Image of Introduction to Calculus - A carefully worded overview of calculus for nonspecialists and the simply curious (written by the man who created Apple's first successful word processor - Apple Writer)

This article provides an overview and introduction to calculus. It's intended for general readers, nonspecialists, and shows the topic's key concepts in a transparent, approachable way.

The article's purpose is to help readers see that calculus is not only relatively easy to understand, but is a useful and accessible intellectual skill without which some aspects of the the modern world are indecipherable - population growth, space travel, climate change, epidemics and control of diseases, economic policy, investment strategies and others - all of which require a knowledge of how and why things change, the kind of knowledge calculus provides.

Some familiarity with secondary school algebra is desirable to be able to follow most of the article's content. Some new topics, such as functions, are introduced and explained before being used.

The article includes tutorial exercises that show concepts without overwhelming the reader in detail, and covers related topics such as differential equations and computer algebra systems.


While introducing calculus, this article tries to demolish a pervasive American myth - that calculus is outside the intellectual grasp of everyday people. In contrast to the practice in many other countries, American students are told that calculus is an advanced mathematical topic, one neither suitable for, nor within the mental abilities of, ordinary people. The author believes this is a crippling and misleading belief, one easily falsified by some exposure to the topic itself. In this article, the author strives to explain the mathematical concepts in the simplest, most accessible way.


The key idea of calculus, on which the entire field stands, is the relationship between a quantity and a rate of change in that quantity. For example, we might want to describe the position of a moving car as time passes:

  • The car's acceleration, controlled by the accelerator pedal, is a rate of change in the car's velocity.
  • The car's velocity, recorded by the speedometer, is a rate of change in the car's position.
  • Expressed in everyday language, position is "where you are", velocity is "how fast you're moving", and acceleration represents "how velocity is changing."
  • These three quantities - position, velocity and acceleration - are bound together in such a way that, with the methods of calculus, we can use any one of them to find the other two.
  • For example, we can note a car's speedometer (velocity) readings over time and use that to reconstruct the car's position, without any other information.

The author Paul Lutus has a very cool background and wrote a great book about circumnavigating the globe in a small sailboat.

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