Calculus is a field of mathematics that deals with rates of change and motion. Suppose that one nation fires a rocket carrying a bomb into the atmosphere, aimed at a second nation. The first nation must know exactly what path the rocket will follow if the attack is to be successful. And the second nation must know the same information if it is to protect itself against the attack. In this example, calculus is used by mathematicians in both nations to study the motion of the rocket.

Calculus was originally developed in the late 1600s by two great scientific minds, English physicist Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716). Both scholars presented their ideas at about the same time, so credit for the invention of calculus must go to both. The debate over credit at the time, however, reached intense levels and sparked bad feelings between the two countries involved (Great Britain and Germany). Over the past 300 years, calculus has become an absolutely essential mathematical tool in every field of science, mathematics, and engineering.

To illustrate the basic principles of calculus, imagine that you are studying changes in population in your hometown over the past 100 years. As you graph the data you collected, you can see that population increased for a number of years, then decreased for a period of time before beginning a second increase. One question you might want to ask is what the rate of change in the population was at any given time, such as any given year. For example, was population increasing at the same rate in 1980 that it was in 1890? One way to answer that question is to locate two points on the curve. The rate of change for this part of the graph, then, is how steeply the curve rises between these two points.

Differential and integral calculus

Calculus can be subdivided into two general categories: differential and integral calculus. Differential calculus deals with problems of the type above, in which some mathematical function (such as population change) is known. From the graphical or mathematical representation of that function, the rate of change can be calculated.

The reverse process can also be performed. For example, it may be possible to find the rate of change for some function. From that rate of change, then, it may be possible to determine the original function itself. This field of mathematics is known as integral calculus.

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