Hi over the festive period i have been trying some past papers, just to get ready for the summer, ive got stuck on a few of the questions and wondering if somebody can help me, the link to the paper is bellow i hope that takes you to it.

https://www.kent.ac.uk/library/onlin...2009/MA301.pdf

The first page is fine, however going on to section B it gets a bit more tricky.

Question 7 using the definition of the derivative i curreny have got the following

a) ((x+h)^-2-(x)^-2))/h

= ((1/(x+h)^2-1/(x)^2)*1/h

=((1/x^2+2hx+h^2)-1/(x^2))*1/h

=((x^2-(x+h)^2)/(x^2(x+h)^2)*1/h

Basically trying to simplify it down in order to get the needed -2x^-3, somehow i need to cancell the h outside, then i think it should be ok as h tends to infinity.

Have the same problem with the other question in 7.

Question 8

Im stuggling with (i) managed to get (ii) via completing the square then using a change of variable.

I know on (i) the function will integrate to something arctan as ive put it into a integration calculator, just wondering if im missing something obvious here.

Also B i wouldnt know where to start.

Question 9 im just wondering if my answers are correct really, i believe i have definately got b, however with a i have:

partial df/dx=e^x(siny) partial df/dy=0 second partial df/dx=0 and second partial df/dy=0

Therefore shwoing laplaces equation = 0

if that is correct, i very much doubt it though, how to obtain the fourth partial derivative without any calculations.

As for question 10 im not sure if i should go there just yet as a few questions have already been asked and might get a bit messy.

Many thanks in advance once more.