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    Thread: Equations 2

    1. #51
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      Quote Originally Posted by esfx View Post
      Pi can be defined using the taylor (macclaurin) expansion of arcsin, arccos, or arctan I think.

      Like, arctan 1 == pi/4, so pi = 4* [1 - 1/3 + 1/5 -1/7 + ...], which is the nice for approximations since it alternates, so error is less than the size of the first dropped term.

      So, with 5 terms, we get pi~= 1052/315

      Ok, maybe there are faster ways to approximate it. This way isn't even linear, but you get the point.
      Yes, but I'm fairly certain that all trig functions are defined in terms of pi, so you can't define pi by them.

    2. #52
      Xei
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      Gr I bracketed it wrong the second time, should've been 2(x)1/2.

    3. #53
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      Quote Originally Posted by Xei View Post
      Gr I bracketed it wrong the second time, should've been 2(x)1/2.
      isnt that just x? or did i need to read your post to know why you put that, or am i talking a load of crap

    4. #54
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      Quote Originally Posted by ninja9578 View Post
      Yes, but I'm fairly certain that all trig functions are defined in terms of pi, so you can't define pi by them.
      No, we're asking "what is the numerical value of pi". We can define pi without knowing its value by saying one full revolution is 2pi.

    5. #55
      Xei
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      Quote Originally Posted by slash112 View Post
      isnt that just x? or did i need to read your post to know why you put that, or am i talking a load of crap
      Coefficients are on the left, powers on the right, so that's the square root of 2x.

    6. #56
      Xei
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      Could somebody please try and integrate lnx/x by parts, using dv/dx=lnx, u=x-1?

      I know there are two much easier ways of doing it, but whenever I try the above method I end up with 0 = -1 and it's bugging me.

    7. #57
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      Quote Originally Posted by Xei View Post
      Could somebody please try and integrate lnx/x by parts, using dv/dx=lnx, u=x-1?

      I know there are two much easier ways of doing it, but whenever I try the above method I end up with 0 = -1 and it's bugging me.
      You mean u=x^(-1), or u=1/x, right? Because you can't seriously think that u=x-1 will look like anything other than x minus 1.

    8. #58
      Xei
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      Yes.

    9. #59
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      Well, I tried it briefly and got something of the form "I = I + stuff", where I is the answer.

      EDIT: Oops nevermind. I had a sign wrong. The answer is (ln(x))^2/2, using u=ln x and dv = (1/x)dx

    10. #60
      Xei
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      Yes, I know. And it's even simpler via the chain rule because clearly the integral is of a product of a function and its deriviative. But when ever I do it setting dv = lnxdx I get I = I + stuff which reduces to 0 = -1 and I can't seem to find the mistake.

    11. #61
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      Quote Originally Posted by Xei View Post
      Yes, I know. And it's even simpler via the chain rule because clearly the integral is of a product of a function and its deriviative. But when ever I do it setting dv = lnxdx I get I = I + stuff which reduces to 0 = -1 and I can't seem to find the mistake.
      Sign.

    12. #62
      Xei
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      S1/x*lnxdx = 1/x(xlnx -x) +S1/x^2*(xlnx -x)dx
      = lnx -1 -lnx +S1/x*lnx
      0 = -1

      Grr.

    13. #63
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      Quote Originally Posted by Xei
      Could somebody please try and integrate lnx/x by parts, using dv/dx=lnx, u=x-1?
      Xei aren't you mean't to be Cambridge material.

      dv/dx=ln(x), then v=xln(x)-x(sadly, I remeber this from summer holidays memorizing integrals).

      Then you can finish this off. Remeber UV- int(V du/dx)dx.

      Quote Originally Posted by Xei
      I get I = I + stuff which reduces to 0 = -1 and I can't seem to find the mistake.
      Don't do that in the exams.

      Quote Originally Posted by Xei
      S1/x*lnxdx = 1/x(xlnx -x) +S1/x^2*(xlnx -x)dx
      = lnx -1 -lnx +S1/x*lnx
      0 = -1
      Int x^-1 lnx= lnx-1 -(-2)( int (x^1)(lnx-1))dx

      Int x^-1 lnx= lnx-1-(-2)( (I -int1))dx

      I=lnx-1+2I-x

      -I=lnx-1-x

      I=1+x-lnx
      Last edited by wendylove; 02-17-2009 at 09:00 PM.
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      The planet Earth exhibits all of these properties and therefore can be considered alive and its own single organism by the scientific definition.
      7. Reproduction: The ability to produce new organisms.
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    14. #64
      Xei
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      Yeah that's what I did above?

    15. #65
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      ∫ (1/x^2)(xlnx - x) dx = ∫ (lnx - 1)/x dx, not lnx/x.
      Or am I totally off?
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    16. #66
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      I can't seem to get the right anwser too. I might have to get paper out.

