Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Enineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 4, November 12/13, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Find an upper and a lower bound for the absolute value of the following function:
f(x) := ex+ 3 sin(x)
e−2x+ 3 , x∈[0,ln(5)] . 2. Sketch the following sets:
A={(x, y)∈R×R| |x|+|y| ≤1}, (1) B =
(x, y)∈R×R| max{|x|,|y|}>1 , (2) C ={(x, y)∈R×R| |x+ 3|<5,|y−2|<3}, (3) D={(x, y)∈R×R|1≤x2+y2 ≤4}, (4) E ={(x, y)∈R×R|x6∈[−2,4], y 6∈[1,3]}, (5) F ={(x, y)∈R×R|x6∈[−2,4] or y6∈[1,3]}. (6) 3. Find allx∈R which solve the following equations:
(x−2)2−4≤0, (7)
(x−1)(x+ 1)(x−2)≤0, (8)
3x+ 2
x−1 <4, (9)
ax−4>2x−1, a ∈R fixed. (10) 4. Find the infimum and supremum of the following set and decide whether they are
the minimum and the maximum of the set:
M :=
2n−3
n+ 1 | n∈N
.
5. Solve the following equations and inequalities:
|x−5|= 1 , (11)
|x−1|+x= 2−x , (12)
|x+ 1| −x≥1, (13)
|x+ 3|<4− |x−2|. (14)
Homework
1. Find an upper and a lower bound for the absolute value of the following function:
f(x) := 1 +x2
2−x3, x∈[−1,1] . 2. Sketch the following sets:
A={(x, y)∈R×R| |x| − |y| ≥1}, (15) B =
(x, y)∈R×R| min{|x|,|y|}>1 , (16) C ={(x, y)∈R×R| |x−2|<4,|y−2|>3}, (17) D={(x, y)∈R×R|5≥x2+y2 ≥4}, (18) E ={(x, y)∈R×R|x∈[−2,4], y 6∈[1,3]}, (19) F ={(x, y)∈R×R|x6∈[−2,4] or y∈[1,3]}. (20) 3. Find allx∈R which solve the following equations:
√2x−4−√
x−1≤1, (21)
x2−3x+ 2≤0, (22)
x2+x−6≤0, (23)
10x+ 2
x+ 5 < 9x+ 3
x+ 4 . (24)
4. Find the infimum and supremum of the following sets and decide whether they are the minimum and the maximum of the set:
M :=
n−1
n+ 1 | n∈N
, S :={y|y= sinx, x∈R}. 5. Solve the following equations and inequalities:
x <|4−2x|, (25)
|3−x|<4−2x , (26)
||x+ 2| − |x−1||= 3 , (27)
|x−1|+|x+ 2|= 3 . (28)
