Delft University of Technology
Faculty of Electrical Engineering, Mathematics and Computer Science
Mekelweg 4, Delft
Examination Calculus I for AE (wi1421LR), part A
Tuesday, January 24, 2017, 9:00 - 11:00 a.m.
It is allowed to use the formulae sheet for Calculus
It is not allowed to use a calculator
1. The volume of the parallelepiped determined by the vectors a = h2, −1, 1i,
b = h−1, 1, 2i and c = h1, 1, −2i equals
A. −20
E.
5
−10
F.
10
B.
C. −5
G. 15
D. 0
H. 20
2. The limit lim
x→∞
√
x2 + 5x −
√
x2 − 4x equals
A. −9
E.
1
B.
− 92
F.
9
2
C.
0
G. 9
D.
1
2
H. ∞
3. An equation of the tangent line to the graph given by 2(x2 + y 2 )2 = 25(x2 − y 2 ) at the
point (3, −1) is
9
A. y − 1 = − 13
(x + 3)
E.
y−1=
9
y + 1 = − 13
(x − 3)
F.
y+1=
B.
9
13 (x
9
13 (x
+ 3)
− 3)
C. y − 1 = −3(x + 3)
G. y − 1 = 3(x + 3)
D. y + 1 = −3(x − 3)
H. y + 1 = 3(x − 3)
√
4. The derivative of y = x arctan(x) − ln( 1 + x2 ) is
A. arctan(x)
1
1 + x2
2x
C. arctan(x) − √
1 + x2
1
1
D. arctan(x) +
−√
2
1+x
1 + x2
B.
arctan(x) − √
1
x
−√
1 + x2
1 + x2
1
2x
F. arctan(x) +
−√
2
1+x
1 + x2
x
2x
G. arctan(x) +
−√
2
1+x
1 + x2
2x
H. arctan(x) +
1 + x2
E.
P.T.O.
arctan(x) +
1
Z
5. The definite integral
−1
(arctan(x))2
dx equals
1 + x2
A. 0
B.
C.
D.
E.
1 2
π
32
1 2
π
16
1 3
π
192
F.
G.
H.
Z
6. For x > 0 the indefinite integral
ln
1 3
π
96
1 3
π
48
1 3
π
32
1 3
π
16
√ x dx is equal to
√
A. x ln( x) + C
√
B. x ln( x) − 2x + C
√
C. x ln( x) − x + C
√
D. x ln( x) − 21 x + C
√
√
x ln( x) − 2x x + C
√
√
F. x ln( x) − x x + C
√
√
G. x ln( x) − 12 x x + C
√
√
H. x ln( x) − x + C
E.
Z
7. The improper integral
∞
√
e−
x
dx is
0
A. divergent
E.
equal to 1
equal to 2
B.
equal to −2
F.
C.
equal to −1
G. equal to 3
D. equal to 0
H. equal to 4
8. A tank contains 100 ` beer with 5 % alcohol. Beer with 7 % alcohol is pumped into the
tank at a rate of 1 `/min. The fluid in the tank is kept thoroughly mixed and drains
from the tank at a rate of 1 `/min. What is the alcohol percentage of the beer in the
tank after 1 hour (60 minutes)?
A.
B.
C.
D.
7 − 2 e−3/5
100
5 − e−3/5
100
5 + e−3/5
100
7 + 2 e−3/5
100
E.
7 − 2 e−3/5
F.
5 − e−3/5
G. 5 + e−3/5
H. 7 + 2 e−3/5
√
( 3 + i)12
9. The complex number √
equals
( 3 − i)9
A. −8i
E.
−8
B.
−4i
F.
−4
C.
4i
G. 4
D. 8i
H. 8
The answers can (soon) be found on Blackboard