A Summer of Maths

Similar things happen in other cases, but I don't really know enough about this to tell you (i) under what conditions noninteger derivatives exist, or (ii) what they are useful for, if anything.

Feynman speaks about half derivatives and other stuff in one of his books.

It is quite interesting though, what information this may provide about the behaviour of a function.. does anyone know Newton's e-mail?

Woops, I wrote that without thinking. I meant to say (e^a)^b = e^(ab) loses validity. Sorry about the confusion, I should head to bed!

Hardy har. I'm not sure how much help he'd be even if he did have email...

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:


Hints:


Full Solution:

I think ill get at least AA in maths and further maths in August and will be starting a maths degree at Nottingham with some luck, and I havent got a clue where to start on the majority of the problems posted on here... how do I go about starting some of these problems that dont 'lead' you through what to do as A level questions do, and should I be a bit worried?

your on part III already- but you understanding is VERY VERY weak!

I don't get how the diagonal of the circle is the diameter? Is that always the case?

Am I? Perhaps, your's is better.

goes through the centre.....

Oh yeah.

To see it, you can consider Thales' theorem (done at A-level!), and notice that you can construct any inscribed rectangle from the right-angled triangle and its replica.

Alternatively, parametrise the point $(x,y)$ on the unit circle (without loss of generality) by $y = \sin\theta$ and $x = \cos\theta$.
By doing so, you can find the diagonal of any inscribed rectangle (under an assumption that does not matter) and show it goes through the origin.

I don't really understand, but thanks.

Which method don't you understand? I can try to give more details, or explain it differently.
The purpose of this thread is to help people learn and understand new things. If it fails to do so, then it is useless.

Well the circle has a radius of 6, so I don't get how you can use the unit-circle co-ordinates with it. Although generally I can see how that would help explain it.

To prove this, you use the angle from the horizontal to the ray from the origin to that point, and nothing else from circle's properties!
This angle does not change when you rescale the graph of the circle; i.e. it is the same for all circles and is called an 'invariant' property (apparently, by construction).

In other words, if you fix $\theta$, then your result will be true for all $(a\cos\theta, a\sin\theta)$ with $a > 0$. If true for all required $\theta$, then it is true for all circles.
The thing is, you have a symmetric choice before choosing the point, but once you choose it, all the rest is fixed about this problem.

Using Thales' theorem is a much simpler (and more interesting) approach to show this feature of the problem.

What is the expression



equal to as a number?

I like this one; it leads to one of those "when you see it you'll **** bricks" moments. Even more so when you discover that it's equal to $\sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}$.