微积分Calculus公式：微积分常用公式及概念总结，重点都在这里了！

微积分是大学数学最最基础的部分。微积分对大学物理，经济，和工程类等课程是最基础的计算工具。一般来说，除个别专业外，大一的微积分都是必修课。这块硬骨头必需啃下，别担心，微积分常用公式及概念总结，重点都在这里了！

微积分calculus概念中英对照

antiderivative: A function F(x) is called an antiderivative of a function f(x) if F'(x) =; f(x) for all x in the domain of f. In words, this means that an antiderivative of f is a function that has f for its derivative.

反导数：如果F’（x）=，则函数F（x）称为函数f（x）的反导数； f域中所有x的f（x）。换句话说，这意味着f的反导数是具有f的导数的函数。

chain rule: The chain rule tells how to find the derivative of composite functions. In words, the chain rule says the derivative of a composite function is the derivative of the outside function, done to the inside function, times the derivative of the inside function.

链规则：链规则告诉如何找到复合函数的导数。换句话说，链式规则说复合函数的导数是外部函数的导数，对内部函数而言，乘以内部函数的导数。

change of variables: A term sometimes used for the technique of integration by substitution.

变量的更改：有时用于通过替换进行集成的技术的术语。

concave downward: A function is concave downward on an interval if f”(x) is negative for every point on that interval.

向下凹入：如果对于该间隔上的每个点f“（x）为负，则函数在该间隔上向下凹入。

concave upward: A function is concave upward on an interval if f”(x) is positive for every point on that interval.

向上凹入：如果f“（x）对于该间隔上的每个点均为正，则函数在该间隔上凹入向上。

critical point: A critical point of a function is a point (x, f(x)) with x in the domain of the function and either f'(x) =; 0 or f'(x) undefined. Critical points are among the candidates to be maximum or minimum values of a function.

临界点：函数的临界点是一个点（x，f（x）），其中x在函数的范围内，且f’（x）=； 0或f’（x）未定义。关键点是函数的最大值或最小值的候选项。

cylindrical shell method: A procedure for finding the volume of a solid of revolution by treating it as a collection of nested thin rings.

圆柱壳法：通过将旋转实体视为嵌套的细环的集合来查找旋转体的体积的过程。

differentiable: A function is said to be differentiable at a point when the function’s derivative exists at that point. A function will fail to be differentiable at places where the function is not continuous or where the function has corners.

可微的：当函数的导数存在于某个点时，该函数被认为是可微的。在该功能不连续或该功能具有拐角的地方，该功能将无法区分。

disk method: A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with circular cross sections.

圆盘法：通过将旋转体视为具有圆形横截面的薄片的集合来查找旋转体的体积的过程。

Extreme Value Theorem: A theorem stating that a function which is continuous on a closed interval [a, b] must have a maximum and a minimum value on [a, b].

极值定理：一个定理指出，在闭合区间[a，b]上连续的函数必须在[a，b]上具有最大值和最小值。

First Derivative Test for Local Extrema: A method used to determine whether a critical point of a function is a local maximum or local minimum. If a continuous function changes from increasing (first derivative positive) to decreasing (first derivative negative) at a point, then that point is a local maximum. If a function changes from decreasing (first derivative negative) to increasing (first derivative positive) at a point, then that point is a local minimum.

局部极值的一阶导数测试：一种用于确定函数的临界点是局部最大值还是局部最小值的方法。如果某个点上的连续函数从增加（一阶导数为正）变为减少（一阶导数为负），则该点是局部最大值。如果某个点的函数从递减（一阶导数为负）变为递增（一阶导数为正），则该点是局部最小值。

general antiderivative: If F(x) is an antiderivative of a function f(x), then F(x) + C is called the general antiderivative of f(x).

通用反导数：如果F（x）是函数f（x）的反导数，则F（x）+ C被称为f（x）的通用反导数。

general form: The general form (sometimes also called standard form) for the equation of a line is ax + by =; c, where a and b are not both zero.

一般形式：直线方程的一般形式（有时也称为标准形式）为ax + by =; c，其中a和b都不都是零。

higher order derivatives: The second derivative, third derivative, and so forth for some function.

高阶导数：某些功能的二阶导数，三阶导数，依此类推。

implicit differentiation: A procedure for finding the derivative of a function which has not been given explicitly in the form “f(x) =;”.

隐式微分：一种隐式求函数的过程，该函数没有以“ f（x）=;”的形式明确给出。

indefinite integral: The indefinite integral of f(x) is another term for the general antiderivative of f(x).

不定积分：f（x）的不定积分是f（x）的一般反导数的另一项。

instantaneous rate of change: One way of interpreting the derivative of a function is to understand it as the instantaneous rate of change of that function, the limit of the average rates of change between a fixed point and other points on the curve that get closer and closer to the fixed point.

