April 11, 2006
Lecturer: Dr Martin Kurth
Trinity Term 2006
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 1
Due: in the Tutorials 21 April / 24 April
“Obvious” is the most dangerous word in mathematics.
Eric Temple Bell (1883–1960)
1. Calculate the following limits using L’Hôpital’s Rule:
sinh x
x ,
cos x
limx→π/2 x−
π ,
2
(a) limx→0
(b)
x2
sinh x ,
limx→∞ (xe−x ).
(c) limx→0
(d)
(4 points)
2. Calculate the following integrals, provided they exist:
R∞ 1
dx,
(a) 1 2x
R∞ 1
(b) 1 2x2 dx,
R1 1
(c) 0 2x
dx,
R2 1
(d) 0 √2x dx.
(4 points)
3. Calculate the following limits and derivatives, provided they exist:
(a) limx→2
x2 −4
x−2 ,
¡
¢
(b) limx→0 x sin x1 ,
(c)
(d)
d
2
dx ln(cosh(x
d √
3x+7
dx x+4 .
− 3x + 4)),
(2 points)
4. What is
Z
0
1
x cosh x − sinh x
dx?
x2
(*)
Questions 1, 2 and 3 should be answered by all students, you will get points for them. Question 4 is more challenging and meant as an exercise for the more mathematically interested
students.