# November 8, 2005 Lecturer: Dr Martin Kurth Michaelmas Term 2005

November 8, 2005
Lecturer: Dr Martin Kurth
Michaelmas Term 2005
Course 1E1 2005-2006 (JF Engineers &amp; JF MSISS &amp; JF MEMS)
Problem Sheet 5
Due: in the Tutorials 18 November / 21 November
Vive la différence!
1. Let f be a function defined on (−∞, ∞), and x any real number. Write
down the definition of the derivative of f at x, ie of f 0 (x).
(2 points)
2. Using this definition, find out if the following two functions are differentiable at x = 0:
(a) f (x) = |2x|,
(b) f (x) = |2x|2 .
(4 points)
3. Using the definition from question 1, calculate the derivative functions of
(a) f (x) = 3x,
(b) f (x) = 3x2 .
(4 points)
4. Consider the following alternative definition of continuity: A function f
is called continuous at any point x0 of its domain, if for all &sup2; &gt; 0 there is
a δ &gt; 0 such that
|x − x0 | &lt; δ =⇒ |f (x) − f (x0 )| &lt; &sup2;.
Why is this definition more general than the one given in the lecture? Can
you find a function that is continuous according the definition above, while
the definition from the lecture does not allow you to determine whether it
is continuous or not?
(*)
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.