# What is calculus? 微積分是甚麼？

Finney, Weir and Giordano, Thomas' Calculus, International Edition, Updated 10th Edition, Addison Wesley, 2003. (東華書局代理)

• 前言 (page xxv) 寫給學生的話 (To the student)
• 先備知識複習 (page 1) Preliminaries: Overview
• 第一章 極限與連續 (page 85) Limits and Continuity
• 第一章 極限的非正規定義 (page 89) Informal Definition of Limit
• 第二章 導數 (page 147) Derivatives
• 第四章 積分 (page 313) Integration

Calculus is the mathematics of motion and change. Where there is motion or growth, where variable forces are at work producing acceleration, calculus is the mathematics to apply. This was true in the beginnings of the subject, and it is true today.

Calculus was first invented to meet the mathematical needs of the scientists of the sixteenth and seventeenth centuries, needs that were mainly mechanical in nature. Differential calculus dealt with the problem of calculating rates of change. It enabled people to define slopes of curves, to calculate velocities and accelerations of moving bodies, to find firing angles that would give cannons their greatest range, and to predict the times when planets would be closest together or farthest apart. Integral calculus dealt with the problem of determining a function from information about its rate of change. It enabled people to calculate the future location of a body from its present position and a knowledge of the forces acting on it, to find the areas of irregular regions in the plane, to measure the lengths of curves, and to find the volumes and masses of arbitrary solids.

Today, calculus and its extensions in mathematical analysis are far-reaching indeed, and the physicists, mathematicians, and astronomers who first invented the subject would surely be amazed and delighted, as we hope you will be, to see what a profusion of problems it solves and what a range of fields now use it in the mathematical models that bring understanding about the universe and the world around us.

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The main building blocks of calculus are functions and graphs. Functions and parametric equations are the major tools for describing the real world in mathematical terms, from temperature variations to planetary motions, from brain waves to business cycles, and from heartbeat patterns to population growth. Many functions have particular importance because of the behavior they describe. Trigonometric functions describe cyclic, repetitive activity; exponential, logarithmic, and logistic functions describe growth and decay; and polynomial functions can approximate these and most other functions.

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The concept of limit is one of the ideas that distinguish calculus from algebra and trigonometry. The outputs of some functions vary continuously as their inputs vary---the smaller the change in the input, the smaller the cahnge in the output. The values of other functions may jump or vary erratically no matter how carefully we control the inputs. The notion of limit gives a precise way to distinguish between these behaviors. We also use limits to define tangent lines to graphs of functions. This geometric application leads at once to the important concept of derivative of a function.

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Let $$f(x)$$ be defined on an open interval about x0, except possibly at x0 itself. if $$f(x)$$ gets arbitrarily close to L for all x sufficiently close to x0, we say that f approaches the limit L as x approaches x0, and we write $\lim_{x\to x_0} f(x) = L$ This definition is "informal" because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context. To a machinist manufacturiing a piston, close may mean within a few thousandths of an inch. To an astronomer studying distant galaxies, close may mean within a few thousand light-years. The definition is clear enough, however, to enable us to recognize and evaluate limits of a number of specific functions.

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The slope of a curve at a point is the limit of secant slopes. This limit, called a derivative, measures the rate at which a function changes, and it is one of the most important ideas in calculus. Derivatives are used widely in engineering, science, economics, medicine, and computer science to calculate velocity and acceleration, to explain the behavior of machinery, to estimate the drop in water level as water is pumped out of a tank, and to predict the consequences of making errors in measurements. Finding derivatives by evaluating limits can be lengthy and difficult. Mathematicians had developed techniques to make calculating derivatives easier.

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The need of calculating instantaneous rates of change led the discoverers of calculus to an investigation of the slopes of tangent lines and, ultimately, to the derivative, to what we call differential calculus. But they knew that derivatives revealed only half the story. In addition to a calculation method (a "calculus") to describe how functions were changing at a given instant, they also needed a method to describe how those instantaneous changes could accumulate over an interval to produce the function. That is, by studying how a behavior changed, they wanted to learn about the behavior itself. For example, from knowing the velocity of a moving object, they wanted to be able to determine its position as a function of time. This is why they were also investigating areas under the curves, an investigation that ultimately led to the second main branch of calculus, called integral calculus.

Once they had the calculus for finding slopes of tangent lines and the calculus for finding areas under curves---two geometric operations that would seem to have nothing at all to do with each other---the challenge for Newton and Leibniz was to prove the connection that they knew intuitively had to be there. The discovery of this connection (called the Fundamental Theorem of Calculus) brought differential and integral calculus together to become the single most powerful tool mathematicians ever acquired for understandnig the universe.

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Created: Oct 24, 2003
Last Revised: 05/09/04, 22/01/02