A Course in Calculus and Real Analysis

Preface

Calculus is one of the triumphs of the human mind. It emerged from investigations into such basic questions as finding areas, lengths and volumes. In the third century B.C., Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of finding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In particular, they showed that the geometric problems of finding the areas of planar regions and of finding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass.

Today, calculus occupies a central place in mathematics and is an essential component of undergraduate education. It has an immense number of applications both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics.

Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notions encountered in calculus, but with greater rigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables.

This book attempts to give a self-contained and rigorous introduction to calculus of functions of one variable. The presentation and sequencing of topics emphasizes the structural development of calculus. At the same time, due importance is given to computational techniques and applications. In the course of our exposition, we highlight the fact that calculus provides a firm foundation to several concepts and results that are generally encountered in high school and accepted on faith. For instance, this book can help students get a clear understanding of (i) the definitions of the logarithmic, exponential and trigonometric functions and a proof of the fact that these are not algebraic functions, (ii) the definition of an angle and (iii) the result that the ratio of the circumference of a circle to its diameter is the same for all circles. It is our experience that a majority of students are unable to absorb these concepts and results without getting into vicious circles. This may partly be due to the division of calculus and real analysis in compartmentalized courses. Calculus is often taught as a service course and as such there is little time to dwell on subtleties and gain perspective. On the other hand, real analysis courses may start at once with metric spaces and devote more time to pathological examples than to consolidating students’ knowledge of calculus. A host of topics such as L’Hopital’s rule, points of inflection, convergence criteria for Newton’s method, solids of revolution, and quadrature rules, which may have been inadequately covered in calculus courses, become pass´e when one studies real analysis. Trigonometric, exponential, and logarithmic functions are defined, if at all, in terms of infinite series, thereby missing out on purely algebraic motivations for introducing these functions. The ubiquitous role of p as a ratio of various geometric quantities and as a constant that can be defined independently using calculus is often not well understood. A possible remedy would be to avoid the separation of calculus and real analysis into seemingly disjoint courses and textbooks. Attempts along these lines have been made in the past as in the excellent books of Hardy and of Courant and John. Ours is another attempt to give a unified exposition of calculus and real analysis and address the concerns expressed above. While this book deals with functions of one variable, we intend to treat functions of several variables in another book.