differentiable

简明释义

英[ˌdɪfəˈrenʃɪəb(ə)l]美[ˌdɪfəˈrenʃəbəl;ˌdɪfəˈrenʃiəb

adj. [数] 可微的；可辨的；可区分的

英英释义

单词用法

同义词

反义词

例句

1.I need to have a vector field that is defined and differentiable — — everywhere in d, so same instructions as usual.

我需要一个确定的向量场，而且它在，D，上是处处可微的，然后和平时一样的做法。

2.Not all continuous functions are differentiable.

函数是连续可微分的。

3.Minimax problem is a sort of non-differentiable optimization problem and the entropy function method provides a efficient approach to solve such kind of problems.

极大极小问题是一类不可微优化问题，熵函数法是求解这类问题的一种有效算法。

4.Because the error transfer function of rough neural network is not differentiable, genetic algorithms are applied for training the network.

由于粗神经网络的误差传递函数不可微，所以采用遗传算法来训练粗神经网络。

5.In this paper, the star-kernel of a quasi-differentiable function in one-dimensional space is studied.

本文在一维情形下对拟可微函数的星核进行了讨论，证明了拟微分星-有界等价子类的存在性；

6.It is also proved as application that any bounded variation function can be seen as a differentiable function beside a set of arbitrary small measure.

作为应用，证明了任意有界变差函数都与某可微函数在除过测度任意小的集合外重合。

7.This article was to offer the method about the complex function's differentiable and holomorphic.

文章针对被积函数是连续函数、可导函数的定积分不等式提出了几种有效的证明方法。

8.The linearized procedure for differentiable nonlinear programming problems can be naturally generalized to the quasi differential case.

可微非线性规划问题的线性化过程可以自然地推广到拟可微的情形。

9.The function f(x) is differentiable at x = 2, meaning it has a defined slope at that point.

函数 f(x) 在 x = 2 时是可微分的，这意味着在该点有一个定义的斜率。

10.To find the maximum value of the curve, we need to determine where the derivative is differentiable.

要找出曲线的最大值，我们需要确定导数在哪里是可微分的。

11.A continuous function is not always differentiable at every point.

一个连续函数并不总是在每一点上都是可微分的。

12.In calculus, we study functions that are differentiable to understand their behavior.

在微积分中，我们研究可微分的函数以理解它们的行为。

13.The theorem states that if a function is differentiable, it is also continuous.

定理表明，如果一个函数是可微分的，那么它也是连续的。

作文

In the field of mathematics, particularly in calculus, the concept of being differentiable is crucial for understanding the behavior of functions. A function is said to be differentiable at a certain point if it has a defined derivative at that point. This means that the function must be smooth and continuous around that point, allowing us to draw a tangent line that accurately represents the function's behavior in that vicinity. If a function is not differentiable, it may have sharp corners, discontinuities, or vertical tangents, which complicate our ability to analyze it effectively. To illustrate this, let’s consider the function f(x) = |x|. This function is continuous everywhere; however, at x = 0, it has a sharp corner. Therefore, while we can evaluate the function at x = 0, the derivative does not exist there, meaning that f(x) is not differentiable at that point. This example highlights the importance of smoothness in determining whether a function is differentiable. The implications of a function being differentiable extend beyond mere mathematical curiosity; they are foundational in various applications across physics, engineering, and economics. For instance, in physics, the concept of velocity is derived from the position function through differentiation. If the position function is differentiable, we can smoothly transition to discussing the motion of an object. Conversely, if the position function has points where it is not differentiable, we encounter difficulties in predicting the object's velocity. Furthermore, in optimization problems, finding maximum or minimum values often requires us to take the derivative of a function. If we want to maximize profit in a business scenario, we need to ensure that the profit function is differentiable so that we can apply techniques such as setting the derivative to zero to find critical points. If the profit function is not differentiable, we might miss out on optimal solutions due to the presence of abrupt changes in the function’s behavior. In addition to practical applications, the study of differentiable functions also leads to deeper insights in theoretical mathematics. For example, the Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point where the derivative equals the average rate of change over that interval. This theorem not only reinforces the connection between continuity and differentiability but also provides a powerful tool for proving other mathematical concepts. In conclusion, the term differentiable encapsulates a fundamental property of functions that plays a pivotal role in both theoretical and applied mathematics. Understanding whether a function is differentiable allows mathematicians and scientists to analyze and predict behaviors effectively. As we delve deeper into calculus and its applications, the significance of differentiable functions becomes increasingly apparent, shaping our approach to solving real-world problems and advancing our mathematical knowledge.

在数学领域，特别是微积分中，differentiable的概念对于理解函数的行为至关重要。如果一个函数在某一点上是differentiable的，这意味着它在该点具有定义的导数。这意味着该函数在该点附近必须是光滑且连续的，从而允许我们绘制一条切线，准确地表示该函数在该区域的行为。如果一个函数不是differentiable的，它可能具有尖角、不连续性或垂直切线，这使得我们有效分析它的能力变得复杂。 为了说明这一点，我们来考虑函数f(x) = |x|。这个函数在任何地方都是连续的；然而，在x = 0时，它有一个尖角。因此，虽然我们可以在x = 0处评估该函数，但导数在这里并不存在，这意味着f(x)在该点不是differentiable的。这个例子突显了光滑性在决定一个函数是否differentiable方面的重要性。 函数成为differentiable的影响超出了单纯的数学好奇心；它们是物理学、工程学和经济学等多个应用领域的基础。例如，在物理学中，速度的概念是通过对位置函数进行微分得到的。如果位置函数是differentiable的，我们可以顺利地过渡到讨论物体的运动。相反，如果位置函数在某些点上不是differentiable的，我们在预测物体的速度时会遇到困难。 此外，在优化问题中，寻找最大或最小值通常需要我们对函数进行导数运算。如果我们想在商业场景中最大化利润，我们需要确保利润函数是differentiable的，以便我们可以应用诸如将导数设为零以找到临界点等技术。如果利润函数不是differentiable的，我们可能会因为函数行为的突然变化而错过最佳解决方案。 除了实际应用之外，研究differentiable函数还带来了更深刻的理论数学见解。例如，均值定理指出，如果一个函数在闭区间上连续，并且在开区间上是differentiable的，那么至少存在一个点，其导数等于该区间上的平均变化率。这个定理不仅强化了连续性与differentiable之间的联系，而且提供了证明其他数学概念的强大工具。 总之，术语differentiable概括了函数的一个基本属性，在理论和应用数学中都发挥着关键作用。理解一个函数是否是differentiable的，使得数学家和科学家能够有效地分析和预测行为。当我们深入研究微积分及其应用时，differentiable函数的重要性愈发明显，塑造了我们解决现实问题和推进数学知识的方式。