End of Term Worksheet Assessment
Note:
Write your Student Number at the top of all pages submitted.
Show all working in the derivations you make and spread your working out well.
I suggest you use A4 sized, lined paper to help with the layout and that you staple the pages.
Hand your solutions in at the School Office, at the latest by
4:00 pm on Thursday 10th December.
Attempt all FIVE questions.
1.
(a) For the first order initial value problem:
, ?0? 5,
2
? xy y ?
dx
dy
use the Euler iteration method i.e.
n n hfn
y ?1 ? y ?
to calculate the values yn+1, for n = 0,1,2 using a step size h = 0.1.
[3 marks]
(b) By separating variables solve the equation exactly and hence compute the errors in the
Euler approximation for the three iteration steps (writing your answers to 6 decimal
places).
[5 marks]
2.
Consider a bar of metal of unit length in which heat transfer between the bar and it
surroundings is assumed to obey the heat equation given by:
, 0 1, 0 2
2
? ? ? ?
?
?
?
?
?
u x t
x
u
t
u
u?0,t? ? u?1,t? ? 0, t ? 0
u?x,0? ? f ?x?, 0 ? x ?1.
(a) By assuming that the solution is of the form:
u?x,t? ? X?x?T?t?,
show that:
X?? ? ?X ? 0
and
T? ? ?1? ??T ? 0
where ? is an unknown constant.
[4 marks]
(b) By considering the X variable equation for all possible ? and applying appropriate
boundary conditions, show that the only possible (non-trivial) solutions are:
X ? A sin?n x?, n ?1,2,3,… n n ?
corresponding to the choices:
, 1,2,3,…
2 2
?n
? ?n ? n ?
where An are unknown constants.
[10 marks]
(c) Hence, obtain the solution:
? ? ? ?
n t
n
n
t
u x t e C n x e
2 2
, sin
1
?
?
?
?
?
?
? ?
where the Cn are left as unknown constants.
[4 marks]
3.
It is estimated that the probability of a rocket exploding during lift-off is 0.02 and that the
chance of its guidance system failing is 0.05, where these events are independent. Find the
probabilities that:
(a) The rocket will not explode during lift-off.
[2 marks]
(b) It will either explode or have its guidance system fail.
[2 marks]
(c) It will not explode and not have a guidance system failure.
[2 marks]
4.
At a service till customers arrive at an average rate of 2.5 per minute.
(a) Write down the statistical model for the number of people arriving each minute.
[1 mark]
(b) Find the probability that at most 3 will arrive in any given minute.
[2 marks]
(c) Find the probability that at least 3 will arrive during an interval of two minutes.
[3 marks]
(d) Use the Normal approximation to find the probability that at least 20 will arrive in an
interval of six minutes.
[3 marks]
5.
The mileage (in thousands of miles) that car owners get from a certain type of tyre is a
random variable x having the probability density:
f (x) = 0 for x < 0
? ?
20
x 20
e
f x
?
? for
x ? 0.
(a) Find the mean mileage achieved with this type of tyre.
[3 marks]
(b) Find the probability that a tyre fails before 10,000 miles have been driven on it.
[2 marks]
(c) Find the probability that it lasts for at least 30,000 miles.
[2 marks]
(d) Find the median lifetime of the tyre.
[2 marks]