Calculus of One Variable

ISBN-13: 9781852339401 / Angielski / Miękka / 2005 / 268 str.

K.E. Hirst

Opis

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

Recenzje

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)

Spis treści

Functions and Graphs.- Limits of Functions.- Differentiation.- Techniques of Differentiation.- Applications of Differentiation.- Maclaurin and Taylor Expansions.- Integration.- Integration by Parts.- Integration by Substitution.- Integration of Rational Functions.- Geometrical Applications of Integration.

O autorze

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.

Blurb

Understanding the techniques and applications of calculus is at the heart of mathematics, science and engineering. This book presents the key topics of introductory calculus through an extensive, well-chosen collection of worked examples, covering;

algebraic techniques

functions and graphs

an informal discussion of limits

techniques of differentiation and integration

Maclaurin and Taylor expansions

geometrical applications

Aimed at first-year undergraduates in mathematics and the physical sciences, the only prerequisites are basic algebra, coordinate geometry and the beginnings of differentiation as covered in school. The transition from school to university mathematics is addressed by means of a systematic development of important classes of techniques, and through careful discussion of the basic definitions and some of the theorems of calculus, with proofs where appropriate, but stopping short of the rigour involved in Real Analysis.

The influence of technology on the learning and teaching of mathematics is recognised through the use of the computer algebra and graphical package MAPLE to illustrate many of the ideas. Readers are also encouraged to practice the essential techniques through numerous exercises which are an important component of the book. Supplementary material, including detailed solutions to exercises and MAPLE worksheets, is available via the web.