Advanced Higher Maths 2012-2013 : Discussion and Help Thread

Hey guys, can anyone explain the sigma notation to me? It keeps coming up in adv maths but my teacher is essentially useless


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For example a sum can be written in the form:

$\displaystyle \sum_{n = 1}^{4} \left(2n + 1\right)$

The bit below the sigma sign is the lower limit, the bit above it is the upper limit, and the bit to the right is called the summand. In the lower limit, in this case $n = 1$, the variable denotes which variable in the summand is to be replaced by some number and this number is taken from the sequence of consecutive numbers that is formed between the lower number (1) and the upper limit (4).

So what this sum says is to count from 1 to 4, and for each number substitute n in the summand by that number and then add all the summands together.

Hence

$\displaystyle \sum_{n = 1}^{4} \left(2n + 1\right) = \left(2 \times 1 + 1\right) + \left(2 \times 2 + 1\right) + \left(2 \times 3 + 1\right) +\left(2 \times 4 + 1\right)$

Ah you're a legend, ukdragon37, what would this thread do without you?

So it's like the integration symbol then, the lowest value on the bottom and the highest on the top?

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Differential Equations:
Missed the damn note on this and subsequently am not quite getting the gist of it:

Oil dripping on a garage floor making ever increasing circular stain.
When radius = 3, the radius is increasing at a rate of 0.2cm/s

Calculate the rate at which the area is increasing at that time

This is related rates. Just different applications of chain and product rules using numbers as well.



Let t be the elapsed time and r be the radius. Hence when r = 3, $\dfrac{dr}{dt} = 0.2$ in cm/s (assuming r is 3cm).



Let A be the area. You want to calculate $\dfrac{dA}{dt}$. This is done by coming up with an equation which connects the unknown dimension (A) with the known dimension (r) and then applying the chain rule to the known facts.

Since the stain is a circle, the area and radius are related by $A = \pi r^2$. There are two ways to proceed that are essentially the same:

1) Differentiating both sides with respect to t, remembering to apply the chain rule to r, gives:

$\dfrac{dA}{dt} = 2 \pi r \dfrac{dr}{dt}$

Substitute in r = 3 and dr/dt = 0.2 gives $\dfrac{dA}{dt} = 2 \pi \times 3 \times 0.2 = 1.2 \pi$ cm^2/s.

2) Differentiate both sides with respect to r gives:

$\dfrac{dA}{dr} = 2 \pi r$

Substitute in r = 3 gives $\dfrac{dA}{dr} = 6 \pi$.

Then $\dfrac{dA}{dt} = \dfrac{dA}{dr} \times \dfrac{dr}{dt} = 6 \pi \times 0.2 = 1.2 \pi$ cm^2/s.

I was wondering, how does STEP compare with first year Cam maths?
I've looked at some of the Syllabus for 1st year and am aware that a lot of 1st year is bringing folk who've not done FM/ AM up to speed.

Another thing, a lot of the stuff the guys are talking about on the STEP thread with integrating using series', which we've not covered yet.

I'm wondering how your STEP prep went even though you'd not covered everything by January (presumably when you started preparing), I'm aware that some of the guys there are on gap years so have covered the course completely.

How's the PhD coming along? Did Cam make you an offer?

what further reading have they been doing? I'd go back and check but it'd take ages to find thee discussion

I was wondering, how does STEP compare with first year Cam maths?
I've looked at some of the Syllabus for 1st year and am aware that a lot of 1st year is bringing folk who've not done FM/ AM up to speed.
Another thing, a lot of the stuff the guys are talking about on the STEP thread with integrating using series', which we've not covered yet.
I'm wondering how your STEP prep went even though you'd not covered everything by January (presumably when you started preparing), I'm aware that some of the guys there are on gap years so have covered the course completely.

Well there's no set progression obviously but you could try reading more in-depth treatments of single topics instead of the general purpose textbook you get for AH. Pick a topic you like and go find a well rated book on amazon; you can read the reviews to gauge whether it's of an appropriate level or not.

STEP is doable without going beyond A-Level knowledge (plus one or two little things) though. You can read the actual spec here. That's not to say additional practice isn't needed, of course.

Are you sitting it, or just interested? Or you won't know for a few days yet?

As ukd says, most of cam maths 1st year is not bringing you up to speed in any way - basically being on top of M+FM is assumed, though you'll get like a week of lectures in some courses that you'll be familiar with. STEP I found pretty easy, 1st year uni I found much harder, but got similar results. It's rather different in style. I'd also say AH to 1st year was an actual jump, whereas I never noticed one in school - take from that what you will!

Basically, to be good at STEP, be very on top of AH, and practice a bunch. You don't really need further techniques for the most part.
If you intend to do STEP III though, there's some stuff involved which isn't in the AH course, so might be worth looking into.

Just thought I'd add another perspective there.

I'm fairly on top of the AH, but things like Laplace I've never heard of and using certain techniques for integration I have literally no idea what they're talking about.

I'm fairly on top of the AH, but things like Laplace I've never heard of and using certain techniques for integration I have literally no idea what they're talking about.

OK, so how did you prepare? Are you a Cam student

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Yes I know that reading much further in depth would increase my knowledge but I'm also looking for some cheeky sledgehammers to crack some of STEP - I think Laplace was one - for a given question, as well as other transforms - but I was wondering where people were learning these, I could go back on the thread and check but it's probably lost in the pages.
I have a few in mind but I'm not sure:
1. Spivak's Calculus
2. Mathematical methods for Physicists and engineers
3. Concise intro to pure maths
I'll be doing it regardless of a Cam offer - I'll be a reapplicant for 2014
Are you a Scottish student there? I've not seen you on this thread.

May i get help with this question? I know it will be simple but i just can't get the correct answer:

Suppose that x and y are differentiable functions of t and that

$\dfrac{d^2y}{dx^2} = t^2 + 1 , \dfrac{dy}{dx} = t^3 + 3t$

Find x(t) given that x(1) = 4

thanks for any help

Keep in mind that to find x(t) you'll need $\frac{dx}{dt}$

Also keep in mind that $\frac{d^2y}{dx^2} = \frac{d}{dt}(\frac{dy}{dx}) \times \frac{dt}{dx}$

Do you know what to do now?