Taylor expansion

简明释义

泰勒展开式

英英释义

例句

1.The Taylor expansion 泰勒展开 of sin(x) provides a useful way to compute its values near zero.

sin(x)的Taylor expansion 泰勒展开提供了一种在零附近计算其值的有效方法。

2.When analyzing the behavior of the function, the Taylor expansion 泰勒展开 gives us insights into its local properties.

在分析函数的行为时，Taylor expansion 泰勒展开为我们提供了对其局部特性的洞察。

3.The Taylor expansion 泰勒展开 allows us to simplify complex functions into polynomials.

通过Taylor expansion 泰勒展开，我们可以将复杂函数简化为多项式。

4.To solve the differential equation, we used the Taylor expansion 泰勒展开 of the exponential function.

为了求解微分方程，我们使用了指数函数的Taylor expansion 泰勒展开。

5.In calculus, the function can be approximated using a Taylor expansion 泰勒展开 around a point.

在微积分中，函数可以通过在某一点附近使用Taylor expansion 泰勒展开来近似。

作文

The concept of Taylor expansion is a fundamental idea in calculus and mathematical analysis. It allows us to approximate complex functions using polynomials, which are much easier to work with. The Taylor expansion of a function provides a way to express the function as an infinite sum of terms calculated from the values of its derivatives at a single point. This powerful tool not only simplifies calculations but also enhances our understanding of how functions behave near that point.To illustrate the significance of Taylor expansion, consider the function f(x) = e^x. This function is known for its rapid growth and complexity. However, by applying the Taylor expansion around x = 0, we can derive a polynomial that closely resembles the behavior of the exponential function near this point. The Taylor expansion for e^x is given by:f(x) = e^0 + (e^0)(x - 0) + (e^0)(x - 0)^2/2! + (e^0)(x - 0)^3/3! + ...This expands to:f(x) = 1 + x + x^2/2 + x^3/6 + ...As we can see, this infinite series converges to e^x for all real values of x. The beauty of the Taylor expansion lies in its ability to provide a polynomial approximation for functions that may otherwise be difficult to analyze or compute.Another important aspect of Taylor expansion is its application in physics and engineering. Many physical phenomena can be modeled using complex functions, and approximating these functions using Taylor expansion helps in simplifying equations and solving problems more effectively. For example, in mechanics, when analyzing the motion of an object under the influence of forces, we often encounter trigonometric functions like sin(x) and cos(x). By employing the Taylor expansion, we can convert these functions into polynomials, making it easier to calculate trajectories and predict outcomes.Moreover, the Taylor expansion has significant implications in numerical methods. In computational mathematics, algorithms frequently rely on polynomial approximations to estimate values of functions. By using the Taylor expansion, we can create efficient algorithms that yield accurate results without requiring extensive computational resources. This is particularly useful in fields such as computer graphics, where rendering complex shapes and animations often necessitates quick calculations.In conclusion, the Taylor expansion is an invaluable mathematical tool that bridges the gap between complex functions and simpler polynomial representations. Its applications span across various domains, including physics, engineering, and computer science. By understanding and utilizing the Taylor expansion, we can tackle challenging problems with greater ease and precision. As we continue to explore the depths of mathematics, the Taylor expansion remains a cornerstone that aids in our quest for knowledge and innovation.

泰勒展开（Taylor expansion）是微积分和数学分析中的一个基本概念。它允许我们使用多项式来近似复杂函数，这些多项式更易于处理。函数的Taylor展开提供了一种方法，可以将函数表示为从单个点的导数值计算得出的无限项的总和。这个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。为了说明Taylor展开的重要性，考虑函数f(x) = e^x。这个函数以其快速增长和复杂性而闻名。然而，通过在x = 0附近应用Taylor展开，我们可以推导出一个在该点附近与指数函数行为相似的多项式。e^x的Taylor展开为：f(x) = e^0 + (e^0)(x - 0) + (e^0)(x - 0)^2/2! + (e^0)(x - 0)^3/3! + ...这扩展为：f(x) = 1 + x + x^2/2 + x^3/6 + ...正如我们所看到的，这个无限级数在所有实数x的情况下收敛到e^x。Taylor展开的美在于它能够为可能难以分析或计算的函数提供多项式近似。Taylor展开的另一个重要方面是它在物理和工程中的应用。许多物理现象可以用复杂函数建模，而使用Taylor展开近似这些函数有助于简化方程并更有效地解决问题。例如，在力的影响下分析物体运动时，我们经常会遇到三角函数，如sin(x)和cos(x)。通过使用Taylor展开，我们可以将这些函数转换为多项式，从而更容易计算轨迹和预测结果。此外，Taylor展开在数值方法中具有重要意义。在计算数学中，算法通常依赖于多项式近似来估计函数的值。通过使用Taylor展开，我们可以创建高效的算法，在不需要大量计算资源的情况下产生准确的结果。这在计算机图形学等领域尤其有用，因为渲染复杂形状和动画通常需要快速计算。总之，Taylor展开是一个无价的数学工具，它架起了复杂函数与更简单多项式表示之间的桥梁。它的应用跨越多个领域，包括物理、工程和计算机科学。通过理解和利用Taylor展开，我们可以更轻松、更精确地解决挑战性问题。在我们继续探索数学深度的过程中，Taylor展开仍然是帮助我们追求知识和创新的重要基石。