November 22, 2005 Lecturer: Dr Martin Kurth Michaelmas Term 2005 Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS) Problem Sheet 7 Due: in the Tutorials 02 December / 05 December It isn’t that they can’t see the solution. It is that they can’t see the problem. from: G. K. Chesterton, The Point of a Pin in The Scandal of Father Brown 1. Calculate the following derivatives: √ d 27 x + 8, (a) dx √ d 27 (b) dx 8x, √ d 27 8 x , (c) dx (d) d √1 dx 27 x8 . (4 points) 2. Calculate the following derivatives: ¡ ¢ d (a) dx sin(2x) sin(x2 ) , ¡√ ¢ d (b) dx 2x cos(3x2 ) , ´ ³√ 3 d (c) dx x2 sin(3x) . (6 points) 3. Let f be a function that is defined on an open interval D. Let f be differentiable and one-to-one on D, and f 0 (x) 6= 0 for all x in D. Let R be the range of f . Show that (f −1 )0 (y) = with y = f (x), for all y in R. 1 f 0 (x) (*) Questions 1 and 2 should be answered by all students, you will get points for them. Question 3 is more challenging and meant as an exercise for the more mathematically interested students.