UNIVERSITY OF DUBLIN
XMA1211
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
JF Mathematics
JF Theoretical Physics
JF TSM, SF TSM
Trinity Term 2008
Course 121
Tuesday, May 27
Goldsmith Hall
9:30–12:30
Dr. P. Karageorgis
Attempt all questions. All questions are weighted equally.
Non-programmable calculators are permitted for this examination.
Log tables are available from the invigilators, if required.
Page 2 of 2
XMA1211
1. Let A, B be nonempty subsets of R such that inf A < inf B. Show that there exists an
element a ∈ A which is a lower bound of B.
2. Let f be the function defined by

 3x + 1
f (x) =
 6 − 2x

if x ∈ Q 
.
if x ∈
/Q 
Show that f is continuous at y = 1.
3. Show that 2e · x2 log x ≥ −1 for all x > 0. Here, e is the usual constant e ≈ 2.718.
4. Compute each of the following integrals:
Z
4x2 − 5x + 2
dx,
x3 − x2
Z
sin3 x dx.
5. Using the mean value theorem, or otherwise, show that
(b − a)ea < eb − ea < (b − a)eb
whenever a < b.
6. Test each of the following series for convergence:
∞
X
2n + 4n
n=1
3n
+
5n
,
∞
X
sin(1/n2 ).
n=1
7. Suppose f, g are integrable on [a, b] with f (x) ≤ g(x) for all x ∈ [a, b]. Show that
Z b
Z b
f (x) dx ≤
g(x) dx.
a
a
8. Letting f (x, y) = log(x2 + y 2 ), find the rate at which f is changing at the point (2, 3)
in the direction of the vector v = h3, 4i.
9. Classify the critical points of the function defined by f (x, y) = x2 + 2y 2 − x2 y.
10. Compute the double integrals (I have included several of those for practice)
Z 1Z 1
Z 2Z 4
Z πZ π
sin y
2
2 xy
dy dx,
x e dx dy,
xey dy dx.
y
0
y
0
x2
0
x
c UNIVERSITY OF DUBLIN 2008
°