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Question from Siklos Booklet

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1. Question from Siklos Booklet

I can't understand how he get from one line to another:
Last edited by Dog4444; 15-07-2012 at 00:39.
2. 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. 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 17:07.
4. 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 is a function then is what you get by substituting for in the expression of , and similarly is what you get by substituting for . And is just .

So for instance if then and .
Last edited by nuodai; 17-07-2012 at 17:28.
5. Re: Question from Siklos Booklet
Alternatively, note that is any function of the variable .

From the above post: is any function of the variable

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

_ _ _ _ _ _ _ _ _

Non-Example:

is not of the form

Example:

Is

of the form ?
6. Re: Question from Siklos Booklet
Example:

Is

of the form ?
Could it be so if ?
7. Re: Question from Siklos Booklet
(Original post by Dog4444)
Could it be so if ?
Yes. To make it less ambiguous, I would use the substitution

So, you have

8. Re: Question from Siklos Booklet
Yes. To make it less ambiguous, I would use the substitution

So, you have

Right, i see.

Is it right that ? Or is it not always the case?
9. 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. Re: Question from Siklos Booklet
(Original post by Dog4444)
Is it right that ? 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 . Then . (It's my favourite counterexample.)
11. Re: Question from Siklos Booklet
(Original post by nuodai)
It's my favourite counterexample.
This one has never occurred to me. Nice. PRSOM
12. Re: Question from Siklos Booklet
Guys, I'm stuck. I tried to substitute
But I failed to show that ,
because

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 23:20.
13. Re: Question from Siklos Booklet
How did you find ?

EDIT: that is meant to be more of a hint than an actual question.
Last edited by jack.hadamard; 18-07-2012 at 23:32.
14. Re: Question from Siklos Booklet
(Original post by Dog4444)
Guys, I'm stuck. I tried to substitute
But I failed to show that ,
because

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 , and then notice that . This ultimately yields , which isn't very ideal because of that pesky .

Or is it? We know , and so . Expanding the LHS gives , which provides you with a means of expressing in terms of .
15. Re: Question from Siklos Booklet
How did you find ?

EDIT: that is meant to be more of a hint than an actual question.
and the rest follows. But i cant see how it helps.
16. Re: Question from Siklos Booklet
(Original post by Dog4444)
and the rest follows. But i cant see how it helps.
17. Re: Question from Siklos Booklet
(Original post by nuodai)
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 02:28.
18. 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 solely in terms of than it is to express in terms of . 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. Re: Question from Siklos Booklet
(Original post by nuodai)
It's much harder to (off the cuff) express solely in terms of than it is to express in terms of . 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. 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 , since , and therefore functions of the form by setting .
Last edited by nuodai; 19-07-2012 at 12:51.

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Last updated: August 13, 2012
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