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    Hey,
    I consider studying maths at uni, and I have few questions about the courses. Basically, I heard that in maths, at uni, you did not demonstrate anything or prove anything in particular during your course. Here, I mean specific demonstrations, not just calculations or reasoning. I'll try to make myself clearer by giving you an example in a second.

    I also have a French friend at Imperial, and he says the same. I wonder how do strangers in general feel about it, or at least people who, during their pre-uni teaching, used to be very rigorous in maths.

    I do not mean that A-levels students are not rigorous, but look at this example.

    Q1: Differentiate f(x)=(3x)/[ln(x)]

    So, according to what I saw, and what I've been told, here is the A-levels method;

    f'(x)=[3.ln(x)-3x/x]/(ln(x)²)
    = 3(ln(x)-1)/(lnx)²

    Now, here is how we have to do it in France (and other countries)

    f(x) is a quotient of two functions u(x)=3x and v(x)=ln(x)
    u(x)=3x, Du=R and v(x)=ln(x), Dv=R+* (Du is the domain of u)

    Now, we need v(x)=/=0 because it is the denominator, therefore ln(x)=/=0 so x=/=1

    Therefore, Df=R+*/{1}

    Now, u(x) is derivable on R as linear function, and v(x) is derivable for all x of Dv, that is for all x>0

    By quotient, f(x) is derivable for all x>0 and x different of 1

    Now, for all x of ]0,1[u]1,+inf[, f is derivable, and we have
    f'(x)=[3ln(x)-3x/x]/(lnx)²
    =3(lnx-1)/(lnx)²


    Of course, I exagerated a bit, but still in France we have to solve it this way. therefore, I wondered how uni was in terms of maths demonstrations.

    Another example would be the demonstration that e^(X+Y)=(e^X)(e^Y) that I could post the French way if you wanted...
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    I'm genuinely confused as to what your point is?

    I can't see any problem with the "A-level" method, at all. The French method is just adding (unnecessary) steps, that's not what rigor is.

    Aside from that, maths at university is more rigorous than pre-Uni. If your course isn't, I would question whether or not understand what's actually being taught (or the university's syllabus)
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    The quotient rule isn't unique to France.
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    Well you need sound understanding if you're looking to get a top mark at uni maths i would imagine.
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    (Original post by boromir9111)
    Well you need sound understanding if you're looking to get a top mark at uni maths i would imagine.
    In England, we do just the same thing. We learn all the basic rules (chain, product, quotient, etc) and show that in answers at first, but once we get the hang of it we just don't bother showing the working, because quite a lot of the time it's obvious. So it's just the same level of understanding (well, theoretically), or at least the same amount being taught. And I'm pretty sure maths at uni is even more rigourous as in school; at least at the better unis, like Imperial. Not sure what the query is about though.
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    I had a friend who was taught in France and was shocked by the 'laziness' of the english system.
    I hear that you'd much rather leave .75 as 3/4 as well.

    As far as I'm aware, English universities don't deviate much, but I hear much more annotation is needed to fully explain methods. No where near as much as what you've described.
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    They are both sound methods. The A Level method is fairly objective, and lets you solve the problem relatively easily. Tbh, that's what maths is - and objective subject, and product differentiation should really be a fairly trivial process in respect to other parts of maths.
    The French method requires you to have a really good understanding of why you get that answer, though it is fairly long-winded and unnecessary, but I can see why it is taught.
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    (Original post by paronomase)
    Hey,
    I consider studying maths at uni, and I have few questions about the courses. Basically, I heard that in maths, at uni, you did not demonstrate anything or prove anything in particular during your course. Here, I mean specific demonstrations, not just calculations or reasoning. I'll try to make myself clearer by giving you an example in a second.
    In a way, ignore what I said before - at A-level what they do in England is teach the method, but not necessarily its derivation, so we have a good understanding of the application, but a very poor understanding of the reasons behind that. (That's true for A-level Maths at least) This would be because of a generally relatively rubbish education here, at least in comparison to some other countries (I would have thought European countries - from what I see definitely France, from what I know definitely Russia). I don't know much about unis, but I would've thought that if you're going to go to one of the best of the English universities, then you'll get the same brilliant quality of education as at the best unis of any other country, so I wouldn't worry.
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    Thank you all for your answers. I was not saying that French were more rigorous, I was just asking a question concerning the maths courses at English unis.

