A Calculus of Ideas:A Mathematical Study of Human Thought
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For example,. If you know how fast some quantity is changing, then how big is it at a certain time? On the highway again, you could imagine trying to figure out how far s you are from home at time t by studying the velocity v speedometer information.
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- Calculus of Ideas, A: A Mathematical Study of Human Thought by Ulf Grenander.
But finding s from v is harder if the velocity is variable during the trip. An intergal analogous to your car's trip odometer which if you set it to 0 when you start out measures s on your dashboard. Instead of a distance, you might be interested in trying to compute the total amount of money in an account if you know how fast it's growing the interest rate , or the total number of people infected with some disease at time t if you know the infection rate.
The same mathematical concept, the integral, is involved in each case. The study of the integral and how it relates to the derivative, is the main theme of Calculus II.
MA005: Calculus I
All of this is hard work. But what did you expect in learning about "one of the great intellectual accomplishments of the human mind? Or, to change the analogy, it's like learning a new language: it can open new vistas but only if you're willing to memorize vocabulary, learn to conjugate verbs, and practice, practice, practice! Introduction: What is Calculus About? The seeds for the main ideas of calculus go back to ancient times but calculus itself, as we think of it today, was invented or discovered?
A Calculus Of Ideas: A Mathematical Study Of Human Thought
Its invention is usually attributed to the English mathematician Sir Isaac Newton and the German mathematician Gottfried Leibniz , working independently. As its power developed, calculus gave scientists a tool to generate remarkable new understandings of the world. Its creation is considered one of the great intellectual accomplishments of the human mind.
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This sweeping assertion is justified not only by the beauty of the subject, but also by the fact that it still retains its fundamental importance, even several centuries after its birth. In fact, its role has become more important than ever as the use of mathematical models has reached beyond areas such as physics and engineering and into such different fields as biology, economics and business.
How fast s is changing at a time t is your velocity v at that time.
Whether a rate of change occurs in biology, physics, or economics, the same mathematical concept--the derivative-- is involved in each case. We'll see indications of some of the diverse applications of calculus during the course. But it's not a course in physics, biology, economics, or business. Many of the most interesting and significant applications you will have to meet elsewhere.
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The ancient origins of calculus
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A Calculus of Ideas: A Mathematical Study of Human Thought
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