Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra 2nd Edition
The remarkable progress that has been made in science and technology during the’ last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in physics, astronomy, engineering, chemistry, geology, biology, and other fields including, rather recently, some of the social sciences.
To give the reader an idea of the many different types of problems that can be treated by the methods of calculus, we list here a few sample questions selected from the exercises that occur in later chapters of this book.
With what speed should a rocket be fired upward so that it never returns to earth? What is the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter L? What volume of material is removed from a solid sphere of radius 2r if a hole of radius r is drilled through the center? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour, by how much will it increase at the end of two hours? If a ten-pound force stretches an elastic spring one inch, how much work is required to stretch the spring one foot?
These examples, chosen from various fields, illustrate some of the technical questions that can be answered by more or less routine applications of calculus.
Calculus is more than a technical tool-it is a collection of fascinating and exciting ideas that have interested thinking men for centuries. These ideas have to do with speed, area, volume, rate of growth, continuity, tangent line, and other concepts from a variety of fields. Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is its unifying power. Most of these ideas can be formulated so that they revolve around two rather specialized problems of a geometric nature. We turn now to a brief description of these problems.
Consider a curve C which lies above a horizontal base line such as that shown in Figure 1.1. We assume this curve has the property that every vertical line intersects it once at most.
The shaded portion of the figure consists of those points which lie below the curve C, above the horizontal base, and between two parallel vertical segments joining C to the base. The first fundamental problem of calculus is this: To assign a number which measures the area of this shaded region.
Consider next a line drawn tangent to the curve, as shown in Figure I. 1. The second fundamental problem may be stated as follows: To assign a number which measures the steepness of this line.
Basically, calculus has to do with the precise formulation and solution of these two special problems. It enables us to define the concepts of area and tangent line and to calculate the area of a given region or the steepness of a given tangent line. Integral calculus deals with the problem of area and will be discussed in Chapter 1. Differential calculus deals with the problem of tangents and will be introduced in Chapter 4.
The study of calculus requires a certain mathematical background. The present chapter deals with this background material and is divided into four parts: Part I provides historical perspective; Part 2 discusses some notation and terminology from the mathematics of sets; Part 3 deals with the real-number system; Part 4 treats mathematical induction and the summation notation. If the reader is acquainted with these topics, he can proceed directly to the development of integral calculus in Chapter 1. If not, he should become familiar with the material in the unstarred sections of this Introduction before proceeding to Chapter I.
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|May 7, 2022|
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