MAS243 Exam 2011/12
(i) The function f (x, y) = (2 + cos(x)(2 + sin(y) has four critical points with 0 x, y < 2.
Find and classify these critical points, and determine the maximum and minimum values of
f . (13 marks)
(ii) Use the method of Lagrange mu
MAS243 PROBLEMS WEEK 2 CONSTRAINED OPTIMISATION
Exercise 1. Show that the function u = x3 + x2 y y 2 2x2 has critical points at (0, 0), (1, 1/2) and
(4, 8), and determine their nature.
Exercise 2. Find all the critical points of the function
u = x3 3xy 2
MAS243 PROBLEMS WEEK 3 INTEGRALS OVER PLANE REGIONS
Exercise 1. Evaluate the following integrals, and sketch the corresponding regions in the (x, y)-plane.
(2x2 + y 2 ) dy dx
x ey dy dx
y 2 dy dx
MAS243 Exam 2010/11
(i) The function f (x, y) = y 2 + 3x4 4x3 12x2 has precisely three critical points, precisely
one of which is a global minimum. Find and classify the critical points, and thus nd the
minimum value of f (x, y). (13 marks)
(ii) Use t
MAS243 EXAM SOLUTIONS 2008/09
(i) The partial derivatives are Fx = 2x + 4xy and Fy = 2y + 2x2 3, so the critical points are the
2x + 4xy = 0
2y + 2x2 3 = 0.
We can rearrange (B) to give y = 2 x2 , and substitute this into (A) to
MAS243 PROBLEMS WEEK 6 VECTOR ALGEBRA AND GRADIENTS
Unless otherwise specied, r refers to the position vector r = (x, y, z), and r = |r| =
x2 + y 2 + z 2 .
Exercise 1. Consider the vectors p = 2i j, q = 4i 2j and r = 2i + 4j. Which of them are parallel to
MAS243 PROBLEMS WEEK 5
CYLINDRICAL AND SPHERICAL INTEGRALS
Exercise 1. The point P has rectangular coordinates x = y = 0 and z = 1, and the point Q has
cylindrical coordinates r, and z. What is the distance from P to Q?
Exercise 2. Let D be a at-topped ci
MAS243 PROBLEMS WEEK 1 OPTIMISATION
Please enter your answers online at http:/aim.shef.ac.uk/AiM by 3AM on Monday February 13th.
To log in there, enter your email address without the @sheffield.ac.uk sux (something like JSmith1),
select MAS243 and click S
MAS243 PROBLEMS WEEK 10 SURFACE INTEGRALS AND THE
Exercise 1. Show that
sin() cos2 () = 4 (sin(3) + sin()
cos3 () =
(You will need these identities in the questions below.)
Exercise 2. Evaluate S F.dA, where F
SCHOOL OF MATHEMATICS AND STATISTICS
Mathematics IV (Electrical)
Solutions to examination questions
The stationary points of the function F (x, y) occur when
In this case we ha
MAS243 PROBLEMS WEEK 8 POLAR FIELDS AND LINE INTEGRALS
There are formulae for div, grad and curl in polar coordinates on the back of this sheet.
Exercise 1. Let u be the vector eld given in spherical polar coordinates by
u = r2 cos()er + r1 e + (r sin()1
MAS243 PROBLEMS WEEK 7 DIV AND CURL
You should enter your answers (for most questions) online at http:/aim.shef.ac.uk/AiM by 3AM
on Monday 26th March.
Exercise 1. Find
u for the following vector elds:
u = (xy, yz, 0)
u = (z, x, y).
MAS243 PROBLEMS WEEK 4
PLANE POLAR INTEGRALS AND VOLUME INTEGRALS
Exercise 1. Consider the integral given in polar coordinates by I =
the corresponding region in the (x, y)-plane, and evaluate the integral.
sin() dr d. Sketch
xy dA, where D
MAS243 PROBLEMS WEEK 9
THE TWO-DIMENSIONAL DIVERGENCE THEOREM AND GREENS THEOREM
Exercise 1. Let D be the disc of radius a centred at (0, 0), and let u be the vector eld (xy 2 , 0). Let
C be the boundary curve of D. Verify the divergence theorem D div(u)
MAS243 PROBLEMS WEEK 11 STOKESS THEOREM
Exercise 1. Let C be the vertical circle given by y = a sin(t) and z = a cos(t) with x = 0. Use Stokess
Theorem to evaluate C (x2 y, z, 0).dr. Check your answer by calculating the integral directly.
Exercise 2. Cons