Ng Tze Beng tbengng@gmail.com 

My Calculus Web

A Place to Learn and to Explore

                               

An Introductory Calculus Course

 

Articles

 
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 cal4.gif (2638 bytes)  Integral of x^(x^n) and x^(-x^n) 
Objectives cal4.gif (2638 bytes)  Do we need Mean Value Theorem? cal4.gif (2638 bytes)  Integration using Differentiation under integral sign  Examples of how to check the condition for using the Theorem on differentiaion under the integral.
Syllabus cal4.gif (2638 bytes)  Extreme Value Theorem cal4.gif (2638 bytes)  Integration Using Power Series Examples of evaluation of (ln(sin(x))^2, (ln(tan(x))^2, etc.
References cal4.gif (2638 bytes)  Intermediate Value Theorem cal4.gif (2638 bytes)  Evaluation of ∫ ln(1+x^4)/(1+x^2Four methods, with one using Fubini's Theorem.
Notes on Derive cal4.gif (2638 bytes)  Boundedness Theorem cal4.gif (2638 bytes) ∫ ln(x)^2/(1+e^x), ∫ xln(x)/(1+e^x),∫ xln(x)^2/(1+e^x), Gamma function and Eta function  New

Proofs of convergence of the improper integrals defining the Gamma function, justification of differentiation under the integration sign, derivatives of Gamma function and Eta function.

General Advice and Learning Guide cal4.gif (2638 bytes)  Monotone Function and Continuity cal4.gif (2638 bytes) On the integrals, ∫ x^s/(1+e^x)^n and ∫ x^s ln(x)/(1+e^x)^n  A recursive relation ie presented , using the relation, you can compute the integral as a finite sum involving the Gamma and eta functions.
Problem-Solving Process cal4.gif (2638 bytes)  Injective Function and Monotone Function cal4.gif (2638 bytes) Integrating  x/(1-e^x) from 0 to infinity without using power series
Tutorials

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? 

 

   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 

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)  Partial Fraction Expansion -- Its proof, a simple application of complex analysis

cal4.gif (2638 bytes)  When is a continuous function on a closed and bounded interval be of bounded variation, absolutely continuous?  Revised

 
    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  

NUS MA3110 2010/11 SEm 1 Exam Comment and Answer

  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

 cal4.gif (2638 bytes)  Algebra refresher Inequalities

 cal4.gif (2638 bytes)   Self Help Centre Quick Reference

 cal4.gif (2638 bytes)  Self Help Centre Refresher

 cal4.gif (2638 bytes)  Self Help Centre Work Books

  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)  

  cal4.gif (2638 bytes) Fourier Series for Even and Odd Functions  
cal4.gif (2638 bytes) Assessment Gallery    cal4.gif (2638 bytes) Riemann Summable everywhere Series, Two Special Cosine series  
cal4.gif (2638 bytes) Comment on A 2019 PSLE Math Question      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    

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   
cal4.gif (2638 bytes) Positive Borel Measure and Riesz Representation Theorem

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 

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  Revised (To include a result that don't require the function f to be bounded on the whole of the domain.)

 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)  A general change of variable theorem for the Riemann integrable New

Generalized Kestelman change of variable theorem - applies to most situation

 cal4.gif (2638 bytes) An Introduction To Measure Theory 

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

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 

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 

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. 

cal4.gif (2638 bytes)Function of Bounded Variation on Arbitrary Set 

Following my previous article, on the image of the total variation function of a function of bounded variation on a closed interval, we now obtain the same result on general arbitrary domain.

 

Denjoy Saks Young Theorem

 cal4.gif (2638 bytes) Arbitrary Function, Lim Sup Inf, Dini Derivates,Lebesgue Density Theorem

cal4.gif (2638 bytes) Functions of Bounded Variation and de La Vallee Poussin's Theorem

 cal4.gif (2638 bytes) Denjoy Saks Young Theorem for Arbitrary Function

 cal4.gif (2638 bytes) Discourse on Monotone Functions

 
cal4.gif (2638 bytes)Absolutely Continuous Function on Arbitrary Domain and Function of Bounded Variation

A continuous function on a closed and bounded domain is absolutely continuous if and only if it is of bounded variation and a Lusin function.  A detail proof of this result is presented and other equivalent formulation with the integrability of the derivative function or the nullity of the image of the set, where the derivative is infinite, positive or negative.  All the technical intermediate results used and their proofs are deliberated.

Topology Article

Explicit decomposition of Steenrod Square 64 by stable secondary cohomology operations and an illustration of proof of non-immersion of quaternionic projective space with obstruction theory

  cal4.gif (2638 bytes)  A de La Vallee Poussin's  Decomposition Theorem

  For arbitrary domain, the classical de La Vallee Poussin's  Decomposition for the outer measure of the image of measurable set under the total variation function.

cal4.gif (2638 bytes)  Lebesgue Stieltjes Measure, de La Vallee Poussin’s Decomposition, Change of Variable, Integration by Parts for Lebesgue Stieltjes Integrals  New with general Change of Variable and general integration by parts with correction terms for discontinuous functions for Lebesgue Stieltjes integrals     

Lebesgue Stieltjes signed measure generated by a function of bounded variation, its total variation measure and Lebesgue Stieltjes measure generated by the total variation of the function. Detail proofs of the de La Vallee Poussin's decomposition including the component given by the Jump function of the function of bounded variation. Decompositions in terms of the Lebesgue measure of the image of the total variation function, positive and negative variation functions of the function of bounded variation are deliberated and proved. Integration by parts and versions of change of variable for Lebesgue Stieltjes integrals.

 

 

COMMENT OR SUGGESTION??

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 tbengng@gmail.com

 

 


        

This page was last updated on 12/02/2024

By Ng Tze Beng