Exam
„Analysis, engl.
für BSc. Mathe PO 2011, BSc. Wima PO 2011“
Fachbereich Mathematik
Prof. Dr. Reinhard Farwig
Summer Term 2019
August 15, 2019
Matriculation number:
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Surname: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
problem
1
2
3
4
5
6
7
8
9
10
11
P
max. points
6
12
6
8
8
12
10
10
12
8
12
104
grade
your points
Important Information
• Fill in the above form in capital letters and in a readable way and write your name and your matriculation
number on each of the sheets you hand in. Please number all the pages consecutively.
• All the statements and results should be proved and justified with the exception of the multiple choice exercises. Simply stating the final result is not sufficient and results in a loss of points.
• Use permanent pens, but not the colour red.
• Allowed auxiliary means: Two sheets of A4 format handwritten by yourself on front and back side are allowed
(or equivalently, four sheets of A4 format used on one side only).
• No calculator, or any other electronic device is allowed.
• Mobile phones must be switched off and stowed away.
• You have 180 minutes for the examination.
• Provide solutions in English. Solutions in German do not score points.
• Please, lay out your student card and a photo identification ready for checking purposes.
• Tip: Get a general idea of the problems before you start.
Good luck!
1
1. Problem (Mathematical induction)
(6 points)
Let n ∈ N. Prove the claim
2n
2n
X
X
1
1
(−1)k−1 =
.
k
m
m=n+1
k=1
2. Problem (Convergence of series)
(3+5+4 points)
i) Determine the radius of convergence of the series
∞
X
2n
n
n=0
x n.
ii) Decide whether the following series converge or diverge:
∞
X
∞
X
n!
,
nn
n=1
n=0
n2
n+4
.
− 3n + 1
iii) Decide whether the following series converges or diverges:
∞
X
(n + 1)n−1
(−n)n
n=1
.
3. Problem (Integration)
(6 points)
Calculate the following integrals
Z
π
Z
9
x cos(x) dx,
0
p
1
1
dx.
p
x(1 + x)2
4. Problem (Differentiation)
(4+4 points)
The function g : R → R is defined by
¨
g(x) :=
0,
if x = 0,
x 2 cos( 1x ), if x 6= 0.
i) Show that g is differentiable at every point x ∈ R and calculate its derivative.
ii) Decide at which points the derivative of g is continuous.
5. Problem (Continuity)
(8 points)
Let f : ( 0, 1] → R be a continuous function.
Show that the limit lim x&0 f (x) exists if and only if f is uniformly continuous.
2
6. Problem (Multiple Choice)
(12 points)
Decide which of the following options is correct. You do not have to justify your answer.
There is exactly one correct answer for each subproblem. Choosing the correct answer and only the correct answer scores
2 points, otherwise 0 points. If you correct an answer, indicate clearly which answer is supposed to be evaluated. In case
of doubt a false answer will be evaluated.
a) Let A, B, C be sets. Then we have
A \ (B ∪ C) = (A \ B) ∪ (A \ C),
A \ (B ∩ C) = (A \ B) ∩ (A \ C),
A ∪ B = A ∪ (B \ (A ∩ B)).
b) The set A = {x ∈ R : x ∈ Q and x 2 ≤ 2} ∪ {x ∈ R : x < 0} satisfies
p
max(A) = 2,
p
sup(A) = 2,
p
inf(A) = − 2.
p
c) The equation x 3 = 4 has the solutions 3 4 and
p p p
p
3 4 − 12 + i 23 , 3 4 23 − i 12 ,
p
p
− 3 4 cos(− π3 ) + i sin(− π3 ) , − 3 4 cos( π3 ) + i sin( π3 ) ,
p
p
3
2π
4π
4π
− 3 4 cos( 2π
3 ) + i sin( 3 ) , − 4 cos( 3 ) + i sin( 3 ) .
d) Which of the following functions does not have a continuous inverse?
f : (0, ∞) → (−∞, 0) ∪ [2, ∞) surjective, discontinuous and strictly monotonically increasing,
g : [0, 1] → [cos(1), 1],
x 7→ cos(x),
¨
x
, if x ∈ [0, 1)
h : [0, 1) ∪ [2, 3] → [0, 2], x 7→
x − 1 , if x ∈ [2, 3].
e) Let (an )n∈N be a strictly monotonically
increasing sequence of natural numbers. Furthermore, let
bn := sin(−an ), cn := exp an1+1 , dn := ln(1 + 2an+1 − an ) for all n ∈ N.
How many of the sequences (bn )n∈N , (cn )n∈N and (dn )n∈N have at least one real cluster point?
exactly two,
exactly one,
none.
f) Let [a, b] ⊂ R be a compact interval and let f : [a, b] → R be a continuous and injective function. Then
f is bijective,
f is strictly monotonic,
f is differentiable.
