An Introductory Calculus Course 

Articles 
General Information
Introductory
Calculus course. Read tutorial guide and the chapters in text book and
then do the tutorial. 

Substitution in Integral and
Differentials 
Objectives 

Do we need Mean Value Theorem? 
Syllabus 

Extreme Value Theorem 
References 

Intermediate Value Theorem 
Notes on Derive 

Boundedness Theorem 
General Advice and Learning Guide 

Monotone Function and Continuity 
ProblemSolving Process 

Injective Function and Monotone Function 
Tutorials 

Riemann Integral and
Bounded Function 
Guide and Comment for Tutorial Assignment 

Riemann Integral and Infinite Series 
Example Sessions 

Derived
functions and Derivative 
Online Quizzes 

Continuity,
Differentiability, Weierstrass' Function 
Precalculus Online Quiz 

Intermediate Value Theorem for
Derived Function 
A
Formula of Euler and Appreciating Calculus


Monotone Function, Bounded Variation,
Fundamental Theorem of Calculus 
Tests and Past Exam Papers 

HeineBorel, Bolzano Weierstrass
Theorems, Uniform Continuity and Riemann Integrability 
Letter to
Students 

Composition and Riemann
Integrability 
Letter to and from a fellow teacher 

Composition and Lebesgue
Integrability 
Link to other Calculus Web sites 

Change of Variable in
Riemann and Lebesgue Integration 
Comment and Errata to
Calculus, an introduction
Calculus, an introduction available from NUS
Coop 

The Cantor Set 


Darboux's Fundamental Theorem of Calculus

Review of 19992000 1st Semester Exam


L Hopital's Rule  And a Generalized Version
Concavity  Definitions and Equivalence 
Books on web: 

Integration By Parts 
Real Numbers?
Newly revised 

Application of integration  arc
length, volume of solid of revolution, area of surface of revolution

Mathematical Analysis, An Introduction
With some tutorials for self study.
Answer
to each individual chapter's exercise is available upon request with your
email New
Now ALL fourteen
chapters come with exercise problems. Intermediate to advanced entry
to mathematical analysis
Comments welcomed
The links to each individual chapter below:
The real numbers,
Sequences,
Continuous functions,
Differenmtiable functions,
Integration,
Series,
Series of functions and Power Series,
Uniform Convergence and differentiation,
Uniform Convergence, Integration and Power Series.
Weierstrass
Approximation Theorem
The
Elementary Functions
Arithmetic of Power Series
Special Test for Convergence  Kummer, Raabe, Gauss and Bertrand's Tests
Improper and Lebesgue Integral


Arc Length, Function of Bounded Variation and Total Variation
Sequences and Series
Change of
Variables Theorems in Integration  a follow up
of "change of Var in Riemann and Lebesgue Integration" shorter
proof.
Kestelman's Change of
Variable Theorem
Functions
Having Finite Derivatives, Bounded Variation, Absolute Continuity, the Banach Zarecki Theorem
and de La Vallee Poussin Theorem
Elementary Proof of de la Vallee Poussin Theorem
Function
of Bounded Variation and Johnson Indicatrix
When is a function
absolutely continuous? The answer and application to generalized
change of variable for Lebesgue integral.
Partial Fraction
Expansion  Its proof, a simple application of complex analysis 


On the primitive of product of two
functions 
A gentle course introducing mathematical analysis Including a week by week
study plan and guide. 

Convergence of ∑
sin(√(n)x)/n and other problems

Advanced Calculus
NUS MA3110
2011/12 Sem 2 Exam Sample Answer
New
NUS MA3110 2010/11 SEm 1 Exam
Comment and Answer New 

Fourier Cosine and Sine Series and Their Convergence 


Ideas of Lebesgue and Perron integration in Uniqueness of Fourier and
Trigonometric series 
Mathematics Diagnostic Testing 

Convergence and summability of Fourier Series 
Basic Skills help:
Algebra refresher
Inequalities 

Second Mean Value Theorem for Integrals and Bonnet Mean Value Theorem 


Abelsummability of Fourier Series and its Derived Series 
Mathematics Assessments for Revision
(Algebra and Calculus AOA level)
New 

Fourier Series for Even and Odd Functions 
Assessment
Gallery
New 

Riemann
Summable everywhere Series, Two Special Cosine series 


An improperly Riemann integrable function that does not give the conclusion
of the Riemann Lebesgue Lemma 
Cantor
Lebesgue Function, Canonical Cantor type function between families of Cantor
sets, Absolute Continuity and Arc Length
New
Included are results on the derivatives of Cantor type
functions over the fat Cantor set and their integrals. 

All About Lim
Sup and Lim Inf New 
Positive Borel Measure and Riesz Representation Theorem
New
Riesz Representation Theorempositive measure version for positive linear
functional. Detail step by step proofs and Lebesgue measure on R^{k} via
Riemann integration and Lebesgue integral. 

Convex Function, L^{p} spaces,
Space of continuous function, Lusin's Theorem
New
A detail introductory exposition of L^{p} spaces and a proof of
Lusin's Theorem including the necessary topological ideas and concepts.

A short proof of the Kestelman change of variable Theorem for Riemann
integral New
A proof using only the properties of absolutely
continuous function and the chain rule for the composition of functions
having finite derivative almost everywhere.
An Introduction To
Measure Theory New A leisurely introduction
to measure theory. A learner's guide to Lebesgue Monotone
Convergence Theorem, Lebesgue Dominated Convergence Theorem, Fatou's Lemma
and complete measure.
Lebesgue
Measure On The Real Numbers and Lebesgue Theorem On Riemann Integrability
New A detail definition of Lebesgue measure on the real
numbers is given. Show that Lebesgue measure is Borel and complete.
Define Riemann integral via step functions, show that it is equivalent to
the Darboux integral and prove the Lebesgue characterization of Riemann
integrability.


Complex
Measure, Dual Space of L^{p} , RadonNikodym Theorem and Riesz
RepresentationTheorems Complex and real versions
New Identification
of the dual of L^{p} spaces and the dual of C_{c}(X) with detail exposition and
proofs. Proofs for both real and complex versions when X is locally
compact as well as the dual for BC(X) the space of bounded continuous real
valued functions when X is normal and Hausdorff are presented. A brief
discussion when X is completely regular and Hausdorff is added.
Convergence In
Measure New
Convergence in measure or in probability, a notion often
used in probability theory. Convergence almost uniformly and convergence
almost everywhere, Egoroff's theorem. As is expected, for a probability
space, convergence almost everywhere implies convergence in measure.
Monotone Convergence theorem, Bounded Convergence THeorem and Dominated
Convergence Theorem for Convergence in measure. Fatou's Lemma. 
Product Measure and
Fubini's Theorem New
This completes the above article: An
Introduction To Measure Theory. A step by step construction of the
product measure space and the definition of the positive product measure
function is given, followed by a detailed elaboration of the proof of the
Fubini's Theorem. The special case when all measure spaces are
required to be complete, is worked through with detail steps and
intermediary results. 

