Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Solutions of Tutorial 2
Q1 We may use the following results without proofs
x
sin x
sin ax a
= 1, lim
lim
= 1, lim
=
x 0 x
x 0 sin x
x 0 sin bx
b
Q1 (b)
0
form
0
1
( a sin ax)
ln(cos ax)
lim
= lim cos ax
x 0 ln(cos bx )
1
x 0
(b sin bx)
cos bx
a sin ax cos
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Q3
Tutorial 1
For cone
radius 3 r
= =
height 6 h
1
r= h
2
(a ) Let V p (t ) be the volume of coffee in the pot at time t
V p = height = 32 h p
base
d
d
V p = 9 h p
dt
dt
d
10 = 9 h p
dt
dh p 10
does not depend on
=
dt
9
pot
hp
(b) Let Vc (t ) be the volum
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Ground Rules
PC1221 Fundamentals of
Physics I
Lectures 9 and 10
The Laws of Motion
A/Prof Tay Seng Chuan
Responsible use of electronic gadgets
No talking while lecture is going on
No gossiping while the lecture is going on
Raise your hand if you have ques
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA1505 MidTerm Test
_
General Information
1. Date: Monday, 29 September 2014.
2. Time: From 8:30pm to 9:30pm.
3. Venue: MPSH 1 and MPSH 2. Please check the sitting
plans which will be available at a later time in the Test
folder in the Workbin at the IVL
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA 1505 Mathematics I
Tutorial 5 Solutions
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 =
an =
1
1
2
bn =
1
x dx =
0
.
4
1 x sin nx cos nx
+
n
n2
x cos nx dx
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA 1505 Mathematics I
Tutorial 6 Solutions
1. V = I R
(i)
=
I =
V
R.
I
1
= .
V
R
If R = 15, then
I
1
=
0.0667 A/V.
V
15
V
I
120
I
= 2 . If V = 120 and R = 20, then
= 2 = 0.3 A/.
R
R
R
20
(iii) By Chain rule,
dI
I dV
I dR
1 dV
V dR
=
+
=
2
.
dt
V dt
R dt
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA 1505 Mathematics I
Tutorial 1 Solutions
1. Note that (g f )(x) =
2. (a) y =
ax + b
,
cx + d
y =
6
3 x  and (f g)(x) =
6
.
3x
a(cx + d) c(ax + b)
ad bc
=
(use quotient rule)
2
(cx + d)
(cx + d)2
(b) y = sinn x cos mx , y = n sinn1 x cos x cos mx m
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Ground Rules
PC1221 Fundamentals of
F d
t l f
Physics I
Lectures 17 and 18
Linear Momentum and Collisions
A/Prof Tay Seng Chuan
Responsible use of electronic gadgets
No talking while lecture is going on
No gossiping while the lecture is going on
Raise you
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA 1505 Mathematics I
Tutorial 3 Solutions
n
1. (a) Let un = (1)n (x+2) .
n
lim
n
(x + 2)n+1
n
un+1
= lim
= x + 2.
n
un
n+1
(x + 2)n
By ratio test, the power series is convergence in x + 2 < 1.
So the radius of convergence is 1.
(b) Let un =
(3x2)n
.
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Remarks of tutorial 4
Q4
Two nonparallel planes always intersect.
Hence the distance between them is zero
2x + 2 y z =
1
In this question, the given two planes are
parallel, since their normals are parallel
4x + 4 y 2z =
5
Choose one point P on one of th
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA 1505 Mathematics I
Tutorial 2 Solutions
1. (a)
lim
x/2
(b)
1 sin x
cos x
sin x
1
= 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 .
lim
= lim
x0 ln(cos bx)
x0 b sin bx
x0 b sin bx cos ax
b2
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Remarks of tutorial 2
Q1( f) Q1( g)
sin x
1
You may use the results lim
x 0
Q1( g)
x
x
lim
1
x 0 sin x
without proofs
x x cos x sin x
x
x cos x sin x
lim
lim
lim
x 0 sin x
x 0 sin x x 0
2 x3
2 x3
Q1( f)
sin x
sin x ln x
lim sin x ln x lim
x ln x lim
1
x
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Ground Rules
PC1221 Fundamentals of
F d
t l f
Physics I
Lectures 15 and 16
Potential Energy
A/Prof Tay Seng Chuan
Responsible use of electronic gadgets
No talking while lecture is going on
No gossiping while the lecture is going on
Raise your hand if you
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Ground Rules
PC1221 Fundamentals of
F d
t l f
Physics I
Lectures 13 and 14
Energy and Energy Transfer
A/Prof Tay Seng Chuan
Responsible use of electronic gadgets
No talking while lecture is going on
No gossiping while the lecture is going on
Raise your ha
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
Chapter 10. Surface Integrals
10.1
Parametric Surfaces
A parametric representation of a surface is given
by the twovariable vector function
r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k
(1)
where u and v are two independent parameters.
The collection of point
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
MA1505
MidTerm Test
Formulae List
1.
The Taylor series of f at a is
k=0
f (k)(a)
(x a)k = f (a) + f (a)(x a) +
k!
f (n)(a)
+
(x a)n +
n!
2.
x
e =
n=0
xn
n!
3.
(1)nx2n+1
(2n + 1)!
sin x =
n=0
4.
cos x =
n=0
(1)nx2n
(2n)!
5.
ln(1 + x) =
n=1
(1)n1xn
n
6.
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
1. Five students measure the mass of an object, each using a
y
different scale. They record their results as follows:
PC1221 Fundamentals of Physics 1
Semester1, AY1314
(10% of Final Score)
2. The quantity with the same units as force times time,
Ft, wi
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
In Physics, you learn models. You are advised
to draw diagrams from the words p
g
presented to
you, and this is a process of formulation.
PC1221 Fundamentals of Physics 1
Semester1, AY1415
You are now receiving tertiary education and
education,
there ar
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
PC1221
1.
(A) 0 00050
0.00050
(B) 0.0050
(C) 5.000
(D) 5000
(E) 0050
Quiz 1
Solutions
Semester1, AY12/13
,
2.
A record player rotates at 50 rpm (revolutions per
minute).
minute) Through how many degrees does it
rotate in 1 second?
(A) 200
(B) 150
(C) 270
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
1.
For the function:
function f = ladder(tetha,alpha)
w1=2; %m
w2=2; %m
f = (w1/sin(tetha)+(w2/sin(pialphatetha);
For the optimization:
clear;clc;clf
alpha=(45*pi)/180):(pi/90):(135*pi)/180); %angles are in radians
ladderlength=zeros(1,numel(alpha); %cr
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
PC1222 Fundamentals of Physics II
Tutorial 8
1. Refraction of Light I.
When a man stands near the edge of an empty drainage ditch of depth 2.80 m, he can barely
see the boundary between the opposite wall and bottom of the ditch as shown. The distance
from
Innovative College of Science in Information Technology
1
ECON 101

Fall 2014
IT1005 Introduction to Programming with MATLAB
Lab 8 Optimization
The objective of this lab is to help you become familiar with and solve simple
optimization problems. The first problem is a very nice everyday example of an
optimization problem: finding t