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.