Question from Siklos Booklet

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  1. Dog4444's Avatar
    1 14-07-2012  23:31
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    Question from Siklos Booklet
    Page 15 of this booklet: http://www.maths.cam.ac.uk/undergrad...tep/advpcm.pdf

    I can't understand how he get from one line to another:
    Last edited by Dog4444; 14-07-2012 at 23:39.
  2. Slumpy's Avatar
    2 14-07-2012  23:34
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    Re: Question from Siklos Booklet
    You just take the first result, and sub in the second one. So on your second line, x_3-x_2 is -(x_2-x_3), which you have an expression for on the first line there.
  3. Dog4444's Avatar
    3 17-07-2012  16:05
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    Re: Question from Siklos Booklet
    There's another one:



    What does f(x^2+1...) means? Does it mean that if we put a polynomial x^2+1 into some function f like (f(x)=2x) so f(x^2+1)=2(x^2+1) )?

    And the same question for (1+t^-2)f(t)
    Last edited by Dog4444; 17-07-2012 at 16:07.
  4. nuodai's Avatar
    4 17-07-2012  16:25
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    There's another one:



    What does f(x^2+1...) means? Does it mean that if we put a polynomial x^2+1 into some function f like (f(x)=2x) so f(x^2+1)=2(x^2+1) )?

    And the same question for (1+t^-2)f(t)
    You should know that if f(x) is a function then f(y) is what you get by substituting y for x in the expression of f(x), and similarly f(x^2+1) is what you get by substituting x^2+1 for x. And af(x) is just a \times f(x).

    So for instance if f(x) = 3\sin x then f(x^2+1)=3\sin(x^2+1) and (1+t^{-2})f(t) = 3(1+t^{-2})\sin t.
    Last edited by nuodai; 17-07-2012 at 16:28.
  5. jack.hadamard's Avatar
    5 17-07-2012  17:02
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    Re: Question from Siklos Booklet
    Alternatively, note that f(x) is any function of the variable x.

    From the above post: f(x^2 + 1) is any function of the variable x^2 + 1


    One could think of those in terms of function composition; e.g. f(g(x)) is any function f of a function g(x).

    _ _ _ _ _ _ _ _ _

    Non-Example:

    3\sin(x^2 + 1) + x is not of the form f(x^2 + 1)


    Example:

    Is

    3\sin(x^2 + 1) + x^2

    of the form f(x^2 + 1)?
  6. Dog4444's Avatar
    6 17-07-2012  17:26
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    Re: Question from Siklos Booklet
    (Original post by jack.hadamard)
    Example:

    Is

    3\sin(x^2 + 1) + x^2

    of the form f(x^2 + 1)?
    Could it be so if f(x)=3sinx+x- 1 ?
  7. jack.hadamard's Avatar
    7 17-07-2012  17:35
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    Could it be so if f(x)=3sinx+x- 1 ?
    Yes. To make it less ambiguous, I would use the substitution t = x^2 + 1

    So, you have

    3\sin(x^2 + 1) + x^2 \ = \ 3\sin(x^2 + 1) + x^2 + 1 - 1

    \qquad \qquad \qquad \qquad\ = \ 3\sin(t) + t - 1

    \qquad \qquad \qquad \qquad\ = \ f(t) \qquad \text{(a function of that variable)}
  8. Dog4444's Avatar
    8 17-07-2012  18:04
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    Re: Question from Siklos Booklet
    (Original post by jack.hadamard)
    Yes. To make it less ambiguous, I would use the substitution t = x^2 + 1

    So, you have

    3\sin(x^2 + 1) + x^2 \ = \ 3\sin(x^2 + 1) + x^2 + 1 - 1

    \qquad \qquad \qquad \qquad\ = \ 3\sin(t) + t - 1

    \qquad \qquad \qquad \qquad\ = \ f(t) \qquad \text{(a function of that variable)}
    Right, i see.

    Is it right that f(a+b)=f(a)+f(b)? Or is it not always the case?
  9. DFranklin's Avatar
    9 17-07-2012  18:07
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    Re: Question from Siklos Booklet
    It is generally not true that f(a+b) = f(a) + f(b).

    E.g. f(x) = x+1, then f(a+b) = a+b+1, while f(a)+f(b) = a+1 + b + 1 = a+b+2.
  10. nuodai's Avatar
    10 17-07-2012  18:14
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    Is it right that f(a+b)=f(a)+f(b)? Or is it not always the case?
    (Original post by DFranklin)
    E.g. f(x) = x+1, then f(a+b) = a+b+1, while f(a)+f(b) = a+1 + b + 1 = a+b+2.
    Or indeed f(x) \equiv 1. Then 1 = f(a+b) \ne f(a)+f(b) = 1+1 = 2. (It's my favourite counterexample.)
  11. jack.hadamard's Avatar
    11 17-07-2012  19:00
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    Re: Question from Siklos Booklet
    (Original post by nuodai)
    It's my favourite counterexample.
    This one has never occurred to me. Nice. PRSOM
  12. Dog4444's Avatar
    12 18-07-2012  21:46
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    Re: Question from Siklos Booklet
    Guys, I'm stuck. I tried to substitute t=(x^2+1)^{1/2}+x
    But I failed to show that 1/2(1+t^{-2})=1/ (x(x^2+1)^{-1/2}+1),
    because dx = dt/(x(x^2+1)^{-1/2}+1)

    Is it the right approach at all? Because I can't get to f(t) otherwise.

