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AP微积分词汇(上篇)

absolutely convergent 绝对收敛

A series with mixed signs is said to converge absolutely if the series obtained by taking the absolute values of its terms converges.

alternating series 交错级数

A series of mixed signs, which has the form:

acceleration 加速度

The rate of change of velocity of an object with respect to time.

alternating series test

交错级数判断法

The alternating series converges if:for all n and the limit of an is 0 as napproaches infinity.

antiderivative 积分

An antiderivative of a function f is a differentiable function F whose derivative is equal to the original function f.

antidifferentiation 积分

The opposite operation of differentiation.

asymptote 渐近线

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

average rate of change 平均变化率

The average value of the rate of change of a function, which evaluates how fast the function is changing on average.

coefficient 系数 The number just before a variable.

composite function 复合函数

A function which is produced by the pointwise application of one function to the result of another function.

concave down 凸的

The line segment between any two points on the graph of the function lies below or on the graph.

concave up 凹的

The line segment between any two points on the graph of the function lies above or on the graph.

conditionally convergent 条件收敛

A series that converges but not absolutely is said to converge conditionally.

continuous 连续的

A function is continuous at a point c, if the value of the function at point c equals to the limit of the function as x approaches c.

converge 收敛

A series converges to a finite number L, if the limit of the sum of the first n terms as n approaches infinity is a finite number L.

convergent 收敛的

A series is convergent to a finite number L, if the limit of the sum of the first n terms as n approaches infinity is a finite number L.

cosine 余弦

A trigonometric function of an angle, defined as the ratio of the length of the adjacent side to the length of the longest side of the triangle.

definite integral 定积分

Given a function f of real variable x, the integral of f over the interval [a,b] has two real numbers: b as the upper limit and a as the lower limit, which is defined as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x=a and x=b.

denominator 分母

Divisor, the number below the fraction bar.

derivative 导数

A function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument.

differential equation 微分方程

A mathematical equation that relates some function with its derivatives.

discontinuity 间断点

A point in the domain where the function is not continuous.

displacement 位移

The shortest distance from the initial to the final position.

distance 距离 A numerical measurement of how far apart objects are.

diverge 发散

A series diverges if the limit of the sum of the first n terms as n approaches infinity is infinity

divergent 发散的

A series is divergent if the limit of the sum of the first n terms as n approaches infinity is infinity

error bound 误差上限

The largest possible value of the error when doing an approximation

Euler"s method 欧拉方法

A first-order numerical procedure for solving ordinary differential equations with a given initial value.

exponential function 指数函数

A mathematical operation written as

exponential growth 指数增长

If a quantity y increases (or decreases) at a rate that is proportional to its present amount at any time, then we can have a differential equation to solve these problems which are called exponential growth.

first term 首项

The first term of a series.

general solution 通解

The family of functions that satisfy a certain differential equation.

general term 同项

The nth term of the series is called the general term.

geometric series 等比级数

A geometric series has a first term a and common ratio of terms r.

global(absolute) maximum 最大值

The maxima (the respective plurals of maximum) of a function is the largest value of the function on the entire domain.

global(absolute) minimum 最小值

The minima (the respective plurals of minimum) of a function is the smallest value of the function on the entire domain.

harmonic series 调和级数

The p-series with p=1 is called the harmonic series.

horizontal asymptote 水平渐近线

The line y=b is a horizontal asymptote of the graph y=f(x) if the limit of f(x) as x approaches positive or negative infinity is b.

improper integral 瑕积分

Two classes of improper integrals: (1) those in which at least one of the limits of integration is infinite.(the interval is not bounded) (2) the integrand function f(x) has a point of discontinuity in the interval.(the function is not bounded)

indefinite integral 不定积分

An integration without the upper and lower limit, which is same as the antidifferentiation.

initial condition 初始条件

A given value of the function at a certain point which can be used to determine the constant when solving the differential equations

instantaneous rate of change 瞬时变化率

The instantaneous value of the rate of change of a function, which evaluates how fast the function is changing at a certain point.

integral test 积分法

If f(x) is a continuous, positive, decreasing function and f(n)=an. then the series converges if and only if the improper integral of f(x) from 1 to infinity converges.

integration by parts 分部积分法

One way to solve the antiderivative by the following formula:

Intermediate Value Theorem 介值定理

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.

interval of convergence 收敛区间

The set of all values of x for which a power series converges is called its interval of convergence.

inverse function 反函数

A function that reverses another function.

integral 反函数

The numerical measure of the area bounded above by the graph of a given function, below by the x-axis, and on the sides by ordinates drawn at the endpoints of a specified interval.

jump discontinuity 跳跃间断点

A discontinuity which is the result of a nonexistent limit at this point, because the left side limit does not equal to the right side limit at this point, or one-sided limit does not exist.

Lagrange error bound 拉格朗日误差上限

An upper bound for the error , the largest possible value of R.

left hand limit 左极限

As x approaches c through values less than c, if the value of the function approaches L, then the left-hand limit of the function at c is L.

left Riemann sum 左矩阵黎曼和

Approximating the function by its value at the left-end point as the height of the rectangle, and the length of the subintervals as the width of the rectangle to find the area.

L "Hospital"s Rule 洛必达法则

For functions f and g which are differentiable on an open interval I, if the limits of functions f and g as x approaches a certain value c are both zero or infinity, and g’(x) ≠0, then the limit of f(x)/g(x) as x approaches c equals to the limit of f"(x)/g"(x) as x approaches c.

limit 极限

The value that a function approaches as the independent variable approaches some value.

limit comparison test 极限比较法

If the limit as n approaches infinity is finite and nonzero, then the two series both converge or both diverge.

line tangent (tangent) 切线

The straight line that just touches the curve at that point.

local linear approximation 切线线性近似

The linear approximation of a function near a point is the tangential line at this point.

local(relative) maximum 极大值

The maxima (the respective plurals of maximum) of a function is the largest value of the function within a given range.

local(relative) minimum 极小值

The minima (the respective plurals of minimum) of a function is the smallest value of the function within a given range.

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