Ng Tze Beng 

My Calculus Web

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A Place to Learn and to Explore

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An Introductory Calculus Course



General Information

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

cal4.gif (2638 bytes)  Substitution in Integral and Differentials
Objectives cal4.gif (2638 bytes)  Do we need Mean Value Theorem?
Syllabus cal4.gif (2638 bytes)  Extreme Value Theorem
References cal4.gif (2638 bytes)  Intermediate Value Theorem
Notes on Derive cal4.gif (2638 bytes)  Boundedness Theorem
General Advice and Learning Guide cal4.gif (2638 bytes)  Monotone Function and Continuity
Problem-Solving Process cal4.gif (2638 bytes)  Injective Function and Monotone Function

cal4.gif (2638 bytes)  Riemann Integral and Bounded Function

cal4.gif (2638 bytes)   Guide and Comment for Tutorial Assignment cal4.gif (2638 bytes)  Riemann Integral and Infinite Series
cal4.gif (2638 bytes)   Example Sessions cal4.gif (2638 bytes)  Derived functions and Derivative
cal4.gif (2638 bytes)   Online Quizzes cal4.gif (2638 bytes)  Continuity, Differentiability, Weierstrass' Function
cal4.gif (2638 bytes)   Precalculus Online Quiz cal4.gif (2638 bytes)  Intermediate Value Theorem for Derived Function
cal4.gif (2638 bytes)  A Formula of Euler and Appreciating Calculus     cal4.gif (2638 bytes)  Monotone Function, Bounded Variation, Fundamental Theorem of Calculus
Tests and Past Exam Papers cal4.gif (2638 bytes)  Heine-Borel, Bolzano Weierstrass Theorems, Uniform Continuity and Riemann Integrability
Letter to Students cal4.gif (2638 bytes)  Composition and Riemann Integrability
Letter to and from a fellow teacher cal4.gif (2638 bytes)  Composition and Lebesgue Integrability
Link to other Calculus Web sites cal4.gif (2638 bytes)   Change of Variable in Riemann and Lebesgue Integration 
cal4.gif (2638 bytes)  Comment and Errata to Calculus, an introduction

Calculus, an introduction available from NUS Coop

cal4.gif (2638 bytes)   The Cantor Set  
cal4.gif (2638 bytes)   Darboux's Fundamental Theorem of Calculus 
Review of 1999-2000 1st Semester Exam   

Download and install the following 3 math fonts to view the review:Click to install  font a  font b font c


cal4.gif (2638 bytes)  L Hopital's Rule - And a Generalized Version 

cal4.gif (2638 bytes)  Concavity - Definitions and Equivalence  

cal4.gif (2638 bytes) Books on web:    cal4.gif (2638 bytes)   Integration By Parts
Real Numbers?  Newly revised


   cal4.gif (2638 bytes)   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.

cal4.gif (2638 bytes)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,

IntegrationSeriesSeries 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

cal4.gif (2638 bytes) Improper and Lebesgue Integral 

  cal4.gif (2638 bytes)   Arc Length, Function of Bounded Variation and Total Variation 

cal4.gif (2638 bytes)   Sequences and Series 

cal4.gif (2638 bytes) Change of Variables Theorems in Integration   - a follow up of  "change of Var  in Riemann and Lebesgue Integration" shorter proof.

cal4.gif (2638 bytes)  Kestelman's Change of Variable Theorem 

cal4.gif (2638 bytes)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

cal4.gif (2638 bytes) Function of Bounded Variation and Johnson Indicatrix

 cal4.gif (2638 bytes)  When is a function absolutely continuous? The answer and application to generalized change of variable for Lebesgue integral.  

cal4.gif (2638 bytes)  Partial Fraction Expansion -- Its proof, a simple application of complex analysis

    cal4.gif (2638 bytes)  On the primitive of product of two functions 
A gentle course introducing mathematical analysis Including a week by week study plan and guide.   cal4.gif (2638 bytes) 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

  cal4.gif (2638 bytes)  Fourier Cosine and Sine Series and Their Convergence   
    cal4.gif (2638 bytes)  Ideas of Lebesgue and Perron integration in Uniqueness of Fourier and Trigonometric series 
cal4.gif (2638 bytes)  Mathematics Diagnostic Testing    cal4.gif (2638 bytes)  Convergence and summability of Fourier Series
Basic Skills help Algebra refresher  Inequalities   cal4.gif (2638 bytes) Second Mean Value Theorem for Integrals and Bonnet Mean Value Theorem 
cal4.gif (2638 bytes)      cal4.gif (2638 bytes) Abel-summability of Fourier Series and its Derived Series  
cal4.gif (2638 bytes) Mathematics Assessments for Revision

    (Algebra and Calculus AO-A level)   New

  cal4.gif (2638 bytes) Fourier Series for Even and Odd Functions
cal4.gif (2638 bytes) Assessment Gallery  New   cal4.gif (2638 bytes) Riemann Summable everywhere Series, Two Special Cosine series
    cal4.gif (2638 bytes) An improperly Riemann integrable function that does not give the conclusion of the Riemann Lebesgue Lemma
 cal4.gif (2638 bytes)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.

  cal4.gif (2638 bytes) All About Lim Sup and Lim Inf  New
cal4.gif (2638 bytes) Positive Borel Measure and Riesz Representation Theorem New

Riesz Representation Theorem-positive measure version for positive linear functional. Detail step by step proofs and Lebesgue measure on Rk via Riemann integration and Lebesgue integral.

  cal4.gif (2638 bytes) Convex Function, Lp spaces, Space of continuous function, Lusin's Theorem  New

A detail introductory exposition of Lp spaces and a proof of Lusin's Theorem including the necessary topological ideas and concepts.    

cal4.gif (2638 bytes)  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.

 cal4.gif (2638 bytes) 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.

cal4.gif (2638 bytes)  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.


  cal4.gif (2638 bytes) Complex Measure, Dual Space of Lp , Radon-Nikodym Theorem and Riesz RepresentationTheorems- Complex and real versions  New

Identification of the dual of Lp spaces and the dual of Cc(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.

cal4.gif (2638 bytes) 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.

cal4.gif (2638 bytes) 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. 




If you have any comment or suggestion regarding this website (for example, you feel that you like something (other than lecture notes!!) to be put on the web. Or if you have found some interesting websites. Or, you particularly like something we have put here.)   I shall be glad to hear from you. You may e-mail Ng Tze Beng at




This page was last updated on 20/10/2018

By Ng Tze Beng