瞬时变化率：解释函数导数的一种方法是将其理解为该函数的瞬时变化率，即固定点与曲线上其他点之间的平均变化率的极限，该点越来越近，更接近固定点。

instantaneous velocity: One way of interpreting the derivative of a function s(t) is to understand it as the velocity at a given moment t of an object whose position is given by the function s(t).

瞬时速度：解释函数s（t）的导数的一种方法是将其理解为位置由函数s（t）给出的对象在给定时刻t的速度。

integration by parts: One of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms.

零件积分：最常用的积分技术之一，用于将复杂的积分减少为基本积分形式之一。

intercept form: The intercept form for the equation of a line is x/a + y/b =; 1, where the line has its x-intercept (the place where the line crosses the x-axis) at the point (a,0) and its y-intercept (the place where the line crosses the y-axis) at the point (0,b).

截距形式：直线方程的截距形式为x / a + y / b =; 1，其中线在点（a，0）处具有x轴截距（线与x轴相交的位置），在点处具有y轴截距（线与y轴相交的位置）（0，b）。

limit: A function f(x) has the value L for its limit as x approaches c if as the value of x gets closer and closer to c, the value of f(x) gets closer and closer to L.

极限：当x接近c时，函数f（x）的极限值为L，如果x的值越来越接近c，f（x）的值越来越接近L。

normal line: The normal line to a curve at a point is the line perpendicular to the tangent line at that point.

法线：点曲线的法线是垂直于该点切线的线。

point of inflection: A point is called a point of inflection of a function if the function changes from concave upward to concave downward, or vice versa, at that point.

拐点：如果函数在该点从凹向上变化到凹向下（反之亦然），则该点称为函数的拐点。

point-slope form: The point-slope form for the equation of a line is y – y1 =; m(x – x1), where m stands for the slope of the line and (x1,y1) is a point on the line.

点斜率形式：直线方程的点斜率形式为y – y1 =; m（x – x1），其中m代表直线的斜率，而（x1，y1）是直线上的一个点。

Riemann sum: A Riemann sum is a sum of several terms, each of the form f(xi)Δx, each representing the area below a function f(x) on some interval if f(x) is positive or the negative of that area if f(x) is negative. The definite integral is mathematically defined to be the limit of such a Riemann sum as the number of terms approaches infinity.

黎曼和：黎曼和是几项的和，形式为f（xi）Δx，如果f（x）为正或负，则每个代表代表函数f（x）下方某个区域的面积如果f（x）为负。当项数趋于无穷大时，定积分在数学上定义为这种黎曼和的极限。

Second Derivative Test for Local Extrema: A method used to determine whether a critical point of a function is a local maximum or local minimum. If f'(x) =; 0 and the second derivative is positive at this point, then the point is a local minimum. If f'(x) =; 0 and the second derivative is negative at this point, then the point is a local maximum.

局部极值的第二阶导数测试：一种用于确定函数的临界点是局部最大值还是局部最小值的方法。如果f’（x）=; 0，此时二阶导数为正，则该点为局部最小值。如果f’（x）=; 0，此时二阶导数为负，则该点为局部最大值。

slope of the tangent line: One way of interpreting the derivative of a function is to understand it as the slope of a line tangent to the function.

切线的斜率：解释函数导数的一种方法是将其理解为与函数相切的线的斜率。

slope-intercept form: The slope-intercept form for the equation of a line is y =; mx + b, where m stands for the slope of the line and the line has its y-intercept (the place where the line crosses the y-axis) at the point (0,b).

斜率截距形式：直线方程的斜率截距形式为y =; mx + b，其中m代表直线的斜率，并且直线在点（0，b）处具有y轴截距（直线与y轴交叉的位置）。

standard form: The standard form (sometimes also called general form) for the equation of a line is ax + by =; c, where a and b are not both zero.

标准形式：直线方程的标准形式（有时也称为通用形式）为ax + by =; c，其中a和b都不都是零。

substitution: Integration by substitution is one of the most common techniques of integration, used to reduce complicated integrals into one of the basic integration forms.

替代：替代积分是最常用的积分技术之一，用于将复杂的积分减少为基本积分形式之一。

tangent line: The tangent line to a function is a straight line that just touches the function at a particular point and has the same slope as the function at that point.

切线：函数的切线是一条直线，它仅在特定点处与函数接触，并且在该点处具有与函数相同的斜率。

trigonometric substitution: A technique of integration where a substitution involving a trigonometric function is used to integrate a function involving a radical.

三角替换：一种积分技术，其中使用涉及三角函数的替换来积分涉及基团的函数。

washer method: A procedure for finding the volume of a solid of revolution by treating it as a collection of thin slices with cross sections shaped like washers.

洗衣机法：一种将旋转体视为一组横截面像垫圈的薄片的集合，以查找旋转体的体积的过程。