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A Summer of Maths

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Original post by Brit_Miller
Thanks

(and none surely?)


Yeah :redface:
Original post by 4ever_drifting
Love these kind of problems :smile: Just a note though - could you not observe that you need the product of the roots of the quadratic equation, which is given by c/a in 0=ax^2+bx+c, so you know the answer is 26 without having to find the roots then multiply them together?

You could in this case because the quadratic simultaneous equations you get when substituting for X and when substituting for Y are the same. I preferred not to take that shortcut on the off-chance that substituting for Y generated a different quadratic than substituting for X.

I think you could probably solve it geometrically as well, the method I posted was just the one I prefer to use.
Original post by DJMayes
At the Nottingham University Open Day there was a "Maths Trail" with several interesting questions on it. The questions ranged from requiring a working knowledge of arithmetic progressions and combinations to lowest common multiples and counting squares; and more emphasis was put on thinking about them than slogging through endless manipulation. I thought I'd share one with you. The question is of the kind that could be set in C1, but is an interesting one:

A rectangle is inscribed inside a circle of radius 6 units such that each of the vertices of the rectangle lie on the circumference of the circle. Given that the perimeter of the rectangle is 28 units, what is the area?

Required Knowledge:

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Hints:

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Full Solution:

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Original post by hassi94

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This one is a bit tricky, but I think it has something to teach; use hints if you get stuck.

{*} Question:

Let f:RR{3}f : \mathbb{R} \to \mathbb{R}\setminus \{3\} be a function with the property that there exists ω>0\omega > 0 such that

f(x+ω)=f(x)5f(x)3\displaystyle f(x + \omega) = \frac{f(x) - 5}{f(x) - 3}

for all xRx \in \mathbb{R}.

Prove that ff is periodic.



{**} Required: (A-level)

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{***} Hints:

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Reply 65
Original post by hassi94

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wasnt this in the online lecture thing? :tongue:
Original post by james22
What is the derivative of y=x^x?

What is the derivatie and inverse of y=x^x^x^x^... (an infinite string of x's)

Here x^x^x=x^(x^x) not (x^x)^x

Also for what values of x does y exist?


Some thoughts under cruel assumptions.

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I wonder how people on here find Analysis in general.

Do you like it (assuming you have seen some), or do you think it is difficult to understand and apply?
Reply 68
Original post by jack.hadamard
This one is a bit tricky, but I think it has something to teach; use hints if you get stuck.

{*} Question:

Let f:RR{3}f : \mathbb{R} \to \mathbb{R}\setminus \{3\} be a function with the property that there exists ω>0\omega > 0 such that

f(x+ω)=f(x)5f(x)3\displaystyle f(x + \omega) = \frac{f(x) - 5}{f(x) - 3}

for all xRx \in \mathbb{R}.

Prove that ff is periodic.



{**} Required: (A-level)

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{***} Hints:

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Didn't seem that hard.

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(edited 11 years ago)
Reply 69
Original post by jack.hadamard
Some thoughts under cruel assumptions.

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You have the right inverse and are right in the simplification of the original equation. I cannot remember the derivative but your answer looks about right. Any luck with the values of x for which this converges?
Original post by Blutooth
...


Can you use a spoiler, please. Thanks.

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Original post by james22
Any luck with the values of x for which this converges?


I don't have a good reason to believe what I derived works, so I have only rough ideas.


Original post by Lord of the Flies
I find analysis a bit boring to be honest. I prefer "even purer maths"... They seem to have more depth and involve more creativity.


I have studied a bit of it, and I find it interesting; some of Cauchy's stuff is amazing and I would say it is creative.
However, I do get stuck from time to time -- I thought I understood uniform continuity two months ago, but now I have to read it again. :biggrin:
Reply 71
Original post by jack.hadamard
Can you use a spoiler, please. Thanks.

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(edited 11 years ago)
Reply 72
Original post by jack.hadamard
{*} Question:

The polynomial x2+1x^2 + 1 is irreducible over R\mathbb{R}.

i) By completing the square, show that x4+1x^4 + 1 is not irreducible over the set of real numbers.

Hence, derive the Sophie Germain algebraic identity

x4+4y4  (x2+2xy+2y2)(x22xy+2y2)x^4 + 4y^4\ \equiv\ (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2)

by starting from the left-hand side.


ii) Evaluate k=1n4k4k4+1\displaystyle \sum_{k=1}^{n} \frac{4k}{4k^4 + 1}


{**} Required:

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Original post by james22
You have the right inverse and are right in the simplification of the original equation. I cannot remember the derivative but your answer looks about right. Any luck with the values of x for which this converges?


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If that is what you meant by convergence?
(edited 11 years ago)
Here's an easy one!

Question

Evaluate limx0sinxnsinnx(n>0)\displaystyle\lim_{x\to0}\frac{ \sin x^n}{\sin^n x}\quad (n>0)
(edited 10 years ago)
Reply 75
Original post by Lord of the Flies

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If that is what you meant by convergence?


That is what I mean by convergence.

You are almost right, but it converges for all 0<x<e^(1/e)
Original post by james22
That is what I mean by convergence.

You are almost right, but it converges for all 0<x<e^(1/e)


Really? I don't see how... When 1<x<e1/e,  f(x)1<x<e^{1/e},\;f(x) returns two values, no?
Original post by Lord of the Flies
Here's an easy one!

Question

Evaluate limx0sinxnsinnx(n>0)\displaystyle\lim_{x\to0}\frac{ \sin x^n}{\sin^n x}\quad (n>0)

Required

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Original post by Brit_Miller

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No, no it is defined. I said L'Hopital's is required, not that by simply applying it you would get the result :tongue:
Original post by Lord of the Flies
No, no it is defined. I said L'Hopital's is required, not that by simply applying it you would get the result :tongue:


Ah, I don't know enough about it to know what to do then. :biggrin:

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