    The only problem would be adaptation, and it would be temporary so it's not a real problem. It's just that when you've learned to solve problems one way, it is challenging to change your method.

    (Original post by oharad)
    ...
    Yeah it is true that we very rarely use decimals in maths (however, we do in physics) For example, we would never write (2/5)²=0,16 but we would write (2/5)²=4/25 (I guess this is just another example of differences.)

    I also agree that proving that a function is derivable is quite trivial and unnecessary. But would you say the same for e^(x+y)=(e^x)(e^y) ? Would you give the solutions of a differential equation such as f'(x)=af(x) without proving how you got that solution? In all our exams, we have an exercise of demonstration (for example the exponentials equivalence mentioned here, or other demonstrations)

    However, I have to agree that if you study Maths in a prestigious university, it will not change a lot if it is in France, Britain, or any other country.
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    I can see the French method being desirable over the Englich method pre university as you don't just start wildly differntiating and jumping into questions with extreme enthusiasm and no thought.

    But adding so much informations is in no way neccessary and at every university the course should result (after some sufficient time) in being more rigorous than your pre university education. If for some questionable reasonable you think this is not so then buying a book will solve your life problems.
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    (Original post by paronomase)
    Basically, I heard that in maths, at uni, you did not demonstrate anything or prove anything in particular during your course.
    This is wrong. Totally wrong. At any university maths course you will find yourself doing rigorous proofs.

    I do not mean that A-levels students are not rigorous, but look at this example. {long example removed}
    Your example is a lot more verbose, but it isn't really more rigourous. The one exception is that you highlight that f isn't differentiable at x = 0. But most A-level students could also tell you that (and why) if you asked them.

    Overall, you've written 10 times as much to very little purpose. If you tried to do that at university, you'd waste an awful lot of time. When doing university maths, you have to be aware of what you need to prove and what you don't. (This can involve a certain amount of mental gymnastics: it's not at all uncommon to sit an exam where one question requires you to prove XYZ and another question is best answered by assuming XYZ without proof).

    To be blunt, pre-university calculus proofs tend to be a waste of time anyhow - it's not so much the proof itself, but that it tends to rely on concepts that haven't been defined properly in the first place. Universities tend to approach things very differently, so you don't really gain anything from having seen a pre-uni proof.
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    (Original post by paronomase)
    I consider studying maths at uni, and I have few questions about the courses. Basically, I heard that in maths, at uni, you did not demonstrate anything or prove anything in particular during your course. Here, I mean specific demonstrations, not just calculations or reasoning. I'll try to make myself clearer by giving you an example in a second.
    The detail of an exposition depends on the course. Proofs usually have a structure; too much detail can obscure the structure of the proof and inhibit understanding.

    In principle, if we wanted we should be able to reduce your problem done to a statement about sets in first-order logic, and then use a formal proof system to derive the result, but that would be absurd and ridiculous. It would introduce detail that is simply not necessary.

    If the goal of mathematics is the understanding of various structures, we should aim to produce expositions that convey understanding in a deductive way that is sufficiently convincing. Often no new understanding is gained by using a more "bare bones" style.
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    If you're doing analysis or a similar pure course, you're going to need to prove everything rigorously as you go along, and state any assumptions you make, where the function is continuous/differentiable and so on.

    If you're doing an applied subject, you typically prove the general case once in lectures (in this example, prove the chain rule for all f(x)/g(x)), and then whenever you need to apply it, then you just do. If you had to write out each step in as much detail as the French system, then you would waste too much time in exams on trivial steps and not really get anywhere.
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    what the hell are you on about
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    (Original post by Tempeststurm)
    If you're doing analysis or a similar pure course, you're going to need to prove everything rigorously as you go along, and state any assumptions you make, where the function is continuous/differentiable and so on.

