Taylor series

简明释义

泰勒级数

英英释义

例句

1.By applying the Taylor series 泰勒级数, we can estimate the value of e^x for small values of x.

通过应用 Taylor series 泰勒级数，我们可以估计小值 x 的 e^x 的值。

2.The Taylor series 泰勒级数 expansion of sin(x) allows us to compute its values more easily.

sin(x) 的 Taylor series 泰勒级数 展开使我们更容易计算其值。

3.The Taylor series 泰勒级数 provides a way to express functions as infinite sums of terms.

Taylor series 泰勒级数 提供了一种将函数表示为无限项和的方法。

4.In calculus, the Taylor series 泰勒级数 is used to approximate functions near a specific point.

在微积分中，Taylor series 泰勒级数 用于在特定点附近近似函数。

5.Engineers often use the Taylor series 泰勒级数 to simplify complex calculations in control systems.

工程师们常常使用 Taylor series 泰勒级数 来简化控制系统中的复杂计算。

作文

The Taylor series is a powerful mathematical tool that allows us to approximate complex functions using polynomials. Named after the mathematician Brook Taylor, this series expands a function around a specific point, providing a way to express it as an infinite sum of terms calculated from the values of its derivatives at that point. Understanding the Taylor series is essential for students and professionals in fields such as engineering, physics, and computer science, where approximations of functions play a crucial role in problem-solving and analysis.To illustrate the concept, let’s consider the function f(x) = e^x, which is the exponential function. The Taylor series expansion of this function about the point x = 0 (also known as the Maclaurin series) can be expressed as:f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...Calculating the derivatives of e^x at x = 0, we find that f(0) = 1, f'(0) = 1, f''(0) = 1, and so on. Therefore, the Taylor series for e^x becomes:f(x) = 1 + x + x²/2! + x³/3! + ...This infinite series converges to e^x for all real values of x, demonstrating how the Taylor series can provide accurate approximations of functions.The significance of the Taylor series extends beyond mere approximation. It also plays a vital role in theoretical mathematics, particularly in calculus and analysis. The ability to represent functions as sums of their derivatives allows mathematicians to explore the properties of functions more deeply. For instance, the convergence of the Taylor series depends on the behavior of the function and the point around which it is expanded. In some cases, the series converges to the function for all values of x, while in others, it only converges within a certain interval.Moreover, the Taylor series has practical applications in numerical methods. Engineers often use this series to solve differential equations, optimize systems, and perform simulations. By approximating functions with polynomials, they can simplify complex calculations, making it easier to analyze and predict system behavior.In addition to its applications in engineering and physics, the Taylor series is also fundamental in computer science. Algorithms that require function evaluations, such as those used in graphics programming or machine learning, often rely on polynomial approximations derived from the Taylor series. This allows for faster computations and improved performance in various applications.In conclusion, the Taylor series is not just a theoretical concept; it is a practical tool that enhances our understanding of functions and their behaviors. Whether in academia or industry, mastering the Taylor series is crucial for anyone looking to delve into advanced mathematics or apply these concepts in real-world scenarios. As we continue to explore the complexities of mathematics, the Taylor series remains a cornerstone of mathematical analysis and application, proving its worth across multiple disciplines.

泰勒级数是一种强大的数学工具，允许我们使用多项式来近似复杂的函数。以数学家布鲁克·泰勒的名字命名，这个级数围绕特定点展开一个函数，提供了一种将其表示为从该点的导数值计算得到的无限和的方式。理解泰勒级数对工程、物理和计算机科学等领域的学生和专业人士至关重要，因为函数的近似在问题解决和分析中发挥着关键作用。为了说明这个概念，让我们考虑函数 f(x) = e^x，即指数函数。该函数在 x = 0 处的泰勒级数展开（也称为麦克劳林级数）可以表示为：f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...计算 e^x 在 x = 0 处的导数，我们发现 f(0) = 1，f'(0) = 1，f''(0) = 1，等等。因此，e^x 的泰勒级数变为：f(x) = 1 + x + x²/2! + x³/3! + ...这个无限级数在所有实数值 x 下收敛，展示了泰勒级数如何提供函数的准确近似。泰勒级数的重要性不仅限于简单的近似。它在理论数学中，尤其是在微积分和分析中，发挥着重要作用。将函数表示为导数的和的能力使数学家能够更深入地探索函数的性质。例如，泰勒级数的收敛性取决于函数的行为以及展开的点。在某些情况下，该级数对所有 x 值收敛，而在其他情况下，它仅在某个区间内收敛。此外，泰勒级数在数值方法中也具有实际应用。工程师经常使用该级数来解决微分方程、优化系统和进行模拟。通过使用多项式来近似函数，他们可以简化复杂的计算，从而更容易分析和预测系统行为。除了在工程和物理中的应用外，泰勒级数在计算机科学中也是基础的。需要函数评估的算法，例如图形编程或机器学习中使用的算法，通常依赖于从泰勒级数派生的多项式近似。这使得各种应用中的计算速度更快、性能更好。总之，泰勒级数不仅是一个理论概念；它是一种实用工具，增强了我们对函数及其行为的理解。无论是在学术界还是工业界，掌握泰勒级数对于任何希望深入研究高级数学或在现实世界场景中应用这些概念的人来说都是至关重要的。当我们继续探索数学的复杂性时，泰勒级数仍然是数学分析和应用的基石，在多个学科中证明了它的价值。