What is maths like at university?

The main difference is that there's much more emphasis on proof. For example, at A Level, you might be asked to difference $x^3$ but at degree you would have to proof that $3x^2$ is actually the derivative. Likewise, you may have to prove that the sequence $(a_n) = \frac{1}{n}$ tends to zero and that a matrix is only invertible if it's determinant is non-zero etc.

how would you do that, out of interest

how would you do that, out of interest

Well firstly let's call $f(x) = x^3$.

The definition of $f'(x)$ at some point $x_0 \in \mathbb{R}$ is $f'(x_0) = \displaystyle\lim_{x \to x_0} \dfrac{f(x)-f(x_0)}{x-x_0}$. Now all you have to do is substitute $x^3$ in for $f(x)$ and $x_0^3$ in for $f(x_0)$ and do a bit of cancelling.

I'm sure there's an even more rigorous way of doing it than this but I have done it yet.

I teach that in C1 .....

I teach that in C1 .....

Well firstly let's call $f(x) = x^3$.

The definition of $f'(x)$ at some point $x_0 \in \mathbb{R}$ is $f'(x_0) = \displaystyle\lim_{x \to x_0} \dfrac{f(x)-f(x_0)}{x-x_0}$. Now all you have to do is substitute $x^3$ in for $f(x)$ and $x_0^3$ in for $f(x_0)$ and do a bit of cancelling.

I teach that in C1 .....

I've always wondered about this after hearing my teacher say its very different from what it is in A-level . I've gone through some prospectuses, in terms of the listed topics, nothing seems out of the ordinary. Anyone care to enlighten me on this?

It's awesome.

Maths at A-Level I found incredibly dull - not because the material is that bad, but because nothing is ever really explained properly.

If you want, you can concentrate your degree on really understanding why the things everyone else takes for granted (calculus, algebra etc.) really works (that's what a 'pure' course is really about).

The great thing about a maths degree is that if that isn't really your thing, your focus can be entirely different. There are more than enough courses on mathematical physics, or statistics, or even plain old methods (that's where you learn to apply more complicated techniques but don't really have to prove they work) to get your fill. And towards the end of your degree you can check out courses in optimisation, mathematical finance, biology etc. etc. if you want to apply stuff you've learnt to other areas.

Its very different from an A-Level but in a good way. I absolutely loved my degree, and there's no reason why it shouldn't be the same for everyone else here too. Don't be fooled by course titles such as 'Calculus' or 'Algebra' or 'Mechanics'. Its definitely not just more of the same

Browse this or this for plenty of examples (as a flavour, you might also want to look at stuff which you'll recognise from A-Levels)

See also a brief list of stuff for uni maths I wrote elsewhere.

You'll also find a lot of stuff which just looks like gobbledegook at the moment - and some courses where you won't even understand what a course is if you just look at the name (does Galois Theory mean anything to you?) But of course, that's all part of the fun. Of course, if you end up in a maths degree and after the end of the course it still looks like gobbledegook, that's when you should be worried

In case anyone's curious, here's what some university exam questions look like

I teach that in C1 .....

Yeah and it's not a proof, it's a heuristic jusification.

It's not a proof because you haven't defined what exactly you mean by "limit". At university mathematics two steps would be taken:

(1) Define rigorously what you mean by "limit"; and
(2) Prove, under this definition, that the limit of (f(x)=f(y)/x-y), as x tends to y, "the derivative of f(x) at y", both exists (limits don't always exist) and is equal to what you say it is equal to. In this case, you are saying that is equal to the function 3x^2.

Even if you have taken step (1), your proof isn't a proof. The generally agreed upon definition of "limit" refers to epsilons and deltas. Here epsilon and delta are variables representing positive real numbers, best thought of as small positive real numbers. Your proof doesn't actually mention anything in the nature of epsilons or deltas.

And you can't prove a theorem if you do not mention part of the statement of the theorem.

Your proof doesn't actually mention anything in the nature of epsilons or deltas.

I haven't given a proof .....

I think we've established that.