The 2013 STEP Prep Thread!

This line, second part of IBP in the bracket. In general $\displaystyle\int \left(\int f dx \right) g dx \ne \int f dx \int g dx$. The inner integral is not definite, as IBP refers to the function integrated.

With STEP papers, can somebody give me a good year so start on? I'm planning on working backwards, and saving the past few years for mocks. Going to focus on STEP I to begin with.

Here is a question I’ve adapted from somewhere else that could be used.

EDIT: I am looking forward to Mechanics and Stats questions. Any ideas for a question that uses Bayes' theorem?

EDIT: I am looking forward to Mechanics and Stats questions. Any ideas for a question that uses Bayes' theorem?

It is a good question, but I think it needs more hints. Also, you could consider turning A3 1998 into a STEP question.
In my opinion, the latter is easier to grasp and explain (i.e. IVT is intuitive) and makes a nice use of the MVT.

EDIT: I am looking forward to Mechanics and Stats questions. Any ideas for a question that uses Bayes' theorem?

'Choose two random numbers from $[0, 1]$ and let them be the endpoints of an interval. Repeat this $n$ times. What is the probability that there is an interval which intersects all others?'

What level of questions are in the Siklo's booklet? I.e STEP I, II or III? If it's all 3, is there a way to tell which?

Another one I like is 'Suppose you play a game with a friend where he writes down randomly two positive integers on separate pieces a paper. You then pick one of the pieces of paper and try to guess whether the number on it is greater than the other number. Exhibit a strategy whereby you guess correctly with probability greater than 0.5 or show that such a strategy does not exist.'

Again I would intuitively say the latter should hold, but there does exist such a strategy.

Hi guys
I've been attempting STEP I & II questions for a while now just going through papers, doing as many questions as I can and as much of a question as I can. But basically I seem to be getting nowhere, I'm lucky if I go through a whole paper and get even 1 question completely right. Most of the time I start a question, write a few lines, get stuck, think about it for forever and eventually resort to looking at the answers.
I thought I would gradually get better but that's not happening. Has anyone got any advice on what to do because I had originally intended to do STEP I, II and III this summer but now I just don't see how I'll manage it when I can barley do any questions and I'm not seeing any signs of improving at all

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The point being, in the end, the questions chosen could be considered "sensible" according to the people here.

Can we somehow get the general opinion for questions? So far, people have said a little about the suggestions.
The point being, in the end, the questions chosen could be considered "sensible" according to the people here.

Any feedback on Shamika's suggestion?

Whoa! Check this beauty out! This is why I call it magic differentiation!

Let $\displaystyle I(\lambda) = \int_{0}^{1}x^{\lambda}\;{dx} = \frac{1}{1+\lambda}.$ Differentiating this w.r.t $\lambda$ we have:

$\displaystyle I^{(n)}(\lambda) = \int_{0}^{1}x^{\lambda}\log^{n}{x}\;{dx} = \frac{(-1)^nn!}{(1+\lambda)^{n+1}}$ -- therefore we have:

$\displaystyle \int_{0}^{1} x^{-x} \;{dx} = \sum_{n \ge 0}\frac{(-1)^n}{n!} \int_{0}^{1}x^n\ln^n\;{dx} = \sum_{n \ge 0}\frac{1}{(1+n)^{n+1}}$

Also $\displaystyle \int_{0}^{1} x^{x} \;{dx} = \sum_{n \ge 0}\frac{1}{n!} \int_{0}^{1}x^n\ln^n\;{dx} = \sum_{n \ge 0}\frac{(-1)^n}{(1+n)^{n+1}}.$

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