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$\noindent It's just asking for the area under the graph of $y=\sin ^{-1} x$ between $x=0$ and $1$. So the answer is $A = \int _{0} ^{1} \sin ^{-1}x \, \mathrm{d}x$. One way to compute this integral is via \textsl{integration by parts}. This is beyond HSC 3U though. The 3U way to do this...

Re: MATH1231/1241/1251 SOS Thread $\noindent Any old set of five vectors in $\mathbb{R}^{5}$ need not be a basis. E.g. if the set contains the zero vector, it won't be a basis (because it's automatically now linearly \emph{de}pendent).$ $\noindent The maximum number of linearly independent...

Re: MATH1231/1241/1251 SOS Thread $\noindent Chuck those polynomials as vectors into a matrix (as the columns of the matrix). Get the matrix into row-echelon form. The vectors (polynomials) will be spanning if and only if the row-echelon form has no zero rows (i.e. every row has a pivot...

Re: MATH1231/1241/1251 SOS Thread $\noindent Recall that we let $x = 2\tan u$ for the substitution. Therefore, $\tan u = \frac{x}{2}$.$

Re: MATH1231/1241/1251 SOS Thread Pythagoras's theorem. Alternatively, recall the identity tan^2 (u) + 1= sec^2 (u).

I think he said that's the only Physics teacher at his school.

Re: Discrete Maths Sem 2 2016 g) The upper bounds of {2, 9} are by definition numbers in the poset that are multiples of both 2 and 9. These are: 36, 54, 72. So U := {36, 54, 72} is the set of upper bounds of {2, 9} here. If there is to be a least upper bound of {2, 9}, by definition it must...

Re: MATH1231/1241/1251 SOS Thread As long as you have the same rows, you'll end up at the same answer (because then you're starting off with the same set of simultaneous equations, just written in a different order, which clearly won't affect the solutions).

Those part c)'s are asking for separate answers for each part. The conditions don't hold at once, they are separate Q's.

Re: MATH1231/1241/1251 SOS Thread In other words, to show a set S is closed under addition, it means we have to show that if a and b are arbitrary elements of S, then so is a+b. To show the set U is closed under scalar multiplication, it means we have to show that if c is an arbitrary scalar...

Re: MATH1231/1241/1251 SOS Thread $\noindent In fact for \emph{any} $m$-by-$n$ matrix $A$, that set (with $\mathbf{x}$ coming from $\mathbb{F}^{n}$, where $\mathbb{F}$ is the field of scalars) is a subspace of $\mathbb{F}^{m}$. This subspace is the \textsl{image} of $A$ (alternatively called...

Re: MATH1231/1241/1251 SOS Thread $\noindent Note$ $$\begin{align*}\lambda \bold{0} + \lambda\bold{0} &= \lambda \left(\bold{0} +\bold{0}\right) \quad (\text{distributivity axiom}) \\&= \lambda \bold{0} \quad (\text{as }\bold{0}+\bold{0} =\bold{0} \text{ due to definition of zero vector}) \\...

I don't think there's really any difference between 'show' and 'prove'. And I think you'd need to prove your claim about the pattern for k. This can be done using induction.

The "there is a pattern" is basically proved using induction (just saying without proof that there is a pattern is kind of assuming the result). And induction is a valid proof method, so you should be allowed to use it (even in the HSC you can I think).

Re: MATH1231/1241/1251 SOS Thread What other method do you have in mind? (Induction helps takes care of it nicely if you want to prove it straight from the axioms.) $\noindent (The statement being asked to prove is not just true by definition like say the statement $nm = \underbrace{m+ m +...

Don't know what method you used to prove f(y) = 2tan(a), but it doesn't look like you assumed the result to do so. So what you did should be correct.

It's pretty stupid if they don't give you the marks just because you did "RHS = LHS rather than LHS = RHS" (but some markers are like this I think unfortunately). Did you assume the result that needed to be proved in doing your proof? That's something that cannot be done for a valid proof.

Re: Multivariable Calculus $\noindent Note $A = \frac{1}{2}ab \sin C$. Percentage change is given by $\frac{\Delta A}{A}$. Note $\Delta A \approx \frac{1}{2}b \sin C \Delta a + \frac{1}{2}a \sin C \Delta b + \frac{1}{2}ab \cos C \Delta C$ (total differential approximation). So dividing through...

Re: First Year Uni Calculus Marathon Basically use induction and the axioms of commutativity and associativity of the addition operation.

$\noindent I assume you got a) out (it's just using the principle of rate of inflow minus rate of outflow).$ $\noindent Now for b), for simplicity, let $a=0.03$. We prove by induction that $m_k = \frac{50a^k}{k!}t^k e^{-at}$ for all $k=0,1,\ldots, n$. The result holds for $k=0$, because we have...