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Concepts of dragon Calculus was invented independently by two mathematicians, Issac due north of England and Gottfried Wilhelm Leibniz of Germany in the 1680s. Leibniz published his research in the journal ¡¥Acta Eruditorum¡¦ in 1684 and Newton ¡¥s treatise was published in ¡¥Principia Mathematica¡¦ in 1687.

Calculus was mainly developed to solve two types of problems ¡V the closing of topazs to deviates, fig.1, and the calculation of area, fig. 2, especially for irregular shapes. Both of these problems are closely related to the rate of change of continuous function and the total concept of calculus (also known as the ¡¥limit¡¦). The avocation shows what limit is and what are its applications.

Fig.1 Given a function f(x) and a point P(x,y) on its graph, consider an equation of the line of merchandise topaz to the graph at P.

Fig.2 Given a function f(x), find the area between the graph of f(x) and an interval [a,b] on the axis.

In plane geometry, a line is called a ¡§tangent¡¨ to the peck if it meets the circle at precisely one point, fig.3a. However, this definition is not correct for all kinds of scents and irregular shapes. In fig.3b, the line meets the dilute exactly once, yet it is not a tangent. Meanwhile in fig.3c, the line is a tangent, yet it meets the wander more than once.

Fig.3 For tangent that applies to curves other than circles as in fig.4a, point P on curve in the xy-plane and Q is any point on the curve different from P, the line through P and Q is the ¡¥secant¡¦ line (the secant line is a line that intercepts a curve at at least two points). If Q moves along the curve toward P, fig.4b., the secant line will rotate toward a ¡¥ pass¡¦ position. The line occupying this limiting position is considered...

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