      Second stab

      int x^(-1) ln(x)= ln(x)-1-[(-2) int ln(x^-1)(lnx-1)]

      I=ln(x)-1+2I-ln(x)

      ???

      Done!

      Which is wrong again.

      Dammit, this is stupid. I thought I had it.

      Where did you find this question?
      Last edited by wendylove; 02-17-2009 at 09:14 PM.
      Xaqaria
      The planet Earth exhibits all of these properties and therefore can be considered alive and its own single organism by the scientific definition.
      7. Reproduction: The ability to produce new organisms.
      does the planet Earth reproduce, well no unless you count the moon.

    17. #67
      Xei
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      Old STEP paper.

      It's very easy to do if you do it by parts the other way or by substitution, but I just can't get the third way to work.

      Either I'm repeatedly fucking it up or this method involves dividing by 0 or something.

    18. #68
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      Did you take into account the absolute value in the logarithm? You may have implicitly logged a negative number, and then that reared its head in the form of a logical impossibility later on.

    19. #69
      Xei
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      Ah... I think you might have it.

      Let me work it out.

    20. #70
      Xei
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      Okay nice question I found today:

      y = 1/(1+x^2) (sketching it helps)

      Find the equation of the line that intersects the curve at the y axis and is tangent to the curve for some x > 0.

      Prove there are no more intersections.

      Consider the area under the curve from 0 < x < 1 and hence show that pi > 3.

      If you can do this in less than 45 minutes you can read maths at Cambridge.

    21. #71
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      Quote Originally Posted by Xei
      If you can do this in less than 45 minutes you can read maths at Cambridge.
      The question is not that hard. The first part is easy just need to be careful, the prove part just needs abit of FP1, the last part needs FP2.

      STEP is unfair as private school people get lessons on it and poor people don't. STEP is abit stupid as it's just a test of technical ability.
      Xaqaria
      The planet Earth exhibits all of these properties and therefore can be considered alive and its own single organism by the scientific definition.
      7. Reproduction: The ability to produce new organisms.
      does the planet Earth reproduce, well no unless you count the moon.

    22. #72
      Xei
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      I think the only FP bit you would need is recognising the arctan integral, which I think is FP2... I haven't done FP2 yet but it's easy enough to work out.

      I'm not really against STEP, and I go to a state school. The only way you can genuinely get good at them is by doing lots of them... there's no special secret or technique that can be taught, it's really just about ability. If anything it weeds out those private school boys who've been coached to the point of perfection to pass A Levels but aren't really very bright.

      Which is necessary considering that some ridiculous number like 80&#37; get As in FM... Labour has devalued qualifications to the point that they are completely and utterly worthless. That's why Cambridge has to make their own exams...

      If I fail STEP I'll retreat magnanimously... I'll just know I'm not good enough to go to Cambridge, so I'll be perfectly happy studying somewhere closer to my level, like Imperial.

      Where are you going / do you have offers from?

    23. #73
      Look away wendylove's Avatar
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      I disagree. If you look at Cambridge it is still just Private school boys doing the Maths degree.

      STEP is crap. It has no proofs hence it can't be a good picture of the Maths you would do at uni.

      Cambridge to me is stupid, look at it's history. GH Hardy said that Cambridge nearly destroyed his love of Maths which says alot. Although, Cambridge is better now. But, I don't see why people assume that if a person goes to Cambridge their Maths skills are at another level.

      A level and STEP has nothing to do with Maths, it's just applying techniques. I guess Cambridge will be full of people that can apply techniques really good, although I don't know what that has to do with Maths.

      I'm going to Manchester uni.
      Xaqaria
      The planet Earth exhibits all of these properties and therefore can be considered alive and its own single organism by the scientific definition.
      7. Reproduction: The ability to produce new organisms.
      does the planet Earth reproduce, well no unless you count the moon.

    24. #74
      Xei
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      They take a number of private schooled people which is in proportion to the total amount of applicants they recieve. If state schools weren't so crap or actually encouraged their students to apply more, it wouldn't be the case.

      In any case it is better than Oxford, where they only have A Levels as a hurdle which are completely worthless. They also have about 10&#37; more private schoolers. Oxford is stuck in the past socially, you should be ashamed for applying there.

      And in no way is STEP 'applying formulae', that is for A Level. STEP requires a lot of free thought to complete most questions. That is the whole point of STEP.

      And of course Cambridge has people with better maths skills. It has the best history for maths in the world, it has the best lecturers and exam results, it has a very strong mathematical community, and is by a very long way the hardest university to get into. Almost every famous British mathematician ever came from Cambridge.

    25. #75
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      It's funny listening to you guys talk about math programs as if it actually makes a difference where you learn math.

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