Larissa Fradkin
Basic Calculus Unveiled
……………………………………………………
14 September 2013
Table of Contents
III. LECTURES
Lecture 13. CALCULUS: Sequences, Limits and Series
13.1 Sequences
13.2 Limit of a sequence
13.3 Series
13.4 Instructions for self-study
Lecture 14. DIFFERENTIAL CALCULUS: Limits, Continuity and Differentiation of Real Functions of One Real Variable
14.1 Limits
14.2 Continuity of a function
14.3 Differentiation of a real function of one real variable
14.4 A historical note
14.5 Instructions for self-study
Lecture 15. DIFFERENTIAL CALCULUS: Differentiation (ctd.)
15.1 Differentiation Table
15.2 Differentiation Rules
15.3 Decision Tree for Differentiation
15.4 The higher order derivatives
15.5 The partial derivatives
15.6 Applications of differentiation
15.7 Instructions for self-study
Lecture 16. DIFFERENTIAL CALCUUS: Sketching Graphs Using Analysis
16.1 Stationary points
16.2 Increasing and decreasing functions
16.3 Maxima and minima
16.4 Convex and concave functions
16.5 Inflexion points
16.6 Sketching rational functions using analysis
16.7 Applications of rational functions and their graphs
16.8 Instructions for self-study
Lecture 17. Application of DIFFERENTIAL CALCULUS to Approximation of Functions: the Taylor and Maclaurin Series
17.1 Approximating a real function of a real variable using its first derivative
17.2 The Maclaurin polynomials
17.3 The Taylor polynomials
17.4 The Taylor series
17.5 The Maclaurin series
17.8 Instructions for self-study
Lecture 18. INTEGRAL CALCULUS: Integration of Real Functions of One Real Variable (Definite Integrals)
18.1 A definite integral
18.2 Notation
18.3 Comparison of a definite Integral and derivative
18.4 Examples of integrable functions
18.5 The Mean Value Theorem
18.6 A definite integral with a variable upper limit - function Φ(x)
18.7 The Fundamental Theorem of Calculus
18.8 Applications of integration
18.9 A historical note
18.10 Instructions for self-study
Lecture 19. INTEGRAL CALCULUS: Integration of Real Functions of One Real Variable (Indefinite Integrals)
19.1 The relationship between differentiation and integration
19.2 An indefinite integral
19.3 Finding an indefinite integral
19.4 Instructions for self-study
Lecture 20. INTEGRAL CALCULUS: Advanced Integration Methods
20.1 Integration of products of trigonometric functions
20.2 Integration by parts (integration of products of different types of functions)
20.3 Partial fractions
20.4 Integration of rational functions
20.5 Decision Tree for Integration
20.6 Instructions for self-study
Lecture 21. INTEGRAL CALCULUS: Applications of Integration
21.1 Mean value of a function
21.2 Electrical and mechanicsl systems
21.3 Rotational systems
21.4 Instructions for self-study
Lecture 22. Ordinary Differential Equations
22.1 Basic concepts
22.2 Order of a differential equation
22.3 Linearity or non-linearity of a differential equation
22.4 Differential equations with constant coefficients
22.5 Homogeneous and inhomogeneous ODEs
22.6 The first order linear homogeneous equation with constant coefficients
22.7 The initial value problems
22.8 Balance equations in chemical engineering
22.9 Ordinary differential equations with complex coefficients
IV. SUMMARIES
Algebra Summary
Functions Summary
Order of Operations Summary
Quadratics Summary
Trigonometry Summary
Complex Numbers
Decision Tree For Solving Simple Equations
Sketching Graphs by Simple Transformations
Finding a Limit of a Sequence
Differentiation Summary
Integration Summary
V. GLOSSARY
VI. STUDY SKILLS FOR MATHS
VII. TEACHING METHODOLOGY (FAQs)
These notes are based on the lectures delivered by the author to engineering students of London South Bank University over the period of 16 years. This is a University of widening participation, with students coming from many different countries, many of them not native English speakers. Most students have limited mathematical background and limited time both to revise the basics and to study new material. A system has been developed to assure efficient learning even under these challenging restrictions. The emphasis is on systematic presentation and explanation of basic abstract concepts. The technical jargon is reduced to the bare minimum.
Nothing gives a teacher a greater satisfaction than seeing a spark of understanding in the students’ eyes and genuine pride and pleasure that follows such understanding. The author’s belief that most people are capable of succeeding in - and therefore enjoying - the kind of mathematics that is taught at Universities has been confirmed many times by these subjective signs of success as well as genuine improvement in students’ exam pass rates. Interestingly, no correlation had ever been found at the Department where the author worked between the students’ qualification on entry and graduation.
The book owns a lot to the authors’ students – too numerous to be named here - who talked to her at length about their difficulties and successes, e.g. see Appendix VII on Teaching Methodology. One former student has to be mentioned though – Richard Lunt – who put a lot of effort into making this book much more attractive than it would have been otherwise.
The author can be contacted through her website www.soundmathematics.com. All comments are welcome and teachers can obtain there the copy of notes with answers to questions suggested in the text as well as detailed Solutions to suggested Exercises. The teachers can then discuss those with students at the time of their convenience.
Good luck everyone!
Throughout when we first introduce a new concept (a technical term or phrase) or make a conceptual point we use the bold red font. We use the bold blue to verbalise or emphasise an important idea. Two major topics are covered in this course, Elementary Algebra and Basic Calculus.
Two major topics are covered in these Notes, Differential Calculus and Integral Calculus. You can understand these topics best if you first study the Notes on Elementary Algebra and Functions. Here is a concept map of Elementary Algebra.
Here is a concept map of Basic Calculus. It is best to study it before studying any of the Calculus Lectures to understand where they are on the map. The better you see the big picture the faster you learn!
