taylor's series

简明释义

泰勒级数

英英释义

例句

1.In calculus, we often use Taylor's series to approximate functions near a specific point.

在微积分中，我们经常使用泰勒级数来近似特定点附近的函数。

2.To solve this differential equation, we can apply Taylor's series expansion.

要解决这个微分方程，我们可以应用泰勒级数展开。

3.The function can be expressed as a sum of its Taylor's series around zero.

该函数可以表示为其关于零的泰勒级数的和。

4.The approximation error decreases as we include more terms in the Taylor's series.

随着我们在泰勒级数中包含更多项，近似误差会减小。

5.When calculating limits, Taylor's series can simplify complex expressions.

在计算极限时，泰勒级数可以简化复杂的表达式。

作文

The concept of Taylor's series is fundamental in the field of mathematics, particularly in calculus. It provides a way to represent complex functions as infinite sums of their derivatives at a single point. This powerful tool not only simplifies calculations but also enhances our understanding of how functions behave near that point. The Taylor's series, named after the mathematician Brook Taylor, allows us to approximate functions that may be difficult to compute directly. To illustrate the significance of Taylor's series, consider the exponential function e^x. While it might seem daunting to calculate e raised to any power, Taylor's series offers a straightforward solution. The series expansion for e^x around x=0 is given by: e^x = 1 + x + (x^2)/2! + (x^3)/3! + ... + (x^n)/n! + ...This series converges for all real values of x, making it an incredibly useful representation. By truncating the series after a few terms, we can obtain a polynomial that closely approximates e^x, which is especially beneficial in numerical methods and engineering applications.Another important application of Taylor's series is in physics, where it helps describe motion and forces. For instance, when analyzing the motion of objects under the influence of gravity, we often encounter nonlinear equations. By applying Taylor's series, we can linearize these equations around a point of interest, simplifying the problem significantly. This technique allows physicists to predict the behavior of systems with greater ease and accuracy.Moreover, Taylor's series plays a crucial role in computer science, particularly in algorithms that require function evaluation. Many numerical methods rely on polynomial approximations derived from Taylor's series to compute values efficiently. For example, in graphics programming, rendering curves and surfaces often involves approximating trigonometric and exponential functions using their Taylor's series expansions.However, it is essential to understand the limitations of Taylor's series. The series converges only within a certain radius of convergence, which means that outside this interval, the approximation may not hold true. Additionally, some functions, like those with discontinuities or sharp turns, may not be well-represented by their Taylor's series. Therefore, while this mathematical tool is powerful, it requires careful application and understanding of its constraints.In conclusion, the importance of Taylor's series in mathematics, physics, and computer science cannot be overstated. It provides a means to approximate complex functions, making them more manageable for analysis and computation. As we delve deeper into the realms of science and technology, the relevance of Taylor's series continues to grow, proving itself to be an indispensable part of mathematical education and practical application. Understanding Taylor's series not only equips students with valuable problem-solving skills but also opens doors to advanced topics in calculus and beyond.

泰勒级数的概念在数学领域，尤其是微积分中至关重要。它提供了一种将复杂函数表示为其在某一点的导数的无穷和的方法。这一强大的工具不仅简化了计算，还增强了我们对函数在该点附近行为的理解。泰勒级数以数学家布鲁克·泰勒的名字命名，允许我们近似那些可能难以直接计算的函数。为了说明泰勒级数的重要性，考虑指数函数e^x。虽然计算e的任意幂似乎令人生畏，但泰勒级数提供了一个简单的解决方案。e^x在x=0附近的级数展开式为：e^x = 1 + x + (x^2)/2! + (x^3)/3! + ... + (x^n)/n! + ...这个级数在所有实数值的x上都是收敛的，使其成为一种非常有用的表示方法。通过在几项之后截断级数，我们可以得到一个多项式，该多项式与e^x密切相近，这在数值方法和工程应用中特别有益。泰勒级数的另一个重要应用是在物理学中，它帮助描述运动和力。例如，在分析物体在重力影响下的运动时，我们经常会遇到非线性方程。通过应用泰勒级数，我们可以围绕感兴趣的点线性化这些方程，从而显著简化问题。这一技术使物理学家能够更轻松、更准确地预测系统的行为。此外，泰勒级数在计算机科学中也发挥着至关重要的作用，特别是在需要函数评估的算法中。许多数值方法依赖于从泰勒级数派生的多项式近似，以高效地计算值。例如，在图形编程中，渲染曲线和表面通常涉及使用泰勒级数展开来近似三角函数和指数函数。然而，理解泰勒级数的局限性也很重要。该级数仅在一定的收敛半径内收敛，这意味着在此区间之外，近似可能不成立。此外，一些具有不连续性或急剧转弯的函数可能无法通过其泰勒级数很好地表示。因此，尽管这一数学工具十分强大，但它需要谨慎应用并理解其限制。总之，泰勒级数在数学、物理学和计算机科学中的重要性不容小觑。它提供了一种近似复杂函数的方法，使其更易于分析和计算。随着我们深入科学和技术的领域，泰勒级数的相关性不断增长，证明了它作为数学教育和实际应用不可或缺的一部分。理解泰勒级数不仅使学生具备宝贵的问题解决能力，还为高级微积分及其他领域打开了大门。