CALCULUS OF ONE VARIABLE by HIRST KEITH E. (PDF)

Description

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 it

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Hirst’s text does an effective job of introducing the techniques and applications of differential and integral calculus. Hirst does this by providing robust examples that enable the reader to develop the skills necessary to solve the quite complicated problems that he poses in the exercises. Since Hirst provides answers to nearly all of these exercises, the text lends itself to self-study. The text feels somewhat incomplete because it does not develop the underlying theory. It also does not cover sequences and series, which Hirst covers in his text

⭐. While its scope is somewhat limited, working through the text is worthwhile since it helps you develop sophisticated problem solving skills.This text is an introduction to college calculus for students who have been exposed to calculus while completing the A level course in pure mathematics. It would be appropriate for American students who have completed an Advanced Placement Calculus course and wish to develop their skills further. The prerequisites covered in the introductory chapter include functions and graphs, domain and range, odd and even functions, composite functions, inverse functions, and piecewise-defined functions. These prerequisites are discussed in the context of polynomial, rational, exponential, logarithmic, trigonometric, and hyperbolic functions and both trigonometric and hyperbolic identities.The next chapter covers limits of functions, including the definition of limits and one-sided limits, algebraic techniques for finding them, properties of limits, infinite limits, limits at infinity, and the Squeeze Theorem. Since some exposure to differential calculus is assumed, Hirst also introduces L’Hopital’s Rule here, although it could be deferred until later without interrupting the flow of the text.Hirst then devotes several chapters to differentiation, covering the definition, properties of the derivative, the Chain Rule, higher derivatives, implicit differentiation, logarithmic differentiation, parametric differentiation, the differentiation of inverse functions, and Leibniz’s Theorem. Once these skills have been introduced, Hirst shows you how to apply them to finding gradients and tangents, maxima and minima, optimization problems, linear motion problems, growth and decay problems, and the Newton-Raphson method of finding the roots of an equation. Related rates are not covered.Hirst concludes his coverage of differentiation with a chapter on Taylor polynomials. The chapter begins with a discussion of linear approximation, the Mean Value Theorem, and quadratic approximation. He then demonstrates how to find the Taylor polynomial and the error associated with the Taylor polynomial approximation of a function. While I generally found this book to be fairly clear, I had to refer to Tom M. Apostol’s text Calculus: Volume I in order to clarify the discussion in this chapter and to learn how to solve some of the problems.The remainder of the text is devoted to integral calculus. After he introduces the anti-derivative (indefinite integral), Hirst discusses the logarithmic integral, integrals with variable limits, infinite integrals, and improper integrals before devoting chapters to integration by parts, integration by substitution, and integration by partial fractions. These chapters are particularly worthwhile since Hirst goes beyond the standard topics in order to cover reduction formulae, the Gamma function, inverse substitutions, trigonometric substitutions (including half-angle substitutions), hyperbolic substitutions, and partial fractions with repeated linear and quadratic factors. The text concludes with a chapter on applications of integration, including calculations of arc length, surface area of revolution, volume of revolution, density and mass, and center of mass.This text would make a good companion to a more theoretical text such as Serge Lang’s

⭐, Tom M. Apostol’s

⭐, or Michael Spivak’s

⭐.

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