first years doing some linear algebra have learned enough to approach this question...it came up in a discussion in a tute I was teaching at ANU. The...

Re: First Year Uni Calculus Marathon The FTC applicable in this case (the first FTC according to wikipedia, although nomenclature varies) tells you that if f(t) is a continuous function on [a,b], then F(x):=\int_a^x f(t)\, dt is differentiable in (a,b), with F'(x)=f(x). I.e all...

Re: MATH1231/1241/1251 SOS Thread $By direct computation: $\\ \\ (U^TU)_{ij} =u_i\cdot u_j =\delta_{ij}\\ \\ $the last equality following from orthonormality.$

Re: MATH1231/1241/1251 SOS Thread Short answer: The Taylor series of a function is a specific power series associated to a given function centred at a given point. A power series is a more general concept and need not be related to a function in the same way that a Taylor series is. Related...

Re: First Year Uni Calculus Marathon The limit you have written in this post is not true, the numerator tends to 0 and the denominator is a nonzero constant, so the fraction you have written converges to 0.

Re: HSC 2016 4U Marathon Well inequalities are usually considered part of Harder 3U, so I would classify them as such. As for what tools from elsewhere in the syllabus are useful to attack these questions, calculus is always good to keep in mind, and the presence of symmetric sums in the...

Re: First Year Uni Calculus Marathon Why have you removed the lim? You haven't evaluated any limits, your expression still depends on n. To evaluate the limit think about what happens to each term and group them accordingly, before making use of basic trig limits such as the high school...

Re: HSC 2016 4U Marathon Higher really*, but some four unit students would definitely be able to do them without assuming anything wildly out of syllabus. (*) By this I mean that the questions as stated would never be on an official MX2 exam because of the lack of guidance, but if one was to...

Re: First Year Uni Calculus Marathon So do that, and see where it leads.

Re: First Year Uni Calculus Marathon So as I said, use the "hint" provided in the question to evaluate the sum and then calculating the limit will be fairly straightforward.

Re: HSC 2016 4U Marathon Has been a while since inequalities have been posted, so here are a couple: $1. Let $n$ be a fixed positive integer, let $x_1,x_2,\ldots,x_n$ be non-negative real numbers and for positive real $p$ define:\\ \\ $\mathcal{A}_p:=\left(\frac{1}{n}\sum_{j=1}^n...

Re: First Year Uni Calculus Marathon sin(x) < x for positive x is a pretty well-known inequality. Depending on your way of defining the sine function, this inequality might either be obvious, or just a consequence of integrating the inequality 1-cos(t) >= 0.

Re: First Year Uni Calculus Marathon...

Re: First Year Uni Calculus Marathon The second equation in your last page should have a +, not a -. You have also lost a pi somewhere as it will appear in the argument of all four trig functions. These mistakes aside, you can make this calculation work. Notice that when you are summing...

Anyone for an ongoing tournament/league? This thread hasn't been active for a while, but I could manage an ongoing tournament that requires you to play your paired opponent at whatever time suits you during the week. Am thinking a double elimination bracket for the sake of there being a grand...

Fair enough, it is pretty much just application of the definitions: T_2(t)=x'(t)=B_1(t) and hence x moves at unit speed in the direction of the Binormal vector of r at time t. To compute the curvature of x, as usual we differentiate T. So k_2(t)N_2(t)=T_2'(t)=B_1'(t)=-t_1(t)N_1(t) where N_1...

What are you finding confusing? You are given an explicit expression for your smooth curve x, so you just need to compute the Frenet data as per the definitions. You will end up with something like: T_2=B_1 N_2=-N_1 B_2=T_1 k_2=t_1 t_2=k_1. Where T,N,B,k,t denote the tangent...

Re: MATH1231/1241/1251 SOS Thread They really aren't hard to remember.

You should probably post the full question / your proof attempt if you want a completely accurate critique here.