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**yonski****Member**- Registered: 2005-12-14
- Posts: 67

Hey,

just wondering whether there's any other ppl on here taking the further maths A-level too? I'm self-teaching it cos there's not a proper class at my school so it's a bit of a lonely experience! Would be nice to get in touch with others who're doing it

Look forward to hearing!

yonski

Student: "What's a corollary?"

Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

I did A-level Further Maths a long, long, long time ago.

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**yonski****Member**- Registered: 2005-12-14
- Posts: 67

Haha cool. I bet it was harder back then too! And I bet you still got an A

Student: "What's a corollary?"

Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 301

Im an American, but might have taken the US equivalent to "Further Maths" What does it entail?

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

im doing further maths on my own too, but not because my school doesn't offer it, just because it clashes with another subject im doing that i can't do on my own, although either way its better for me to do it on my own because it goes far too slow in the classes, i taught myself the full Alevel maths course and i already know alot of stuff on the further maths course, and stuff above alevel aswell so i work much faster on my own ^^

The Beginning Of All Things To End.

The End Of All Things To Come.

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

Hello.

I'm doing further maths as well, but i am doing it at school. I'm in year 12 though.. so I'm on the normal A level at the moment. How are you finding it so far? Are you in year 12 or 13?

*Last edited by Daniel123 (2007-09-22 11:05:29)*

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

bossk171 wrote:

Im an American, but might have taken the US equivalent to "Further Maths" What does it entail?

Well, topics covered in further maths include (copied from the syllabus):

Inequalities, series, complex numbers, numerical solution of equations, first order differential equations, second order differential equations, polar coordinates, coordinate systems, hyperbolic functions, integration, matrix algebra, vectors, Maclaurin and Taylor series, numerical methods, proof

*Last edited by Daniel123 (2007-09-22 11:04:24)*

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 301

With the exception of "numerical methods" which is simply a term I haven't heard before, those are all things I've covered before. Interestingly, in a few different classes, not one.

Can anyone give an example of a "numerical methods" question?

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

I couldn't give you an example, as I don't know what it is either , but, in more detail, the syllabus describes it as: "equations of the form f(x)=0 solved numerically by i) interval bisection ii) linear interpolation and iii) the Newton-Raphson process.

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 301

I think that's something I don't understand. The other stuff is all stuff I've seen before, but in a bunch of different classes, Not just one.

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Numerical methods is basically just number crunching, the kind of tedious, repititious process that we normally leave to computers.

It's used to approximate values that can't be found analytically (square roots, for example).

(i) Interval bisection

This is probably the most boring of the three, because it is the slowest to get you to a certain accuracy. It involves making an initial interval that you know the answer is in, and then halving it repeatedly.

For example, let's try to find :sqrt2. The equation in this case would be x²-2=0.

The first interval is sometimes given to you, but if you have to find your own, you're looking for two numbers a and b such that f(a)<0 and f(b)>0.

A good first interval in this case would be (1,2).

1²-2 = -1 and 2²-2 = 2, so that works as an interval.

Now we find the halfway point [(a+b)/2] and put that into our function.

1.5²-2 = 0.25. This is on the "f(2) side" of 0, and so it replaces 2.

The new interval is (1,1.5).

Now do that lots of times until your interval is small enough.

f(1.25) = -0.4375 --> (1.25,1.5)

f(1.375) = -0.109375 --> (1.375,1.5)

f(1.4375) = 0.06640625 --> (1.375,1.4375)

...

Note that it's important to keep the as much of the number as you can. Rounding will cause inaccuracies in later stages. You'll have to keep going until one of two things happen. The question will either say to keep going until your interval is a certain size, or until you can state the answer to a certain accuracy.

That last interval would be enough for you to say that :sqrt2 ≈ 1.4, because both values give that when rounded to 1dp.

(ii) Linear interpolation

Similar to interval bisection, but a bit cleverer, harder and quicker. In the above example, we initially had that f(1) = -1 and f(2) = 2. As f(1) is closer to 0, you might want to guess the next value as, maybe, 1.3 instead of just blindly halving the interval all the time, and that's what this method does.

Graphically, to do this you would mark the points x=a and x=b on the graph of f(x), then draw a straight line between them. The place where this line crosses 0 is the next x-value you would try.

Algebraically, the next number to try would be b - [(b-a)(f(b))/[(f(b)-f(a)]].

A bit horrible, but it works better than halving.

Going back to the interval (1,2) (with f(1)=-1, f(2)=2), we get this:

2 - [(2-1)(2)/[2-(-1)]] = 2 - 2/3 = 1 1/3.

f(4/3) = -0.222...<0, so the new interval is (4/3, 2).

A few more iterations of this get:

(1.4,2)

(24/17,2)

(24/17,461/290)

I've put them as fractions there for ease of keeping the accuracy, but you can put them as rounded decimals as long as you keep the whole thing in your calculator. The last interval is roughly (1.41, 1.59).

This method usually gets accuracy quicker than interval bisection.

(iii) Newton-Raphson

This one is slightly different because it only uses one value, not an interval. It's similar to linear interpolation, but instead of using another point to draw a line, it uses the gradient of one point.

That means that to use this method, the function has to be differentiable. It is also by far the quickest method of the three though, and if given the choice I definitely recommend using this one.

Graphically, the tangent to an initial point would be drawn and wherever that line crosses 0 would become the new point.

Algebraically, the next point is given by a - [f(a)/f'(a)], where f'(x) is the derivative of f(x).

Our f(x) is still x² - 2, which means that in this case f'(x) = 2x.

