November 8, 2005 Lecturer: Dr Martin Kurth Michaelmas Term 2005 Course 1E1 2005-2006 (JF Engineers & JF MSISS & 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 ² > 0 there is a δ > 0 such that |x − x0 | < δ =⇒ |f (x) − f (x0 )| < ². 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.