An Introduction to Calculus |
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An Introduction to Calculus |
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What is "calculus"? Actually, "calculus" simply refers to a set of rules for calculating. One might talk about a calculus of arithmetic -- rules for adding, subtracting, multiplying, and dividing real numbers. Computer scientists speak of a "predicate calculus" -- a set of rules observed in the formation of computer languages. These are not what we have in mind in a "calculus" course, though. It's actually about the use and application of some ideas that were first systematically studied in the 17th Century by Isaac Newton and Gottfried Leibniz.
Calculus 1 covers that strand of mathematics commonly referred to as the "differential calculus" and introduces the branch of mathematics known as the "integral calculus". We'll be studying the definition, properties, and applications of the derivative and of the integral. Before we discuss differential and integral calculus, we first need to consider the concept of a limit.
As the references listed below indicate, Gottfried Wilhelm Leibniz and Sir Isaac Newton are believed to have independently discovered calculus. Since Leibniz was the first to publish material on calculus, the notation and terminology we use today more closely resemble his than Newton's.
In differential calculus we approximate a (possibly) nonlinear function f(x) at a specific point (x0 , y0 ) with a line. The slope of this line is called the "derivative of f at x0". Since the slope of this line represents the rate of change of f(x), an understanding of the derivative enables us to analyze problems involving quantities that change. So, in a nutshell, differential calculus deals with slope, rates of change, and related applications.
In integral calculus, we begin by considering the amount of area bounded by a curve y = f(x) and the x-axis over a given interval a < x < b. This area is called the "definite integral of f over [a, b]". An amazing theorem (known as the Fundamental Theorem of Integral Calculus) tells us that this area is related to the "anti-derivative" of f.
People currently study calculus for a wide variety of reasons; in a sense, calculus is the real "language of science". Whereas pre Seventeenth Century mathematicians mostly studied static problems, we use calculus to model dynamically changing quantities. Biologists, chemists, economists, engineers, financial analysts and physicists use calculus to study problems in their fields. Without calculus, it would be much harder to put satellites into orbit, build stable bridges, design efficient machinery, or prescribe appropriate drug doses. The contributions of calculus to human understanding make the subject worthy of study by any college student.
For more information, see these references:
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