    If you're doing an applied subject, you typically prove the general case once in lectures (in this example, prove the chain rule for all f(x)/g(x)), and then whenever you need to apply it, then you just do. If you had to write out each step in as much detail as the French system, then you would waste too much time in exams on trivial steps and not really get anywhere.
    mmhh, I think you're right. you wouldn't take the problem with the same approach if you're doing pure maths or if you're only applying it to a problem of physics.

    (Original post by DFranklin)
    Your example is a lot more verbose, but it isn't really more rigourous. The one exception is that you highlight that f isn't differentiable at x = 0. But most A-level students could also tell you that (and why) if you asked them.
    Ok, I exagerated a bit for this one. But I bet that, at a first glance, most A-levels students do not exclude the x=1 possibility, and therefore they write f'(x)...
    But imagine that just after, you have a question concerning the derivative of the function at x=1. If you did not write the preliminary stuff, then you'd end up giving a wrong answer.

    In addition, I'm not at uni yet, but I guess that the level is going to be much harder than a simple differentiation. Therefore, when you start to have really complex problems, if you never proved them at least once, then you're going to apply some concepts, rules, and principles that you did not understand (it's a bit like understanding an exercise by looking only at the correction). Or at least this is my point of view, and it's why I asked for other opinions...
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    (Original post by paronomase)
    Ok, I exagerated a bit for this one. But I bet that, at a first glance, most A-levels students do not exclude the x=1 possibility, and therefore they write f'(x)...
    But imagine that just after, you have a question concerning the derivative of the function at x=1. If you did not write the preliminary stuff, then you'd end up giving a wrong answer.
    Firstly, that question would never appear in an A-level exam. Nobody is going to ask you the derivative of a function at a point where it's not even defined; it would be horrendously unfair. Secondly, even if that did happen, I'm quite sure most people would write "f'(x) = 3(log x - 1)/(log x)^2, so f'(1) = -3/0", and then spot that something was wrong.

    (Original post by paronomase)
    In addition, I'm not at uni yet, but I guess that the level is going to be much harder than a simple differentiation. Therefore, when you start to have really complex problems, if you never proved them at least once, then you're going to apply some concepts, rules, and principles that you did not understand (it's a bit like understanding an exercise by looking only at the correction). Or at least this is my point of view, and it's why I asked for other opinions...
    Without meaning to patronise you... if you're not at university, you probably don't understand differentiation either. You can give some sort of washy account of it, but nothing rigorous, so you're basically on the same footing as an A-level student. I'm quite certain all A-level students know about the domain and range of functions, have a vague idea of how the quotient rule works, know that a function explodes if its denominator goes to zero, etc. I'm also quite certain that neither you nor the average A-level student could prove the quotient rule for me, so all rigour goes out of the window there...

    From your example, it seems to me not that French students know more, but simply that they write more.
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    Perhaps one of the issues with the current state of play in the A-level and FM system is that for many topics no understanding whatsoever is actually needed, the ability to jump through hoops will suffice. Of course this will then lead to the further argument that Maths at this level is designed for both future mathematicians and non-mathematicians alike; and that the undergraduate Mathematician will encounter the desired level of rigor whilst studying Bsc/ Ba.
    A large part of the understanding and depth will of course come from the competence of the teacher and the textbook/ reference book used. I must say that at the FM level I would appreciate a little more depth; whilst recently reviewing several fp2 texts on the topic of 2nd order DE's, not one of them gave any form of explanation as to why y=e^(ax)(Px+Q) was the solution to the said equation when the Auxillary eqn. contained a repeated root. In fact the Oxford textbook gave no theory whatsoever ... it just said ... if the roots are this then the General solution is this ......

