MH1100/MTH112: Calculus I.
Tutorial in the final week.
This weeks topics:
LHospitals rule.
Limits to positive and negative infinity.
Basics of antiderivatives and indefinite integrals.
Substitution method.
Integration by parts.
The problems that will
MH1100/MTH112: Calculus I.
Problem list for Week #1.
This weeks topics:
The basic theory of functions. Chapters 1.1 through 1.3 in Stewart.
Your tutor will aim to discuss: Problem 1, 3, 5, 7(b), 11, 12, 14, 16, 17.
Problem 1: (Problem 1.1.35 from [St].)
Outline
1
Continuous Functions
Tang Wee Kee (Division of Mathematical Sciences School
MH1810
of Physical
Mathematics
and Mathematical
1 Part 2
Sciences Nanyang
Semester
Technological
1 2013/14
University)
1/1
Continuous Functions
Very often, we are intere
Nanyang Technological University
Division of Mathematical Sciences, SPMS
AY 2015/2016 Semester 2
MH1810 Mathematics 1
Tutorial 8
Question 1
Let u = i + j 5k and v = 2i + j k. Find kuk, kvk, u v, u v, v u, and projv u.
Note that we have the formula:
) v
=
MH1810 Mathematics 1
Lecture 12
Rafael M. Siejakowski
NTU
Last time
Last week, we talked about complex numbers numbers of the
form x + yi with x, y R and i 2 = 1.
We defined algebraic operations +, , , on complex
numbers.
We also discussed different ways
Nanyang Technological University
Division of Mathematical Sciences, SPMS
AY 2015/2016 Semester 2
MH1810 Mathematics 1
Please be remined that there will be a 10-minute quiz during this tutorial session.
Tutorial 9
Question 1
Consider vectors u = (1, 1, 2)
Nanyang Technological University
Division of Mathematical Sciences, SPMS
AY 2015/2016 Semester 2
MH1810 Mathematics 1
Tutorial 6
Question 1
1. Evaluate each of the following integrals by an appropriate substitution.
Z
Z
Z
arctan x
u2
(arctan x)2
1
(a)
dx
Nanyang Technological University
Division of Mathematical Sciences, SPMS
AY 2015/2016 Semester 2
MH1810 Mathematics 1
Tutorial 10
Reference: Thomas Calculus, Appendix 7.
Question 1
Evaluate the expression and write your answer in the form a + bi, a, b R.
Nanyang Technological University
Division of Mathematical Sciences, SPMS
AY 2015/2016 Semester 2
MH1810 Mathematics 1
Tutorial 7
Question 1
Z
cosn x dx for n = 0, 1, 2, 3, .
Let In =
(a) Prove the reduction formula
In =
1
n1
cosn1 x sin x +
In2 for n 2.
n
MH1810 Mathematics 1
Lecture 6
Rafael M. Siejakowski
NTU
Last time
In the previous lecture, we studied applications of derivatives.
We learnt that the derivative f 0 contains a wealth of useful
information about the original function f . In particular, it
MH1810 Mathematics 1
Lecture 8
Rafael M. Siejakowski
NTU
Last time
Last week, we talked about properties of integrals and methods
of integration:
Integration by substitution
Integration by parts
Integration of rational functions via partial fraction
decom
PH1104/CY1305/SM2: Mechanics
Semester 1 2014
Mid-Term Exam 2
Nov 4, 2014
Answer ALL THREE (3) questions. Take g = 9.8 m/s2.
1.
A 5.00-g bullet moving with an initial speed of = 400 m/s is fired into and passes
through a 1.00-kg block as shown in the figur
PH1104: Mechanics
Tutorial 7
Qualitative questions
1
Quantitative questions
Note: mass of electron is 9.10938215(45) 1031 kg
2
Challenging question
3
Extra question - Falling rope
A rope of length L lies in a straight line on a frictionless table, except
PH1104: Mechanics
Tutorial 4
Qualitative questions
Quantitative questions
1
2
3
Challenging question
Extra question for thought:
As a cart is sliding down a frictionless incline, it ejects a ball upward into
the air in a direction that is normal to the in
PH1104: Mechanics
Tutorial 11
Qualitative questions
1
Quantitative questions
2
3
Extra question
A uniform solid sphere with mass = 2.0 kg and radius = 0.10 m is
set into motion with angular speed o = 70 rad/s. At = 0 the sphere is
dropped a short distance
MH1100/MTH112: Calculus I.
