Skip to main content
main-content

Über dieses Buch

The purpose of a first course in calculus is to teach the student the basic notions of derivative and integral, and the basic techniques and applica­ tions which accompany them. The very talented students, with an ob­ vious aptitude for mathematics, will rapidly require a course in functions of one real variable, more or less as it is understood by professional is not primarily addressed to them (although mathematicians. This book I hope they will be able to acquire from it a good introduction at an early age). I have not written this course in the style I would use for an advanced monograph, on sophisticated topics. One writes an advanced monograph for oneself, because one wants to give permanent form to one's vision of some beautiful part of mathematics, not otherwise ac­ cessible, somewhat in the manner of a composer setting down his sym­ phony in musical notation. This book is written for the students to give them an immediate, and pleasant, access to the subject. I hope that I have struck a proper com­ promise, between dwelling too much on special details and not giving enough technical exercises, necessary to acquire the desired familiarity with the subject. In any case, certain routine habits of sophisticated mathematicians are unsuitable for a first course. Rigor. This does not mean that so-called rigor has to be abandoned.

Inhaltsverzeichnis

Frontmatter

Review of Basic Material

Frontmatter

Chapter I. Numbers and Functions

Abstract
In starting the study of any sort of mathematics, we cannot prove everything. Every time that we introduce a new concept, we must define it in terms of a concept whose meaning is already known to us, and it is impossible to keep going backwards defining forever. Thus we must choose our starting place, what we assume to be known, and what we are willing to explain and prove in terms of these assumptions.
Serge Lang

Chapter II. Graphs and Curves

Abstract
The ideas contained in this chapter allow us to translate certain statements backwards and forwards between the language of numbers and the language of geometry.
Serge Lang

Differentiation and Elementary Functions

Frontmatter

Chapter III. The Derivative

Abstract
The two fundamental notions of this course are those of the derivative and the integral. We take up the first one in this chapter.
Serge Lang

Chapter IV. Sine and Cosine

Abstract
From the sine of an angle and the cosine of an angle, we shall define functions of numbers, and determine their derivatives.
Serge Lang

Chapter V. The Mean Value Theorem

Abstract
Given a curve, y = f(x) we shall use the derivative to give us information about the curve. For instance, we shall find the maximum and minimum of the graph, and regions where the curve is increasing or decreasing. We shall use the mean value theorem, which is basic in the theory of derivatives.
Serge Lang

Chapter VI. Sketching Curves

Abstract
We have developed enough techniques to be able to sketch curves and graphs of functions much more efficiently than before. We shall investigate systematically the behavior of a curve, and the mean value theorem will play a fundamental role.
Serge Lang

Chapter VII. Inverse Functions

Abstract
Suppose that we have a function, for instance
$$ y = 3x - 5$$
.
Serge Lang

Chapter VIII. Exponents and Logarithms

Abstract
We remember that we had trouble at the very beginning with the function 2 X (or 3 X , or 10 X ). It was intuitively very plausible that there should be such functions, satisfying the fundamental equation 2 x+y =2 x 2 y for all numbers x, y, and 2° = 1, but we had difficulties in saying what we meant by \( 2^{\sqrt 2 }\) (or 2 { π .
Serge Lang

Integration

Frontmatter

Chapter IX. Integration

Abstract
In this chapter, we solve, more or less simultaneously, the following problems: (1) Given a function f(x), find a function F(x) such that
$$ F'\left( x \right) = f\left( x \right)$$
Serge Lang

Chapter X. Properties of the Integral

Abstract
This is a short chapter. It shows how the integral combines with addition and inequalities. There is no good formula for the integral of a product. The closest thing is integration by parts, which is postponed to the next chapter.
Serge Lang

Chapter XI. Techniques of Integration

Abstract
The purpose of this chapter is to teach you certain basic tricks to find indefinite integrals. It is of course easier to look up integral tables, but you should have a minimum of training in standard techniques.
Serge Lang

Chapter XII. Applications of Integration

Abstract
Mathematics consists in discovering and describing certain objects and structures. It is essentially impossible to give an all-encompassing description of these. Hence, instead of such a definition, we simply state that the objects of study of mathematics as we know it are those which you will find described in the mathematical journals of the past two centuries, and leave it at that. There are many reasons for studying these objects, among which are aesthetic reasons (some people like them), and practical reasons (some mathematics can be applied).
Serge Lang

Taylor’s Formula and Series

Frontmatter

Chapter XIII. Taylor’s Formula

Abstract
We finally come to the point where we develop a method which allows us to compute the values of the elementary functions like sine, exp, and log. The method is to approximate these functions by polynomials, with an error term which is easily estimated. This error term will be given by an integral, and our first task is to estimate integrals. We then go through the elementary functions systematically, and derive the approximating polynomials.
Serge Lang

Chapter XIV. Series

Abstract
Series are a natural continuation of our study of functions. In the preceding chapter we found how to approximate our elementary functions by polynomials, with a certain error term. Conversely, one can define arbitrary functions by giving a series for them. We shall see how in the sections below.
Serge Lang

Functions of Several Variables

Frontmatter

Chapter XV. Vectors

Abstract
The concept of a vector is basic for the study of functions of several variables. It provides geometric motivation for everything that follows. Hence the properties of vectors, both algebraic and geometric, will be discussed in full.
Serge Lang

Chapter XVI. Differentiation of Vectors

Abstract
Consider a bug moving along some curve in 3-dimensional space.
Serge Lang

Chapter XVII. Functions of Several Variables

Abstract
We view functions of several variables as functions of points in space. This appeals to our geometric intuition, and also relates such functions more easily with the theory of vectors. The gradient will appear as a natural generalization of the derivative. In this chapter we are mainly concerned with basic definitions and notions. We postpone the important theorems to the next chapter.
Serge Lang

Chapter XVIII. The Chain Rule and the Gradient

Abstract
In this chapter, we prove the chain rule for functions of several variables and give a number of applications. Among them will be several interpretations for the gradient. These form one of the central points of our theory. They show how powerful the tools we have accumulated turn out to be.
Serge Lang

Backmatter

Weitere Informationen