April 18, 2006
Lecturer: Dr Martin Kurth
Trinity Term 2006
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 2
Due: in the Tutorials 28 April / Lecture 02 May
Every human activity, good or bad, except mathematics, must come to an end.
Paul Erdös (1913–1996)
1. Calculate the following integrals, provided they exist:
¢
R∞¡
(a) 1 cosh x − 12 ex dx,
R∞
(b) 1 1+x
x2 dx,
R2 1
(c) 1 x−1 dx,
¢
R2¡
(d) 0 sin( x1 ) − x1 cos( x1 ) dx.
(4 points)
Hint for (d): Calculate the derivative of x sin(1/x) first.
2. Calculate the limits of the following sequences {an }, provided they exist:
(a) an =
(b) an =
4
n2 ,
n+3
n ,
(c) an = (−1)n ,
(d) an =
1
n2
sin(n) cos(n).
(4 points)
3. Consider the function f defined by
f (x) = x4 − x2 .
Find the critical points of f , and determine whether they are minima,
maxima, or neither.
(2 points)
4. Prove the Continuous Function Theorem: If {an } is a sequence with
limn→∞ an = L and f is a function that is continuous at L and defined at
all an , then limn→∞ f (an ) = f (L).
(*)
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.