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The error wouldn't stay the same throughout, it'd have to change. E.g. consider the raising to the k-th power step. $\noindent If we have some quantity $x$ estimated by $\hat{x}$ with error $\Delta x$, we can write $x = \hat{x}+\Delta x$. Then (with $k$ a fixed positive integer say), we have...

If the RHS of the integral approximation line was what you wanted clarification on, that's just the Simpson approximation to the given integral with one parabola: ((b-a)/6) * (f(a)+ 4*f(0) + f(b)).

By the way, seanieg89 wrote a bit about numerical integration, including proofs of error bound formulas for various numerical integral methods and some exercises. It may be found here: http://community.boredofstudies.org/14/mathematics-extension-2/342900/numerical-integration.html, and could be...

Assuming you mean for the error bound, the main thing is to make sure you're clear on what the symbols all refer to. In that particular formula, n refers to the number of parabolas used in the Simpson's Rule approximation. In your one, exactly one parabola is used, so n = 1. Also, a refers to...

Which RHS do you mean? The one in the integral's approximation? Or the one referring to the bound on the Simpson's Rule error?

Did you mean "there does" exist such a method?

You don't need to know how to derive it. If you're curious though, a derivation may be found here: http://www.musada.net/papers/appendix2.pdf .

Re: MX2 2016 Integration Marathon $\noindent Yes: $\int \left(u^\prime v +uv^\prime\right)\text{ d}x =uv + C$.$

Re: MX2 2016 Integration Marathon It's basically $$\int \frac{u^\prime v - uv^\prime}{v^2}\text{ d}x = \frac{u}{v}+\text{constant}$ ($u,v$ being functions of $x$).$

Re: MATH1131 help thread Have you typed the equations correctly? For the first one, you seem to have written + 4*x twice.

Re: MX2 2016 Integration Marathon $\noindent Find $\int _0 ^{2\pi} \frac{x \cos x}{1 + \sin^2 x} \text{ d}x$.$

$\noindent The given statement with only two sets, i.e. $\mathbb{P}\left(A \cap B\right) \geq 1 - \left[\mathbb{P}\left(A^c\right)+\mathbb{P}\left(B^c \right) \right]$, with $A=A_1 \cap \cdots \cap A_{n}$ and $B=A_{n+1}$.$

Re: MATH1131 help thread There shouldn't be an i (the roots are real). Even if there was an i, it wouldn't be something like ((non-zero) real number)*i, unless the equation was of the form where a purely imaginary solution existed, i.e. z^2 + a = 0 for some a > 0 (and the given equation isn't...

Re: HSC 2016 4U Marathon The question as asked requires us to assume U2 = gR, and then prove from this assumption that the projectile reaches the height R above the Earth's surface.

Re: MATH1131 help thread Ah yeah, then the roots are real (no i). What do the answers say?

Re: MATH1131 help thread $\noindent When we take the square root of a negative number, $i$ must appear (the answer has to be purely imaginary). E.g. $\sqrt{-16}=4i$.$

For the inductive step, write it as (A1 ∩ … ∩ An) ∩ A_(n+1). Then apply the two set result to this (can assume using strong induction). Then remember Pr((A1 ∩ … ∩ An)c) = 1 – Pr(A1 ∩ … ∩ An). Then use the n-set result (inductive hypothesis).

$\noindent A useful probability identity is $\Pr \left(A\right) = \Pr \left(A \cap B\right) + \Pr \left(A \cap B^c\right)$ (follows from the Law of Total Probability; using set difference notation, can also be written as $\Pr \left(A\right) =\Pr \left(A \cap B\right) + \Pr \left(A - B\right)$)...

Let B1 be the event that a black ball was transferred from Urn 1 to Urn 2. Let R1 be a similar event for red (red transferred from Urn 1 to Urn 2). Let B2 and R2 respectively be the events that black and red are drawn from Urn 2 after the transfer from Urn 1 has happened. We want to find...

So you basically meant for proving results like Vandermonde's convolution ( https://en.wikipedia.org/wiki/Vandermonde%27s_identity )? You probably should provide justification for your steps in the HSC if the question is asking for a proof of that kind of result.