# The Fundamental Theorem of Calculus

**Episode #3 of the course “Equations that changed the world”**

Among its purposes, the mathematical system of calculus is used to calculate the slope of a curve or the rate of change in a system. This is called the “derivative,” and it is essential in calculus. The idea of the derivative was developed by Sir Isaac Newton in his formulation of the classical mechanics, and it applies to a number of functional equations used across science and mathematics. If the derivative is graphed on an x and y axis, it may be necessary to calculate a surface area below a section of the curve. Or, it can be used to determine certain changes that occurred in a system during a particular period of time. This surface area or section of change in a system is the “integral,” which is the other half of the Fundamental Theorem of Calculus.

Essentially, the concept behind the mathematics is that small, incremental changes over time add up to a large total change over more time. This is demonstrable in an infinite number of ways: small deteriorations from the environment turn cities into ruins, or individual drops of water in a bucket will eventually fill and overflow it. These changes can be plotted geometrically, as well as calculated algebraically, and both approaches were explored separately as far back as ancient Greece. In the 17th century, the first equations and proofs were published connecting the two. By the 18th century, the theorem had developed the form that continues today.

The Fundamental Theorem is broken in two—the First Fundamental Theorem of Calculus and the Second Fundamental Theorem of Calculus. Essentially, they tie the concepts of derivatives and integrals together as part of the same system. If a derivative is a solid line on its graph, without interruptions or intervals, its properties can be explored with the First Fundamental Theorem. If a graphed derivative line is not solid because there are interruptions over time, which create breaks in the graph, mathematicians use the formula of the Second Fundamental Theorem of Calculus to account for those intervals.

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