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Calculus is a branch of mathematics, developed from algebra and geometry, built on two major complementary ideas.
The first idea, called differential calculus, is about a vast generalization of the slope of a line. It is a theory about rates of change, defining differentiation. It permits velocity, acceleration, and the slope of a curve at a given point all to be discussed on a common conceptual basis.
The second idea, called integral calculus, is about a vast generalization of area. It is a theory about accumulation of small, even infinitesimal, quantities, defining integration. Though originally motivated by area, it includes related concepts such as volume and even distance.
The two concepts differentiation and integration define inverse operations in a sense made precise by the fundamental theorem of calculus. Therefore, in teaching calculus either may in fact be given priority, but the usual educational approach (nowadays) is to introduce differential calculus first.
Often what determines whether calculus or simpler mathematics is required to solve any given problem is not what ultimately needs to be accomplished. Rather, it is whether the requisite formula is provided or not. For example, finding the circumference of a circle does not require calculus provided the following is given: However, if one has only a related formula such as for the area of a circle, then calculus must be used to derive the formula for the circumference. For students studying calculus, this formula is usually the final answer to the problem, and no further input is requested.
It must be realized, though, that calculus is not about formulas. The subject applies in many situations where the relevant functions and answers do not have formulas, which is the usual situation in real world applications. More precisely, any function whose graph is smooth enough to have tangent lines can be investigated with differential calculus. And any function whose graph has no breaks can be approached with the integral calculus.
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