MH1100/MTH112: Calculus I.
Tutorial in the week beginning November the 11th.
This weeks topics:
LHospitals rule.
Limits to positive and negative innity.
Basics of antiderivatives and indenite integrals.
Substitution method.
Integration by parts.
The
After last lecture a few students asked me
Question.
Is this a
contradiction?
Answer.
Question.
Answer.
1
x sin( ) x 0
f ( x) =
x
0
x=0
And this
limit
doesnt
This lecture:
This lecture will focus on the first ideas about antiderivatives
and indefinite integrals.
What are anti-derivatives and indefinite integrals?
How many anti-derivatives can a function have?
Elementary anti-derivatives.
Two basic integration
In the previous lecture I discussed the question
Answer:
Ill discuss three explanations of this fact:
1.
An intuitive explanation.
2.
A very intuitive explanation.
3.
An explanation based on the definition of derivative.
#1: The intuitive explanation:
#1:
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
Comment
This form
This lecture:
concave down
on this interval
concave up
on these two intervals
Horizontal asymptotes: a first example.
Horizontal asymptotes
Horizontal asymptotes
A round of the game:
+
M=2
Lets watch a few rounds
Player A.
Player B.
Lets watch a few round
This lecture.
This lecture we are going to focus on the calculus of the
trigonometric functions.
Trigonometric functions: the definitions?
Recall the main idea of these functions
Trigonometric functions: the definitions?
The precise definition (wont be te
This lecture.
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 extreme valu
This lecture:
This lecture will be on the topic of The Mean Value Theorem
(The MVT), and related topics.
Why this class of theorem is interesting.
The first step: Rolles Theorem, an important special case of
the MVT.
Using Rolles Theorem to prove the full
This lecture.
This lecture:
The chain rule. A careful statement, and some
examples.
Why is the chain rule true? To give some explanation,
well first think about linear approximations of
differentiable functions.
Differentiating compositions: the chain rul
This lecture.
Recall:
Example.
=0.20=0.70
=0.15=0.65
=0.25=0.75
=0.30 =0.85
=0.35 =0.90
=0.50
=0.45
=0.40
=0.55
=0.80
=0.60
Functions can only have one limit
Now that we have a rigorous definition of limit, we can set
about building up the theory of calc
This lecture.
The rules for differentiation.
Recently weve been thinking carefully about the precise
meaning of derivatives using the concept of a limit.
But, of course, to actually calculate a derivative, we
(usually!) just use some standard rules.
Now t
This lecture.
The definition of and topics related to the important
concept of Continuity.
The definition and first examples.
Important classes of continuous functions:
q
Polynomials
q
Rational functions
q
Trigonometric functions
q
Exponential functions
U
The precise definition of limit.
Recall the idea of limit
?
Recall the intuitive definition we gave earlier.
Right now well:
Play with this definition a bit to get a feeling for what it
says.
Then well obtain the precise mathematical definition by
simply
This lecture.
The precise definitions of some other types of limits:
Infinite limits.
One-sided limits.
Limits to infinity.
Limits to infinity (cont.).
Limits to infinity (cont.).
Actually this definition is very similar to the definition of limit
we have
Here is part of a question from the quiz that many students
got wrong
An incorrect solution:
To understand compositions clearly, you have to think in
terms of the machine recall from the 1st lecture:
To understand compositions clearly, you have to think i
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:
Limits
The contents o
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:
You
read this express
Andrew Kricker
[email protected]
SPMS building, 4th
floor,
office number MAS
04-18.
Your final mark
1.
2.
3.
4.
60% from the final exam.
20% from the mid-term exam. (On 15th October, in the
normal lecture slot.)
10% from a quiz. (On 19th or 20th Septem
This lecture.
The main topics of this lecture are the general exponential and
logarithmic functions.
What is an exponential function, exactly?
The calculus of exponential functions.
What is the constant e ? Where does it come from?
Logarithmic functio
This lecture.
The main topics of this lecture are the general
exponential and logarithmic functions.
What is an exponential function, exactly?
The calculus of exponential functions.
What is the constant e ? Where does it come from?
Logarithmic functions a