    EDIT: sry for my latex, i dunno whats wrong with it.
    EDIT by nuodai: fixed for you - you closed a { bracket with a ) bracket!
    Last edited by Dog4444; 18-07-2012 at 22:20.
  13. jack.hadamard's Avatar
    13 18-07-2012  22:31
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    Re: Question from Siklos Booklet
    How did you find \displaystyle \frac{dx}{dt}?

    EDIT: that is meant to be more of a hint than an actual question.
    Last edited by jack.hadamard; 18-07-2012 at 22:32.
  14. nuodai's Avatar
    14 18-07-2012  22:40
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    Guys, I'm stuck. I tried to substitute t=(x^2+1)^{1/2}+x
    But I failed to show that 1/2(1+t^{-2})=x(x^2+1)^{-1/2}+1,
    because dx = dt/(x(x^2+1)^{-1/2}+1)

    Is it the right approach at all? Because I can't get to f(t) otherwise.
    It's one approach, which is different from the one that Siklos uses.

    Start by multiplying top and bottom by (x^2+1)^{1/2}, and then notice that (x^2+1)^{1/2} = (x+(x^2+1)^{1/2}) - x. This ultimately yields \displaystyle \int \dfrac{t-x}{t} f(t)\, dt, which isn't very ideal because of that pesky x.

    Or is it? We know t-x = (x^2+1)^{\frac{1}{2}}, and so (t-x)^2 = x^2+1. Expanding the LHS gives t^2-2xt=1, which provides you with a means of expressing x in terms of t.
  15. Dog4444's Avatar
    15 18-07-2012  23:41
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    Re: Question from Siklos Booklet
    (Original post by jack.hadamard)
    How did you find \displaystyle \frac{dx}{dt}?

    EDIT: that is meant to be more of a hint than an actual question.
    dt=x(x^2+1)^{-1/2}+1 dx and the rest follows. But i cant see how it helps.
  16. nuodai's Avatar
    16 19-07-2012  00:12
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    dt=x(x^2+1)^{-1/2}+1 dx and the rest follows. But i cant see how it helps.
    Psst... read my reply...
  17. Dog4444's Avatar
    17 19-07-2012  01:09
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    Re: Question from Siklos Booklet
    (Original post by nuodai)
    Psst... read my reply...
    lol a little bit sleepy, it works out, thanks.

    For the second part, why would you bother about the first part if you can just substitute the same thing in? I mean, what's the point of the first part?
    Last edited by Dog4444; 19-07-2012 at 01:28.
  18. nuodai's Avatar
    18 19-07-2012  10:28
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    For the second part, why would you bother about the first part if you can just substitute the same thing in? I mean, what's the point of the first part?
    It's much harder to (off the cuff) express (x^2+1)^{\frac{1}{2}} solely in terms of t than it is to express x in terms of t. As a general rule, if you make an expression look cleaner then you're moving in the right direction. Siklos's method applies this philosophy straight away, and ends up reaching the solution faster, but all is not lost by doing it your way.
  19. Dog4444's Avatar
    19 19-07-2012  11:44
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    Re: Question from Siklos Booklet
    (Original post by nuodai)
    It's much harder to (off the cuff) express (x^2+1)^{\frac{1}{2}} solely in terms of t than it is to express x in terms of t. As a general rule, if you make an expression look cleaner then you're moving in the right direction. Siklos's method applies this philosophy straight away, and ends up reaching the solution faster, but all is not lost by doing it your way.
    Ok, and why would you consider a fucntion f((x^2+1)^{1/2}...) rather than (x^2+1)^{1/2}... on it's own? Is it to to generalise some integrals?
  20. nuodai's Avatar
    20 19-07-2012  11:47
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    Re: Question from Siklos Booklet
    (Original post by Dog4444)
    Ok, and why would you consider a fucntion f((x^2+1)^{1/2}...) rather than (x^2+1)^{1/2}... on it's own? Is it to to generalise some integrals?
    I think this problem is quite contrived for the purposes of setting a STEP question, but it's not completely useless: for instance, it provides a starting point for integrating \sinh^{-1} x, since \sinh^{-1} x = \ln (x + (x^2+1)^{\frac{1}{2}}), and therefore functions of the form g(\sinh^{-1} x) by setting f = g \circ \sinh^{-1}.
    Last edited by nuodai; 19-07-2012 at 11:51.
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