October 11, 2005
Lecturer: Dr Martin Kurth
Michaelmas Term 2005
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 1
Due: in the Tutorials 21 October / 24 October
It is easier to square the circle than to get round a mathematician.
Augustus De Morgan (1806-1871)
1. For each of the following pairs of points P1 and P2 , find the point-slope
equation and the slope-intercept equation of the line L passing through
P1 and P2 , provided they exist. If possible, write down the slope-intercept
equations for the lines parallel to L and perpendicular to L passing through
(4, 2).
(a) P1 = (1, 3),
P2 = (3, 1),
(b) P1 = (−2, −1),
P2 = (0, 4),
(c) P1 = (8, 4),
P2 = (10, 2),
(d) P1 = (3, 6),
P2 = (3, 4),
(e) P1 = (2, 4),
P2 = (6, 4).
(5 points)
2. For each of the following functions, write down its natural domain and its
range.
(a) f (x) = x3 ,
(b) f (x) = 1/x2 ,
(c) f (x) = 4,
√
(d) f (x) = x,
√
(e) f (x) = 1/ x.
(5 points)
3. Let f be a function with domain (−∞, ∞). Prove the following:
(a) There exist an even function feven and an odd function fodd , such
that
f (x) = feven (x) + fodd (x)
for all x ∈ (−∞, ∞).
(b) This decomposition is unique.
(*)
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.