Calculus is a branch of mathematics used to describe physical properties of the universe by approximating curves or functions to determine their rate of change, area, or volume. Sir Isaac Newton and Gottfried Leibniz independently developed calculus in the 18th century. Differential calculus finds the derivative of a function, while integral calculus finds the original function given its derivative. Both can be used to solve problems in physics, such as determining the speed of a moving object or the surface area of a complex object.
The branch of mathematics called calculus originates from describing the basic physical properties of our universe, such as the motion of planets and molecules. The calculation approaches the paths of moving objects as curves, or functions, and then determines the value of these functions to calculate their rate of change, area or volume. In the 18th century, Sir Isaac Newton and Gottfried Leibniz simultaneously but separately described calculus to help solve problems in physics. The two divisions of calculus, differential and integral, can solve problems such as the speed of a moving object at a given moment or the surface area of a complex object such as a lampshade.
All computation is based on the fundamental principle that you can always use approximations of increasing accuracy to find the exact answer. For example, you can approximate a curve with a series of straight lines—the shorter the lines, the more like a curve they are. You can also approximate a spherical solid with a set of cubes, getting smaller and smaller with each iteration, that fit inside the sphere. Using calculus, you can determine that approximations tend toward the precise end result, called the limit, until you have accurately described and reproduced the curve, surface, or solid.
Differential calculus describes the methods by which, given a function, its associated rate-of-change function, called the “derivative”, can be found. The function must describe an ever-changing system, such as the change in temperature over the course of a day or the speed of a planet around a star during one rotation. The derivative of those functions would give you the rate at which the temperature changed and the acceleration of the planet, respectively.
Integral calculus is like the opposite of differential calculus. Given the rate of change in a system, you can find data values that describe the input to the system. In other words, given the derivative, such as acceleration, you can use integration to find the original function, such as velocity. Additionally, integration is used to calculate values such as the area under a curve, the surface area, or the volume of a solid. Again, this is possible because you start by approximating an area with a series of rectangles and make your guess more and more accurate by studying the limit. The limit, or the number toward which the approximations tend, will give you the precise surface.
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