antiderivative

简明释义

英[ˌæntɪdɪˈrɪvətɪv]美[ˌæntidɪˈrɪvətɪv;ˌæntaɪˈrɪvət

n. 不定积分；反式衍生物

英英释义

单词用法

同义词

反义词

例句

1.The indefinite integral is an antiderivative.

不定积分是反导数。

2.You're not really asking whether it has an antiderivative, but whether you know the name of that function.

所以这道题真正问的并不是“此函数是否有不定积分”，而是“你是否知道此函数的名称”。

3.You're not really asking whether it has an antiderivative, but whether you know the name of that function.

所以这道题真正问的并不是“此函数是否有不定积分”，而是“你是否知道此函数的名称”。

4.The antiderivative 不定积分 of a constant function is a linear function.

常数函数的antiderivative 不定积分是一个线性函数。

5.To find the area under the curve, we need to calculate the antiderivative 不定积分 of the function.

要找到曲线下的面积，我们需要计算该函数的antiderivative 不定积分。

6.Finding the antiderivative 不定积分 can sometimes be challenging without proper techniques.

如果没有适当的技巧，寻找antiderivative 不定积分有时会很困难。

7.In calculus, we often use the antiderivative 不定积分 to solve problems involving motion.

在微积分中，我们经常使用antiderivative 不定积分来解决涉及运动的问题。

8.The Fundamental Theorem of Calculus connects differentiation and antiderivative 不定积分 in a profound way.

微积分基本定理以深刻的方式将微分和antiderivative 不定积分联系起来。

作文

In the field of calculus, the concept of an antiderivative is fundamental to understanding how functions behave. An antiderivative of a function is essentially the reverse process of differentiation. While differentiation involves finding the rate at which a function changes, taking an antiderivative seeks to uncover the original function from its derivative. This relationship is crucial in many applications of mathematics, especially in physics and engineering, where we often need to determine the original position or velocity from the acceleration of an object.To grasp the idea of an antiderivative, consider a simple example. If we have a function f(x) = 2x, its derivative, according to the rules of calculus, is f'(x) = 2. Conversely, if we are given the derivative f'(x) = 2, we can find the antiderivative by asking what function would result in this derivative. The answer is f(x) = x^2 + C, where C is a constant. This constant is essential because the process of differentiation loses information about constant values; hence, there are infinitely many antiderivatives for a single function, all differing by a constant.The notation for an antiderivative is typically represented using the integral sign. For instance, we write ∫f'(x)dx = f(x) + C. This notation not only denotes the process of finding an antiderivative but also emphasizes the connection between integration and differentiation. In fact, the Fundamental Theorem of Calculus states that differentiation and integration are inverse processes. This theorem provides a powerful tool for evaluating definite integrals, allowing us to compute the area under a curve using antiderivatives.Understanding antiderivatives opens up various avenues in calculus, including solving differential equations and analyzing motion in physics. For example, when studying the motion of an object, we often start with its acceleration function. By finding the antiderivative of the acceleration, we can determine the velocity function. Further, by taking the antiderivative of the velocity, we arrive at the position function. Thus, the chain of relationships among these functions illustrates the practical importance of antiderivatives in real-world scenarios.Moreover, the concept of antiderivatives extends beyond mere calculations. It encourages critical thinking and problem-solving skills. When faced with a mathematical challenge, recognizing the need to find an antiderivative can lead to deeper insights into the behavior of functions and their applications. Additionally, it allows students and professionals alike to appreciate the beauty and interconnectedness of mathematical concepts.In conclusion, the antiderivative is a vital concept in calculus that serves as a bridge between differentiation and integration. It is not just a theoretical construct but a practical tool that aids in solving real-world problems. By mastering the idea of antiderivatives, one gains a deeper understanding of the mechanics of change and the relationships between different mathematical functions. As we continue to explore the vast realm of mathematics, the significance of antiderivatives will undoubtedly remain a cornerstone of our analytical toolkit.

在微积分领域，反导数的概念对于理解函数的行为至关重要。一个函数的反导数本质上是求导的逆过程。求导涉及找到一个函数变化的速率，而求反导数则试图从其导数中揭示原始函数。这种关系在数学的许多应用中至关重要，尤其是在物理和工程中，我们常常需要从物体的加速度中确定原始位置或速度。为了理解反导数的概念，考虑一个简单的例子。如果我们有一个函数f(x) = 2x，根据微积分的规则，它的导数是f'(x) = 2。相反，如果我们给定导数f'(x) = 2，我们可以通过询问哪个函数会产生这个导数来找到反导数。答案是f(x) = x^2 + C，其中C是一个常数。这个常数是必不可少的，因为求导过程会丢失关于常数值的信息；因此，对于一个单一的函数，存在无限多个反导数，它们之间只相差一个常数。反导数的记号通常使用积分符号表示。例如，我们写作∫f'(x)dx = f(x) + C。这种记号不仅表示寻找反导数的过程，还强调了积分与求导之间的联系。实际上，微积分基本定理指出，求导和积分是逆过程。这个定理为评估定积分提供了强有力的工具，使我们能够使用反导数计算曲线下的面积。理解反导数为微积分打开了各种途径，包括解决微分方程和分析物理中的运动。例如，在研究物体的运动时，我们通常从其加速度函数开始。通过找到加速度的反导数，我们可以确定速度函数。进一步地，通过对速度进行反导数运算，我们得到了位置函数。因此，这些函数之间的关系链展示了反导数在现实场景中的实际重要性。此外，反导数的概念超越了单纯的计算。它鼓励批判性思维和解决问题的能力。当面临数学挑战时，认识到需要找到反导数可以引导人们深入理解函数的行为及其应用。此外，它使学生和专业人士都能欣赏数学概念的美丽和相互联系。总之，反导数是微积分中的一个重要概念，作为求导与积分之间的桥梁。它不仅是一个理论构造，更是一个实用工具，帮助解决现实世界的问题。通过掌握反导数的概念，人们可以更深刻地理解变化的机制以及不同数学函数之间的关系。随着我们继续探索广阔的数学领域，反导数的重要性无疑将始终是我们分析工具箱中的基石。