MH1100 and SM2.
Problem set #10.
Solutions
Problem 1:
Which of the following functions are 1-1. Briey justify.
(i) f (x) = x2 2x.
(ii) f (x) = 10 3x.
(iii) g(x) = 1/x.
(iv) g(x) = |x|.
(v) h(x) = 1 + cos x.
(vi) h(x) = 1 + cos x, for 0 x .
Solution
Well g
MH1100 and SM2
Solutions for the Week 8 tutorial problems.
Solutions
Problem 1:
Differentiate the function u(t) =
5
t + 4 t5 .
Solution
d
[u(t)] =
dt
=
=
=
=
i
d h
5
t + 4 t5
dt
1 i
d h 1
t 5 + 4 t5 2
dt
i
5
d h 1
t 5 + 4t 2
dt
5
1 1 1
5
t 5 + 4 t 2 1
5
This lecture.
The definition of and topics related to the important concept
of Continuity.
The definition and first examples.
Important classes of continuous functions:
Polynomials
Rational functions
Trigonometric functions
Exponential functions
U
MH1100 and SM2.
Problem Set #9.
SOLUTIONS
Problem 1:
Sketch the graph of a function which is continuous on [1, 5], and has all the
following properties:
It has an absolute maximum at 5.
It has an absolute minimum at 2.
It has a local maximum at 3.
It
This lecture.
The definition of and topics related to the important concept
of Continuity.
The definition and first examples.
Important classes of continuous functions:
Polynomials
Rational functions
Trigonometric functions
Exponential functions
U
This lecture.
This lecture we are going to focus on the concept of the
derivative of a function.
The definition of the derivative of a function at a point .
The meaning of the word differentiable.
The derivative function.
How are the concepts of continuou
This lecture.
In this lecture:
Inverse functions. Review of the basic concepts.
Calculus of inverse functions:
o Continuity of the inverse.
o Derivative formula for the inverse function.
Inverse trigonometric functions.
Inverse functions
Given a functi
This lecture.
This lecture we are going to focus on the calculus of the
trigonometric functions.
Some comments on some alternative approaches to a precise
definition of and .
Two basic limits well assume.
The functions and are continuous at every point.
T
This week.
The topics this week split into two parts:
1. Inverse functions.
2. Exponential and logarithmic functions.
This lecture: Part 1.
In this lecture:
Inverse functions. Review of the basic concepts.
Calculus of inverse functions:
o Continuity of
This lecture: Part 1.
Reminder of the precise definition of .
Some more advanced aspects of this definition:
Using the definition to prove some basic properties of
limits. For example, the fact that a function can have at
most one limit at some .
Prov
MH1100 and SM2
Tutorial in the final week.
This weeks topics:
LHospitals rule.
Basics of antiderivatives and indefinite integrals.
Substitution method.
Integration by parts.
The tutor will select a few problems to focus on.
Problem 1: (Various from Se
MH1100 Calculus I and SM2 Mathematics I.
Problem Set for the Week 8 Lectures. (Tutorial held during Week 9.)
This weeks topics:
Differentiation rules.
The calculus of the trigonometric functions.
The chain rule.
The tutor will aim to discuss: 2, 8, 9,
This lecture: Part 1
This lecture will be on the topic of extreme values.
The main definitions: absolute versus local maxima and
minima.
When does a function have absolute extreme values? The extreme
values theorem.
Where should you look to find the ex
This lecture.
This lecture we are going to focus on the concept of the
derivative of a function.
The definition of derivative at a point .
The meaning of the word differentiable.
The derivative function.
How are the concepts of continuous and differentiab
The precise definition of limit.
Recall the idea of limit
As
?
gets closer and closer to 1, gets
closer and closer to 1.
Recall the intuitive definition we gave earlier.
Temporary,
intuitive definition of limit.
Let be a function, let , and let .
We say t
Andrew Kricker
ajkricker@ntu.edu.sg
SPMS building, 4th floor,
office number MAS 04-18.
The textbook
It is:
Calculus
By James Stewart
The 8th, metric edition.
I recommend buying this because:
This course will be based closely on this book.
It will als
This lecture. Part 1:
This lecture will be on the topic of extreme values.
The main definitions: absolute versus local maxima and
minima.
When does a function have absolute extreme values? The extreme
values theorem.
Where should you look to find the e
This lecture: Part 1
Reminder of the precise definition of lim = .
Some more advanced aspects of this definition:
Using the definition to prove some basic properties of
limits. For example, the fact that a function can have at
most one limit at some .
Limits
Our main task in this lecture is to start thinking about limits.
This is one of the most fundamental concepts in calculus (and,
in fact, is one of the most important ideas in all of
mathematics).
Here is how we denote a limit:
lim
You read this ex
MH1100 Calculus I and SM2 Mathematics I.
Problem list for Week #6.
This weeks topics:
The definition of the derivative of a function at a point a.
The derivative function.
Differentiability at a point implies continuity at that point.
The tutor will ai
This lecture:
This lecture is all about understanding the behaviour of functions
the variable becomes a large positive or negative number.
The concept of a horizontal asymptote.
Limits of functions towards or .
The precise definitions.
Proofs.
The limit l
This lecture:
This lecture is all about understanding the behaviour of functions
() as the variable becomes a large positive or negative number.
The concept of a horizontal asymptote.
Limits of functions towards or .
The precise definitions.
Proofs.
The l
This lecture:
This lecture is all about understanding the behaviour of functions
() as the variable becomes a large positive or negative number.
The concept of a horizontal asymptote.
Limits of functions towards or .
The precise definitions.
Proofs.
The l
MH1100 and SM2.
Solutions to Problem Set #11.
Problem 1:
Evaluate the following limit:
lim
x
12x3 5x + 2
.
1 + 4x2 + 3x3
Solution:
The square root function is continuous. So we can take the limit towards
infinity inside the square root:
12x3 5x + 2
12x3 5
This lecture.
Two main topics:
Part 1: The calculus of trigonometric functions.
Part 2: The chain rule.
This lecture: Part 1
This lecture we are going to focus on the calculus of the
trigonometric functions.
Some comments on some alternative approaches
MH1100 and SM2
Problem set for teaching week 11.
This weeks topics:
Limits to positive and negative infinity.
The tutors will select a few problems to discuss.
Problem 1: (#3.4.8 from [Stewart].)
Evaluate the following limit:
lim
x
12x3 5x + 2
.
1 + 4x2
This lecture. Part 1:
This lecture will be on the topic of extreme values.
The main definitions: absolute versus local maxima and
minima.
When does a function have absolute extreme values? The extreme
values theorem.
Where should you look to find the e
MH1100 and SM2 Mathematics I.
Solutions to the Week #2 problem set.
Solution to Problem 1.
In this problem we are asked to guess the slope of the tangent to the graph
x
at the point P (1, 12 ) by looking at the slopes of
of the function f (x) = 1+x
variou
MH1100 and SM2.
Problems accompanying the Week 10 lectures.
This weeks topics:
Basic concepts of inverse functions.
Derivatives of inverse functions.
Inverse trigonometric functions.
Exponential and logarithmic functions.
The tutor will aim to discuss
MH1100 Calculus I and SM2 Mathematics I.
Solutions to the Week #1 problem set.
Problem 1: (Problem 1.1.35 from [St].)
What is the domain of
f (x) =
4
1
?
5x
x2
Solution:
This expression makes sense when x2 5x > 0. To understand when this is
true, we beg