UNIVERSITY OF DUBLIN
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Natural Science
SF Human Genetics
SF Chemistry with Molecular Modelling
SF Physics and Chemistry of Advanced
Materials
SF Medicinal Chemistry
Trinity Term 2010
Module MA22S1 - Sample Paper
Dr. D. Kitson
Attempt THREE questions.
Log tables are available from the invigilators, if required.
Non-programmable calculators are permitted for this examination,—please indicate the make
and model of your calculator on each answer book used.
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1. (a) Find the length of the space curve with parametric equations
x = sin t − t cos t, y = t2 , z = cos t + t sin t, 0 ≤ t ≤ π
(b) Compute the curvature for the smooth space curve with vector equation
r(t) = h
t3 t2
, , t i, −1 ≤ t ≤ 1
3 2
at the point P (0, 0, 0).
(c) Let C be the planar curve with vector equation
r(t) = h2t, 4t2 i, −1 ≤ t ≤ 1
Find the centre and radius of the osculating circle at the point P (0, 0).
2. (a) What is the maximum rate of change of the function
f (x, y, z) = yexz
at the point (0, 1, 2)? Find a scalar equation for the tangent plane to the level
surface f (x, y, z) = 1 at (0, 1, 2).
(b) Find the critical point(s) of the function
f (x, y) = y 2 + 2xy + 4x − 6y + 3
and determine if each critical point is a local maximum, local minimum or saddle
point.
(c) Use the method of Lagrange multipliers to find the maximum and minimum values
that the function
f (x, y, z) = 2x + 6y + 10z
takes subject to the constraint
x2 + y 2 + z 2 = 35.
3. (a) Use the component test to determine if the following vector field is conservative:
F(x, y, z) = h y, x, xyz i
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(b) Compute the line integrals
R
i. C 2xy ds
R
ii. C y dx − x dy
where C is the circle with centre (0, 0) and radius 2.
(c) Find a potential for the vector field
F(x, y, z) = h 2xz + y 2 , 2xy, x2 + 3z 2 i
and use it to evaluate the line integral
Z
F · dr
C
where C is the smooth space curve with vector equation
r(t) = h t2 , t + 1, 2t − 1 i, 0 ≤ t ≤ 1.
4. (a) Evaluate the following integral by reversing the order of integration.
Z 1Z 1
x
e y dy dx
0
x
(b) Compute the triple integral
ZZZ p
x2 + y 2 + z 2 dV
E
where E is the solid region
E = {(x, y, z) ∈ R3 : 1 ≤ x2 + y 2 + z 2 ≤ 4 and z ≥ 0}.
(c) Use the transformation (u, v) → (x, y) with x = 2u+v and y = u+2v to evaluate
the double integral
ZZ
4x − y dA
R
where R is the region in R2 bounded by the triangle with vertices (0, 0), (1, 2) and
(2, 1).
c UNIVERSITY OF DUBLIN 2010