Calculus 2
May 31, 2021, 13:00-15:00
• Write your student-number and name CLEARLY VISIBLE on every
exam paper you use (in CAPITAL LETTERS).
• This exam has 4 questions. On the last page you find a list of
formulas that you can use.
• The use of a graphical calculator or programmable calculator is
prohibited. A simple scientific calculator is allowed.
• Motivate every answer you give with some calculation or argumentation.
• On the left in the margin you will find the maximum points available for every exercise. Your exam grade will be calculated as (points
gained)/10.
30
10
10
1.
a) Determine the points of intersection of the graphs of f (x) = x6 and
g(x) = x, sketch the area that is enclosed by the graphs of f (x) and
g(x) and determine the surface of this area.
b) Determine the antiderivatives of f (x) =
Z 100
f (x)dx.
proper integral
x
√ ex
e −1
and calculate the im-
0
10
25
5
10
c) Determine the antiderivatives of (3x + 5) cos 4x.
2.
Given is the function f (x, y) = x8 − 8xy + y 8 .
a) Prove that f (x, y) = (x4 − y 4 )2 + 2(x2 y 2 − 1)2 + 4(xy − 1)2 − 6.
b) Determine ∂f
∂x ,
points of f .
∂f
∂y ,
and show that (0, 0), (1, 1), (−1, −1) are the critical
1
2
10
25
6
6
c) Check for each of the critical points from b) whether f takes on a
maximum or minimum, or whether it is a saddle point of f . Check
for the eventual maxima or minima if they are global or local.
3.
a) Write
(3+2i)(5−i)
6+i
in the form a + bi.
√
b) Write (5 3 − 5i)171 in the form a + bi.
6
c) Determine the solutions of iz 2 + (1 + 3i)z + 3 = 0 and write the
solutions in the form a + bi.
7
d) Determine the solutions of z 12 = 1012 i and write them in the form
ρ(cos ψ + i sin ψ) with ρ > 0.
20
10
10
4.
∞ X
(−2)n
(−4)n
a) Calculate
+2 n
.
n
3
5
n=0
√
∞
X
3 10 n + 4
b) Check whether
converges or diverges. You are allowed
2+1
n
n=1
∞
X
n−s converges if s > 1 and diverges if s ≤ 1.
to use that
n=1
Trigonometric formulas
sin(x + y) = sin x · cos y + cos x · siny;
cos(x + y) = cos x · cos y − sin x · sin y;
sin 0 = cos π2 = 0; sin π2 = cos 0 = 1;
√
√
sin π6 = cos π3 = 12 ; sin π3 = cos π6 = 12 3; sin π4 = cos π4 = 12 2.