Labs and project overview for EE1003
EE1003, introductions to signals and communications, is a core module in department of
Electrical and Computer engineering. The fundamental conceptions in signals and
communications are introduced in lectures. These co
MA 1505 Mathematics I
Tutorial 2 Solutions
1. (a)
lim
x/2
1
1 sin x
cos x
sin x
= lim
= lim
=.
1 + cos 2x x/2 2 sin 2x x/2 4 cos 2x
4
a sin ax
2
ln(cos ax)
cos ax = lim a sin ax cos bx = a .
(b) lim
= lim
x0 b sin bx
x0 b sin bx cos ax
x0 ln(cos bx)
b2
c
MA 1505 Mathematics I
Tutorial 3 Solutions
1. (a) Observe that sec2 x > 0 and 4 sin2 x 0 on [/3, /3].
/3
Area =
/3
1
sec2 x (4 sin2 x) dx
2
1
tan x + (2 2 cos 2x) dx
2
/3
= tan + (2x sin 2x)
3
/3
4
4
= 3 + 2 sin = .
3
3
3
=
/3
/3
(b) The points of inte
MA 1505 Mathematics I
Tutorial 4 Solutions (AY2009/2010 Semester 2)
1. Rewrite the function:
1
f (x) = (x + |x|) =
2
0 < x < 0
x 0<x<
The Fourier series of f (x) is given by
a0 +
(an cos nx + bn sin nx).
n=1
a0 =
1
2
1
an =
1
bn =
x dx =
0
.
4
1 x sin nx
1
Tutorial 1 Comments
Q1
f (x) = x|x|. To nd f (0), we have by denition
f (x) f (0)
= lim x|x|
x0
x0
x
f (0) = lim
if the limit EXISTS.
Now consider the function g (x) = x|x|. This function is continuous for all x R.
Recall that for a function g (x) which
1
Tutorial 2 Answers + Comments
Q1
This question uses the Intermediate Value Theorem. In it's most general form, we have
Thorme 0.1 Let I be an interval and let f : I R be continuous on I . If a, b I and
if k R satises f (a) < k < f (b), then there exists
i u i w v w u x w t g x w s i g u v u v u y q g t w g s g g i t g s g g v g p g v s y g x w v u t s r g q p i g h g f e
I
T d 3( 7 "'9 $ 7! @ c "
A
A
I
6 & ' & % " & ' " 2 ' ) " $ @ ! % " & 2 4 3 3 ( $ " 0 ( ' ( 0 ) 2 " ! " ' % ( & ) ! & " &bS ! $ & $ ' %
1
Extra Integrals
Show that for the following integrals (all of them exist and are nite):
1
dx = ln( 2 + 1) + ln 2,
1
1
2
x
x 1+x
0
/2
cos x ln(tan x)dx = ln 2,
0
/2
0
x sin x cos x
( 2)
dx =
,
2
2
tan x + cot x
32
+
sin3 x
3
dx = ln 3.
2
x
4
0
As an e
consultation schedule at
9-10
E1A-03-6
11-12
12-1
1-2
Du LL
Mon
10-11
Gao R
Gao B
Cai
Tue
Wed
Zhou
Hu HN
Li SR
Fri
Du ZK
Zhu
3-4
4-5
Zheng Yu
Zhang XW
Hou
Tang
Thu
2-3
Ngo
Zhang XT Gao Fan
Ma
Huang
Wu
Chapter 10. Surface Integrals
10.1
Parametric Surfaces
A parametric representation of a surface is given by the two-variable vector function r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k where u and v are two independent parameters. The collection of points wi
Chapter 9. Line Integrals
9.1
9.1.1
Introduction
Work Done I
(i) Let F be a constant force acting on a particle
in the displacement direction as shown in gure
(i) above. Suppose the distance moved by the
particle is s. The work done is given by
W = F s.
(
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 4
1. Find the area of the following region.
1
sec2 x, y = 4 sin2 x, x = and x = .
2
3
3
12
(b) The region in the rst quadrant bounded by y = x, y = x and below y = 1
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 6
1.
Write the vector 3i+2j+k as a sum of two vectors u+v such that u is parallel to w = i+3j+4k
and v is perpendicular to w. (Hint: Use projection)
Ans.
2.
1
2 (i
1
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 7
1. Imagine you are visiting a country in the winter season. Let T (x, y ) = 36 1 [x2 + (y 5)2 ]
5
be the temperature at location (x, y ) in a 10ft 10ft hotel room
Chapter 1. Functions: Limits and Continuity
1.1
Functions
It is common that the values of one variable depend
on the values of another. E.g. the area A of a region on the plane enclosed by a circle depends on
the radius r of the circle (A = r2, r > 0 .) M
Chapter 3. Integration
3.1
Indenite Integral
Integration can be considered as the antithesis of differentiation, and they are subtly linked by the Fundamental Theorem of Calculus. We rst introduce indenite integration as an inverse of differentiation.
3.1
Chapter 4. Sequences and Series
4.1
Innite sequences
An innite sequence (or sequence) of real numbers
is an innite succession of numbers which is usually
given by some rule.
We shall denote an innite sequence by
a1 , a 2 , a 3 , , a n , ,
and we shall oft
Chapter 5. Fourier Series
5.1
Periodic functions
A function f (x) is called periodic if it is dened for
all real x and if there is some positive number p such
that
f (x + p) = f (x) for all x.
(1)
The number p is called the period of f (x).
5.1.1
Graphs o
Chapter 8. Multiple Integrals
8.1
8.1.1
Double Integrals
Denition
The denition of denite integral in chapter 3 can be
extended to functions of two variables.
Let R be a plane region in the xy -plane.
Subdivide R into subrectangles Ri (i = 1, ., n).
Let Ai
NATIONAL UNIVERSITY OF SINGAPORE
Department of Mathematics
MA 1505 Mathematics I
Tutorial 3
1. Evaluate the following denite integrals.
22
s+ s
(a)
ds .
s2
1
4
(b)
|x| dx .
4
1
(cos x + | cos x|) dx .
02
(d)
sin2 1 +
d .
2
0
Ans. (a) 1 + 2 23/4 (b) 16
(c)
Page 12
The CAD les
A case of 4D and fraud
The Business Times, Monday. FebruaryB. 200-1
A bank was cheated'of millions, the evidence lled an entire room from floor to ceiling, and over 1,400 counts of cheating charges
were filed against one person a rec
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v
Public Prosecutor
[1999] SGHC 65
High Court Magistrates Appeal No 108 of 1998
Yong Pung How CJ
2 February; 22 March 1999
Criminal
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Public Prosecutor and another matter
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(District Arr
Running Head: What Managers Believe as Important
Master Thesis
International Hotel and Tourism Leadership
What Managers Believe as Important while
Recruitment in Fast Food Industry within
Norway: A case study of
Burger King & McDonalds
Authors: Rashida Sa
Chapter 2
Attracting and Retaining Staff: The Role
of Branding and Industry Image
Michelle Wallace, Ian Lings, Roslyn Cameron and Neroli Sheldon
Abstract In increasingly competitive labour markets, attracting and retaining talent has become a prime concer
Recruitment & Training at McDonalds
Stock Control
Finance
Education
Franchising
Franchising
Stock Control
Finance
Finance
Glossary
I.T.
Recruitment
As the biggest family restaurant business in the world,
McDonalds vision is to provide the best family re
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v
Asia Pacific Breweries (Singapore) Pte Ltd and another
and another appeal
[2011] SG