A thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof, this book is appropriate for a beginning honors calculus course assuming high school calculus or a “bridge course” using basic analysis to motivate and illustrate mathematical rigor. It can serve as a combination textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in the context of a coherent logical structure, building on familiar properties of real numbers and teaching methods of proof by example along the way. Numerous examples reinforce both practical and theoretical understanding, and extensive historical notes explore the arguments of the originators of the subject.
No previous experience with mathematical proof is assumed: rhetorical strategies and techniques of proof (reduction ad absurdum, induction, contrapositives, etc) are introduced by example along the way. Between the text and the exercises, proofs are available for all the basic results of calculus for functions of one real variable.
Table of Contents
Sequences and their Limits
The Rhetoric of Mathematics (Methods of Proof)
Answers to Selected Problems
About the Author
Excerpt: Ch. 6.1 Local Approximation of Function by Polynomials (p. 368)
Even though in principle a function unambiguously specifies the output value ƒ(x) resulting from a given input x, this doesn't guarantee that the in practice this value can be located as a point on the number line. In Chapter 2, we faced a situation of this sort for the square root function ƒ(x)= √x evaluated at x = 2, and the procedures we came up with for locating √2 could easily be adapted to finding √x for any particular positive number x, provided that x itself is properly located on the number line. However, it would hardly be practical to carry out such a procedure every thime we needed a square root. And for other functions, like sin x, we would have to come up with an entirely different evaluation procedure.
Fortunately, it is possible using tools from calculus to come up with a systematic procedure for evaluating most of the functions we deal with. This is based on a somewhat different point of view: instead of the trial-and-error approach we used in Chapter 2, we try to identify a new function ø which on one hand is relatively easy to calculate and, on the other, yields values acceptable close to those of our desired function ƒ. We refer to such a function ø as an approximation to the function ƒ; the size of their difference | ƒ(x) – ø(x) | will be referred to as the error of the approximation of x. Of course, we shouldn't expect to actually calculate the error (just as we can't actually calculate the error in a decimal approximation to √2), but we hope to establish an error bound, showing that the error stays within ceratin bounds (at least for certain values of x), indicating the accuracy of the approximation. We shall, in this section, be intersted in local approximations, that is, to approximations that have good accuracy when x is near some particular value x = a.
Print ISBN: 978-0-88385-758-8
325 pp., 2010
List Price: $15.00
Contains solutions for all of the problems contained in the textbook.
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