Analysis 2 (MATS3103)
Mathematics and Computer Science - MCS
Semester: Resit
Level: 300
Year: 2015
RESIT EXAMINATION 2015/2015 SESSION
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: MATS3103 COURSE TITLE: Analysis II
DURATION: 2 HRS Answer all questions.
All the steps must be justify.
Exercise 1 Calculate the following limits.
1) a)
b)
(x), with u(x)
, v(x) =
(sin
) ln (
)
2) f is a numerical function
;
a and l
a) By using the definition that If f converge in a to l, then ……..
b) Give an example to show that the converse of a) is wrong.
Exercise 2 Let f be a real function define in an interval (a, b) of , where x ! "
#$%f is said Lipschitzion in (a,b) if there exist a constant c independent of x such that:
Vx,x’ (a,b), |f(x) – f (x’)| &
#
'
(
#
%
1) Suppos e d t h a t f i s derivable i n (a, b) and is der i v a t i v e f’ is continuo u s
and bounded in (a,b). show that f is Lipschitzan in (a, b). One can use
the equality f(x) –f(x’) =
)
*
(
+
,+-
(
for all x, x’ in (a,b).
2) Show th a t t h e f u n ction f(x) = Arctan (x) is Lipschitzian in . It is
uniformly continuous in ?
3) We consider the function f (x) = ln (x) define in
|
*.
i) Show that the function f is uniformly continuous in all closed segment content
in
+
*.
ii) For all n ., simplify the quantity In (c
-n-1
) — ln (e
-n
)
iii) Show that for all a > 0, the function f is not uniformly continuous in ]0,/% .
4) Compute cos (Arctan x), sin (Arctan x), tan (2Arctan x), One can set y = Arctan x
and express x in term of cosy and siny then express. cosy in terms of x
2
.
Exercise 3
We call hyperbolic cosine function and hyperbolic sine function and
denote respectively ch and sh the functions define in
by ch(x) =
0
and sh(x) =
0
.
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1) Show that the function ch is even and the function sh is odd.
2) Show that for all x
, we have: ch
2
x — sh
2
x = 1 and ch
2
x sh
2
x = ch (2x .
3) Show that those two functions are continuous and derivables.
4) a) Show that the function sh :
is strictly nondecreasing.
b) Deduce that sh :
admit a converse denoted Arg sh.
c) Draw the graph of the functions sh :
and Arg sh :
in the same
orthonormal landmark.
d)Show that for all x
[1, +
∞
[
one has the equality Argsh (x) = in (x +
1
2 3) .
5) Do a study similar to question
4)
for the function hyperbolic tangent denoted
th and define in
by th (x)=
4
54
Each question is on 3/70 marks
Examiner: Fotso Tachago Joel (Msc).
Try, Try again, Try for ever. Good Luck!
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