    I view such omissions from the core syllabus as both unfortunate and indeed unfair on the bright students that would enjoy the approach from 1st principles....

    etc.
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    (Original post by generalebriety)
    Firstly, that question would never appear in an A-level exam. Nobody is going to ask you the derivative of a function at a point where it's not even defined; it would be horrendously unfair. Secondly, even if that did happen, I'm quite sure most people would write "f'(x) = 3(log x - 1)/(log x)^2, so f'(1) = -3/0", and then spot that something was wrong.


    Without meaning to patronise you... if you're not at university, you probably don't understand differentiation either. You can give some sort of washy account of it, but nothing rigorous, so you're basically on the same footing as an A-level student. I'm quite certain all A-level students know about the domain and range of functions, have a vague idea of how the quotient rule works, know that a function explodes if its denominator goes to zero, etc. I'm also quite certain that neither you nor the average A-level student could prove the quotient rule for me, so all rigour goes out of the window there...

    From your example, it seems to me not that French students know more, but simply that they write more.

    Thanks for you answer. However, you are wrong when you say that 1) I probably don't understand differentiation, and 2) that I could not prove you the product rule.

    Just to tell you, I proved the product rule last year, that is 2 yrs before uni, and it was an exercise similar to the ones we get for our final exams.
    In case you do not believe me, here is how I would do it:
    Let a be a fixed real number, and u and v two functions defined and derivable in a.

    Then, lim [(uv)(a+h) - a(uv)] /h (as h tends to 0)
    ( I remove the lim so that it is clearer and easier to write)
    = [u(a+h)v(a+h)-u(a)v(a)] /h
    = u(a+h)v(a+h)-u(a)v(a+h)+u(a)v(a+h)-u(a)v(a) / h

    (I can add what is in red since it is equal to 0)

    = lim (u(a+h)v(a+h)-u(a)v(a+h)/)h + lim (u(a)v(a+h)-u(a)v(a))/h

    For the first limit, factorise by v(a+h) and for the second by u(a)
    It gives you v(a+h)*u'(a)+u(a)*v'(a+h), and using the fact that h tends to 0, you have u'(a)v(a)+u(a)v'(a)

    If it was an exam, I would have been more rigorous lol
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    (Original post by paronomase)
    Then, lim [(uv)(a+h) - a(uv)] /h (as h tends to 0)
    ( I remove the lim so that it is clearer and easier to write)
    = [u(a+h)v(a+h)-u(a)v(a)] /h
    = u(a+h)v(a+h)-u(a)v(a+h)+u(a)v(a+h)-u(a)v(a) / h

    (I can add what is in red since it is equal to 0)

    = lim (u(a+h)v(a+h)-u(a)v(a+h)/)h + lim (u(a)v(a+h)-u(a)v(a))/h

    For the first limit, factorise by v(a+h) and for the second by u(a)
    It gives you v(a+h)*u'(a)+u(a)*v'(a+h), and using the fact that h tends to 0, you have u'(a)v(a)+u(a)v'(a)
    Not bad at all, but all you've done there is the superficial work. Can you even tell me what a limit is (rigorously)? Can you tell me why taking the limit in two different ways (e.g. along two different sequences converging to the same point) doesn't give you two different answers (and hence why writing "u'(a)" even makes sense)? Can you use all of your machinery above to prove that the derivative of e^x is e^x?

    (I'm not necessarily doubting you can, by the way, but I'd be very surprised if you could, since it takes most university students a full lecture course in analysis to even get these basic facts down. If you can, and you're a typical baccalauréat student, the English education system has a lot to learn from you.)
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    (Original post by Mrm.)
    not one of them gave any form of explanation as to why y=e^(ax)(Px+Q) was the solution to the said equation when the Auxillary eqn. contained a repeated root.
    For what it's worth (and I'm a third year undergrad), I still don't know the answers to questions like that. I always suspected that questions like this required some very specialised theory to answer on anything more than a superficial "look, substitute it in, it works" kind of way.
 
 
 
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