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1、函数In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.极限In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches some value.The concept
2、 of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually denoted lim as in limn c(an) = L, and the fact of approaching a limit is represented by the right arro
3、w () as in an L.Suppose f is a real-valued function and c is a real number. The expression means that f(x) can be made to be as close to L as desired by making x sufficiently close to c.无穷小InfinitesimalIn common speech, an infinitesimal object is an object which is smaller than any feasible measurem
4、ent, but not zero in size; or, so small that it cannot be distinguished from zero by any available means.无穷大连续函数In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output.介值定理In mathematical analysis, the intermed
5、iate value theorem states that if a continuous function f with an interval a, b as its domain takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. This has two important specializations: If a continuous function
6、 has values of opposite sign inside an interval, then it has a root in that interval (Bolzanos theorem).1 And, the image of a continuous function over an interval is itself an interval.导数The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function or d
7、ependent variable) which is determined by another quantity (the independent variable).Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y. This relationship can be written as y = f(x). If f(x) is the equation for a s
8、traight line, then there are two real numbers m and b such that y = m x + b. m is called the slope and can be determined from the formula:， where the symbol (the uppercase form of the Greek letter Delta) is an abbreviation for change in. It follows that y = m x. A general function is not a line, so
9、it does not have a slope. The derivative of f at the point x is the slope of the linear approximation to f at the point x.微分罗尔定理In calculus, Rolles theorem essentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point so
10、mewhere between them; that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero.If a real-valued function f is continuous on a closed interval a,b, differentiable on the open interval (a,b), and f(a)= f(b), then there exists a c in the open inte
11、rval (a,b) such that . This version of Rolles theorem is used to prove the mean value theorem, of which Rolles theorem is indeed a special case. It is also the basis for the proof of Taylors theorem.拉格朗日中值定理Lagranges mean value theorem柯西中值定理Cauchys mean value theorem Cauchys mean value theorem, also
12、 known as the extended mean value theorem, is a generalization of the mean value theorem. It states: If functions f and g are both continuous on the closed interval a,b, and differentiable on the open interval (a, b), then there exists some c (a,b), such that ；Of course, if g(a) g(b) and if g(c) 0,
13、this is equivalent to: 。洛必达法则LHpitals ruleIn calculus, lHpitals rule (pronounced:lopital) uses derivatives to help evaluate limits involving indeterminate forms. Application of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit.In its simplest
14、form, lHpitals rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I: If orand exists, andfor all x in I with x c, thenThe differentiation of the numerator and denominator often simplifies the quotient and/or converts it to
15、a determinate form, allowing the limit to be evaluated more easily.泰勒公式Taylors theorem Statement of the theoremThe precise statement of the most basic version of Taylors theorem is as follows: Taylors theorem.Let k1 be an integer and let the function f: R R be k times differentiable at the point a R
16、. Then there exists a function hk: R R such that This is called the Peano form of the remainder. 不定积分AntiderivativeIn calculus, an antiderivative, primitive integral or indefinite integral1 of a function f is a differentiable function F whose derivative is equal to f, i.e., F = f. The process of solving for antiderivatives is called antidiff
