MTH118 Analysis II 2021/22
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MTH118
2nd SEMESTER 2021/22 FINAL EXAMINATION
BACHELOR DEGREE - Year 2
Analysis II
Questions
Q 1. (a) (10 marks) Define a function f : [0, 
] → R by
f (x) = 
if x ← Q;
otherwise.
Let P = f0 = x0 < x1 < . . . < xi-1 < xi = 
} be a partition of [0, 
]. Find the upper sum U (f, P) and the lower sum L(f, P) (the answers can be given in terms of the numbers xi for i = 0, 1, 2, . . . , n).
(b) (20 marks) Determine, providing a proof or a counterexample, whether the following state- ments are true or false:
(i) If f : [0, 1] → R is bounded and integrable on [0, 1], then |ef(。)| is integrable on [0, 1]. (ii) If f : [0, 1] → R is integrable, then there exists F : [0, 1] → R such that F\ = f .
[30 marks]
|
Q 2. |
(a) (15 marks) Let f : R → R be a continuous function and let 。 F (x) = xf (t)dt. 0 Quoting clearly any theorems used, find the derivative F\ (x). (b) (20 marks) Using L’Hospital’s Rules, or otherwise, find the value of the limit
|
and hence determine whether or not the improper integral
( 一 tan-1 (x))dx
converges.
[35 marks]
Q 3. (a) (15 marks) Justify whether or not the following exchange in order of integration and sum-
mation yields a correct equality:
1 o xi o 1 xi
-1 i=1 n2 i=1 -1 n2
(b) (20 marks) The Fourier series of the 2π-periodic function defined by f (x) = |x| on the interval [一π, π] is given by
o
f (x) ~ 一
i=1
Justifying the convergence of the series differentiated term by term at a suitable point or otherwise, find the sum of the following infinite alternating sign series:
S = 1 一 
+ 
一 
+ 
一 . . .
[35 marks]
2022-07-27