Taking the initial point as 1, the next one would be given by 1 - [(-1)/2] = 1.5.

Then, 1.5 - (0.25/3) = 1.41666...

Continuing this gives (as far as my calculator will go):

1.414215686...

1.414213562...

And from there it stays on that. So in just three stages, there's already an answer to 9 decimal places.

Some important things:

The functions that these are used on must be continuous (or, in the case of the first two, continuous within the initial interval). Also, the methods might start acting strangely if there is more than answer in the interval.

Why did the vector cross the road?

It wanted to be normal.

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**bossk171****Member**- Registered: 2007-07-16
- Posts: 301

mathsyperson wrote:

Numerical methods is basically just number crunching, the kind of tedious, repititious process that we normally leave to computers.

Well, that doesn't seem so bad. Maybe a bit useless, but not so difficult to learn.

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**yonski****Member**- Registered: 2005-12-14
- Posts: 67

Daniel123 wrote:

Hello.

I'm doing further maths as well, but i am doing it at school. I'm in year 12 though.. so I'm on the normal A level at the moment. How are you finding it so far? Are you in year 12 or 13?

Heya. That's cool, i'm in year 13. So which modules are you working on right now / doing over the rest of this year?

I'm not finding it too difficult so far. Did C1-C3 + M1, M2 and S1 last year and it was all fairly staright-forward. M2 was easily the most difficult of the lot for me, but I was pleasantly surprised by my grade when it came through My worst mark came in S1 simply cos it's far too boring to study!

Are there many other ppl taking it at your school?

Student: "What's a corollary?"

Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

yeh, S1 is terribly boring, im starting S2 after january exams: im in yr 12, but i started early.

so far ive done C1-C4, M1, M2, S1 and in january im taking FP1 and D1

in my school there are 4 people taking further maths (excluding me) and they're all the people i wouldn't have thought would take it, i would have thought the sort of 'intellectual' type people would, but its been the opposite, the people in my class who seemed to care the least when it came to academics have all done further maths ^^

The Beginning Of All Things To End.

The End Of All Things To Come.

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

At the moment I'm doing C1, C2 and M1, which are all really straightforward, and I have the C1 and M1 exams in january. By the end of this school year I will have done C1-C4, M1 and S1. Yeah Ive been told that M2 is hard, and M3 is apparently a nightmare.

Actually we have quite a few taking further maths... 13 including me in my year, and about 7 in the year above.

What other subjects are you doing? Are you going to / have you applied for uni?

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**yonski****Member**- Registered: 2005-12-14
- Posts: 67

luca-deltodesco wrote:

so far ive done C1-C4, M1, M2, S1 and in january im taking FP1 and D1

Haha, I took one look at that D1 book and thought there's no way i'm ploughing through this lot! How are you finding FP1?

Over the summer i've done C4, FP1 and FP2, and i've almost finished M3 now. I'm gonna sit all these in january so i'll only have 2 to do in the summer, plus any resits if necessary. I've actually found M3 to be quite good, really interesting topics (well i find them interesting anyway) on centres of mass of solids and stuff. It seems about the same difficulty as M2 to me, just with different material. I think i'm gonna give M4 and maybe even M5 a go - the books are nice and slim From what i've done so far of the further maths material, the integration in FP2 is the fcuker - hopefully the questions in the exam papers won't be as tough as some of those in the book.

Wow there are lots of people doing further maths in your schools!! Where i'm at, there are only 5 people in my regular A2 math class, and one of them doesn't show up much!

I'm doing physics and ICT with the maths. Wanna do a theoretical physics degree at uni.. maybe Warwick or Imperial with any luck. I'm off to the Warwick open day tomorrow actually You two decided what you're gonna study and where yet?

*Last edited by yonski (2007-09-25 09:29:46)*

Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

I'm nervous for the later modules, because so far nothing has come that is a challenge... I'm doing physics, english and economics with the Further Maths... so much work! Do you have many different maths teachers that teach your class? We have three.. which gets really annoying.

I'm not really sure what I want to study yet.. but I was thinking about Warwick.. and Nottingham. I might do physics with philosophy - looks really interesting.

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

: How are you finding FP1?

tiss easy, i already know all of it and more which probably helps

The Beginning Of All Things To End.

The End Of All Things To Come.

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**yonski****Member**- Registered: 2005-12-14
- Posts: 67

So how come you're only doing the 2 modules in january then?

*Last edited by yonski (2007-09-26 04:15:03)*

Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

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**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

teachers still want me to cover it all formally and have written work they can mark me on to make sure im not going to just go in and fail.

The Beginning Of All Things To End.

The End Of All Things To Come.

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**SimonYu****Member**- Registered: 2008-07-15
- Posts: 6

luca-deltodesco wrote:

: How are you finding FP1?

tiss easy, i already know all of it and more which probably helps

how comes you already know must of it?

can u give me a hand on series then please? i dont get a word they are tralkin about!

Sunje Sy

xxx

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

SimonYu wrote:

luca-deltodesco wrote:: How are you finding FP1?

tiss easy, i already know all of it and more which probably helps

how comes you already know must of it?

can u give me a hand on series then please? i dont get a word they are tralkin about!

It would be easier for us if you posted specific questions.

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**zetafunc.****Guest**

Bumping this because I'm curious if there are any other people doing this...

I did the C1, C2, C3 and M1 modules in Year 11, and C4, S1-4, M2-5, FP1-3 and D1-2 modules in Year 12, which when cashed in should yield 3A* grades but they are withholding the cashing in because they want me to consider doing re-sits for modules I didn't do too well in (like D2).

Is anyone else doing FM?

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