For Q. 4(a) the first thing you should do is draw as sketch of the situation and convince yourself that the conclusion must be true. Now look at the hint: consider the function . Now and are both continuous so must be also. Now note that

and

Now as satisfies the Intermediate Theorem and goes from +1 to -1 between 0 and 1 there must be a point where . Now note that from whence it follows that ; i.e. the curves intersect at .

Look at Autumn 2012 and Summer 2012 for your optimisation practise.

To find an asymptotic of

just take the highest powers above and below, for example

so that is an asymptotic for .

My advice for the exam is to answer the question that is asked.

Regards,

J.P.

I have a few queries about my revision for the repeats, if you wouldn’t mind answering them.

Looking through the past exams, in particular Q.4 (a) in the autumn 2012. What do I do? I don’t just state the intermediate value theorem do I?

On the summer 2012 paper there is an optimisation question and I haven’t found any others. My notes are lacking in places and I’m just wondering did we do optimisation?

Also how do I find an asymptotic for f(x) ?

Do you have any advice for the exam itself?

Thanks.

]]>Not quite. The function is smooth (and thus continuous) for and . But what about ?

For continuity at you want that

and to do this you usually just check that the limit exists by proving that the left-hand limit is equal to the right-hand limit (usually one of these will equal ) so we look at

and

.

You need these equal for continuity so you need .

Now if and then you are not continuous and hence not differentiable.

Regards,

J.P.

I have a question relating to an exam question on the 2012 paper Q3 (a).

Am I right in thinking that the function is continuous for all values of and as when is less than zero, it is just a line with a constant slope and when is greater than zero, it is a smooth curve?

Thanks for your help.

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