Larissa Fradkin

Differential Calculus Simplified to the Bone


14 September 2013

Table of Contents




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 CALCULUS: 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


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





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 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.

One major topic is covered in these Notes, Differential Calculus. You can understand this topic best if you first study the Notes on Elementary Algebra and Functions. Here is a concept map of Elementary Algebra.

Differential Calculus is part of Basic Calculus. 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!