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Calculus is the mathematical study of continuous change.
The two basic problems of calculus from a geometric point of view are the tangent line problem and the area problem. The first asks to find the slope of the tangent line to the graph of a function at a given point. The second asks to find the area under the graph of a function over a given interval.
The tangent line problem is solved by using an idea/calculation called the derivative of the function. The area problem is solved by using an idea/calculation called the integral of the function. Essential to both the derivative and the integral is the idea of a limit. Although initially it looks like derivatives and integrals are extremely difficult to calculate and have nothing to do with each other, it turns out that there are systematic methods for calculating derivatives and integrals, and that they are basically opposites of each other. The importance of calculus comes from the enormous variety of applications for derivatives and integrals beyond these motivating geometric ideas.
Calculus was invented in the 17th century by a number of people (Isaac Newton being the best known) to solve certain problems in physics. With calculus, the mathematical description of the physical universe became possible for the first time and modern science was born.
Throughout the 18th and 19th centuries calculus was used to describe an enormous variety of physical phenomena from the motion of planets to electromagnetic radiation, in many cases through the closely related topic of differential equations. Today calculus remains at the heart of the way we think of the physical world (science), our methods for manipulating it (technology), and our language for describing it (mathematics). As a result all students in science, engineering, mathematics, and computer science are required to take a course in calculus.