3
7. Problem
(4+4+2 points)
Consider
F : R4 → R2 , (w, x, y, z) 7→
− arctan(x)w2 + sin(z) y
.
x 2 + sin(w y) + 2 cos(z)
i) Show that there exists a differentiable function g = (g1 , g2 ) defined on a neighbourhood U ⊂ R2 of (1, 0) such that
g(1, 0) = (0, π2 ) and F (w, x, g(w, x)) = 0 for all (w, x) ∈ U .
ii) Calculate g 0 (1, 0) with g as in i).
iii) Is there a function h : R2 → R2 such that F −1 {(0, 0)} = {(w, x, h(w, x)) : w, x ∈ R}?
8. Problem
(3+1+6 points)
R
i) We consider the set M := (x, y) ∈ R2 : x 2 + y 2 ≤ 1, x ≥ 0, y ≤ 0 . Calculate M y d(x, y).
R∞
ii) Let f : R → [0, ∞) be a continuous function such that the improper Riemann integral 1 f ( y) d y exists.
R∞
a) Show that 1 f ( y) p1y d y exists.
b) Show that the limit
lim"&0
R1R
"
1
x2
1
f ( y) d y d x exists and that it coincides with
9. Problem
R∞
1
f ( y) p1y d y .
(5+3+4 points)
R
i) Let F := (u, v , cosh(u) sin(v )) ∈ R3 : u ∈ [0, 1], v ∈ [0, π2 ] . Calculate F H dσ with
z
H : R3 → R, (x, y, z) 7→ Æ
.
2
cosh (x) + cos2 ( y)
Hint: cosh(x) = 12 (e x + e−x ) and sinh(x) = 21 (e x − e−x )
for all x ∈ R.
ii) Let Ω ⊂ R3 be a C 1 - normal domain. Calculate the surface integral
Z
arctan( y)z
exp(x 2 ) · N dσ.
∂ Ω cos(x 4 + y)
iii) Let ∂ D be the boundary of D := [−1, 1] × [0, 1] ⊂ R2 that is parametrised by a piecewise C 1 -path X : [0, 1] → ∂ D
with negative orientation. Calculate
Z
5yx
· dX .
x3
∂D
10. Problem
(8 points)
For every n ∈ N let f n : [0, 1] → R be a monotonically increasing function. Furthermore, the sequence of functions
( f n )n∈N converges pointwise to a continuous function f : [0, 1] → R. Show that ( f n )n∈N converges uniformly to f .
Hint: First show that for every " > 0 there exists a k ∈ N \ {0} and an N ∈ N with the following properties:
i) For all x, y ∈ [0, 1] with |x − y| ≤
1
k
it holds | f (x) − f ( y)| < 2" .
ii) For all j ∈ {0, . . . , k} and all n ∈ N with n ≥ N it holds | f n ( k ) − f ( k )| < 2" .
j
j
4
11. Problem (Multiple Choice)
(12 points)
Decide which of the following options is correct. You do not have to justify your answer.
There is exactly one correct answer for each subproblem. Choosing the correct answer and only the correct answer scores
2 points, otherwise 0 points. If you correct an answer, indicate clearly which answer is supposed to be evaluated. In case
of doubt a false answer will be evaluated.
a) Which of the following functions has an antiderivative on R2 ?
y e x y +x 10
f (x, y) = x e y x +cos( y)
g(x, y) =
ey
ex
h(x, y) =
π+x 2
x2
b) The Fourier series of | · | : [−π, π] → R, x 7→ |x| is given by
P∞
a0
2 +
n=1 an cos(nx) with a0 6= 0.
P∞
n=1 an cos(nx).
P∞
n=1 an sin(nx).
c) Let b > 0 and γ : [0, b] → R2 be a rectifiable path. We define γ̃ : [−b, b] → R2 via γ̃(t) := γ(|t||)| for all t ∈ [−b, b].
Then the path γ̃ is
not rectifiable.
rectifiable with L(γ̃) = L(γ).
rectifiable with L(γ̃) = 2L(γ).
d) Consider f : R2 → R, (x, y) 7→ x y and g : R2 → R, (x, y) 7→ x 2 + 4 y 2 − 2.
Then the following point is an extremum of f under the constraint g(x, y) = 0:
p
( 2, 0)
(1, 21 )
(0, 0)
e) The function
Z
f : R → R,
2
x 7→
1
is not differentiable.
is differentiable with f 0 (x) =
sin(2x)
x
−
is differentiable with f 0 (x) =
sin(2x)
2
− sin(x) for all x ∈ R.
sin(x)
x
sin(t x)
dt
t
for all x ∈ R\{0}.
f) Let D be the open unit disk in R2 . Which of the following subsets D j of D is Jordan measurable?
D1 = D\{(r cos(ϕ), r sin(ϕ)) : r ∈ [0, 1) ∩ Q, ϕ ∈ [0, 2π)}
D2 = D\{(r cos(ϕ), r sin(ϕ)) : r ∈ [0, 1), ϕ ∈ [0, 2π) ∩ Q}
1
1
D3 = D\ 1+t
cos(t), 1+t
sin(t) : t ∈ (0, ∞)
5