Problem list for Week #2.
Tutorial in the week beginning 26th of August.
This weeks topics:
Some motivations for limits: tangents and velocities. (Section 1.4).
The basic concept of a limit. (Section 1.5).
Using the limit law
MH1100/MTH112: Calculus I.
Tutorial problems for Week #9.
This weeks topics:
Types of extreme values.
The Closed Interval Method.
Rolles Theorem and The Mean Value Theorem.
The tutor will aim to discuss problems: 3, 5, 9, 11, 15, 16, 18, 21, 22, 23,
an
PH1104: Mechanics
Tutorial 9
Qualitative questions
Quantitative questions
1
2
Numerical Answers
8.82
8.95
8.104
8.106
8.110
29.5 cm
(a) 9.35 m/s (b) 3.29 m/s to the left
0.105 m/s to the right
1.29 m to the left
8.63 km from launch point; 5.33 105 J
3
PH1104/CY1305/SM2-18: Mechanics
Semester 1 2014
Mid-Term Exam I
Answer ALL TWO (2) questions. Take g = 9.8 m/s2.
1.
Another zookeeper is trying to catch a runaway monkey found hanging on a tree h = 12
m from the ground (see figure below). This zookeeper i
PH1104/SM2-B19: Mechanics
Homework 2
Due date: Thursday 8th October 2015
Homework solutions should be submitted during the first 10 minutes of Thursday
lecture on 8th October 2015 at the respective LTs according to Theta/Phi group.
Marks will be deducted
PH1104/SM2-B19: Mechanics
Homework 1
Due date: Thursday 17th September 2015
Homework solutions should be submitted during the first 10 minutes of Thursday
lecture on 17th September 2015 at the respective LTs according to Theta/Phi
group.
Please write the
PH1104/SM2-B19: Mechanics
Homework 3
Due date: Monday 26th October 2015
Homework solutions should be submitted during the first 10 minutes of Monday
lecture on 26th October 2015 at the respective LTs according to Theta/Phi
group. Marks will be deducted fo
MH1810 Mathematics 1
Lecture 11
Rafael M. Siejakowski
NTU
Last time
Last week, we discussed matrix algebra.
More specifically, we defined the operations of addition and
multiplication of matrices, whenever they can be performed.
We also defined the invers
MH1810 Mathematics 1
Lecture 9
Rafael M. Siejakowski
NTU
Analytic geometry
In this lecture, we are going to revise basic methods of analytic
geometry. This is material from Chapter 12.
What distinguishes analytic geometry from synthetic geometry is
that a
Nanyang Technological University
MH1810 (FE1006) Mathematics 1
Tutorial 3
Basic Mastery Questions
Do the following problems without the use of calculators.
1. Find the matrix product AB
1 1
1 1
1 2
;B =
3 4
0
1
0
1
1 1 2
1 1 2
1 1 A
(b) A = @ 1 2 1 A ; B
Nanyang Technological University
MH1810 (FE1006) Mathematics 1
Tutorial 4
Announcement There is a 15-minutes Quiz 1 (10%) during tutorial class (TIME: XX:05 XX:20). The rst quiz contains 2-3 problems on topics discussed in Tutorials 1, 2 and 3.
Section A:
Nanyang Technological University
MH1810 (FE1006) Mathematics 1
Tutorial 6 (Solutions to Discussion Questions)
Discussion Questions
1. Evaluate the limit, if it exists.
(a)
lim
t 4
cos 2t
cos2 t sin2 t
(cos t sin t) (cos t + sin t)
= lim
= lim
cos t sin t
a3. Find the domain of function f (x) =
x2 +x12
.
x2 x2
Solution: The denominator cannot be zero. Note x2 x 2 = 0 leads to x = 1, x = 2. Hence,
the domain of f (x) is cfw_x R : x = 1 or x = 2.
1
a1. Find the domain of function f (x) =
x2 +x6
.
x2 1
Solution: The denominator cannot be zero. Note x2 1 = 0 leads to x = 1, x = 1. Hence, the
domain of f (x) is cfw_x R : x =
1.
1