November 15, 2005
Lecturer: Dr Martin Kurth
Michaelmas Term 2005
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 6
Due: in the Tutorials 25 November / 28 November
Igitur sensus lemmatis est, ut, si quantitatum quarumcunque perpetuo motu
crescentium vel decrescentium A, B, C, &c. momenta, vel his proportionales
mutationum velocitates dicantur a, b, c, &c. momentum vel mutatio geniti rectanguli AB fuerit aB + bA, . . .
(Wherefore the sense of the Lemma is, that if the moments of any quantities A,
B, C, &c. increasing or decreasing by a perpetual flux, or the velocities of the
mutations which are proportional to them, be called a, b, c, &c. the moment or
mutation of the generated rectangle AB will be aB + bA, . . . )
from: Isaac Newton, Philosophiae Naturalis Principia Mathematica
1. Consider the following functions
f1 (x) = x2 − 1,
1
f2 (x) =
2
(x − 4)
f3 (x) = x + 3,
f4 (x) = x.
(|x| =
6 2),
Calculate
(c)
d
dx (f1 (x)f2 (x))
d
dx (f3 (x)f4 (x)),
d f3 (x)
dx f2 (x) ,
(d)
d f4 (x)
dx f1 (x)
(e)
d
dx (f1 (x)f2 (x)f3 (x)f4 (x))
(a)
(b)
(|x| =
6 2),
(|x| =
6 1),
(|x| =
6 2).
(6 points)
2. Calculate
(a)
(b)
d
dx
d
dx
tan x,
cot x.
(4 points)
3. Calculate
d752
d999
sin x + 752 cos x.
999
dx
dx
(*)
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.