Calculus of One Variable

1. Front Matter

   Pages i-xi

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2. Functions and Graphs

   Pages 1-45

3. Limits of Functions

   Pages 47-77

4. Differentiation

   Pages 79-92

5. Techniques of Differentiation

   Pages 93-109

6. Applications of Differentiation

   Pages 111-132

7. Maclaurin and Taylor Expansions

   Pages 133-151

8. Integration

   Pages 153-171

9. Integration by Parts

   Pages 173-183

10. Integration by Substitution

    Pages 185-199

11. Integration of Rational Functions

    Pages 201-216

12. Geometrical Applications of Integration

    Pages 217-239

13. Back Matter

    Pages 241-267

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The development of the di?erential calculus was one of the major achievements of seventeenth century European mathematics, originating in the work of N- ton, Leibniz and others. Integral calculus can be traced back to the work of Archimedes in the third century B. C. Since its inception, calculus has dev- oped in two main directions. One is the growth of applications and associated techniques,indiverse?eldssuchasphysics,engineering,economics,probability and biology. The other direction is that of analytical foundations, where the intuitive and largely geometrical approach is replaced by an emphasis on logic and the development of an axiomatic basis for the real number system whose properties underpin many of the results of calculus. This approach occupied many mathematicians through the eighteenth and nineteenth centuries, c- minating in the work of Dedekind and Cantor, leading into twentieth century developments in Analysis and Topology. We can learn much about calculus by studying its history, and a good starting point is the St Andrews’ History of Mathematics website www-history. mcs. st-and. ac. uk/history/ Thisbookisdesignedforbeginninguniversitystudents,boththosestudying mathematics as a major subject, and those whose main specialism requires the use and understanding of calculus. In the latter case we would expect that lecturers would customise the treatment with applications from the relevant subject area. Thepre-universityschoolmathematicscurriculaofmostEuropeancountries all include some calculus, and this book is intended to provide, among other things, a transition between school and university calculus. In some countries suchastheU. K.

• Calculus of a single variable
• Integration
• Real functions
• calculus
• real analysis

From the reviews:

"The book is aimed at beginning university students as a introductory calculus textbook … . Undoubtedly it is a clearly written exposition of the most typical topics of calculus ... full of good examples, with plenty of exercises and answers to them. ... Summing up, the reviewed book is a decent item, which might be recommended not as a main textbook for an introductory calculus course, but as a additional source mainly for practising typical procedures of differentiation and integration." (Piotr Zarzycki, The Mathematical Gazette, Vol. 92 (525), 2008)

• School of Mathematics, University of Southampton, Southampton, UK

  Keith E. Hirst

Keith Hirst is an experienced author and lecturer, with many years’ experience teaching undergraduate courses. His research interests include Mathematics Education and Teaching Methods in Higher Education, with particular emphasis on the school / university interface. Keith also researched the area of learning problems in calculus as part of an investigation into undergraduates’ conceptions of mathematical ideas. In 2000, he was awarded one of the first prestigious National Teaching Fellowships by the Institute of Learning and Teaching, to develop a database of undergraduate teaching resources.

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