point of tangency

简明释义

切点

英英释义

例句

1.At the point of tangency, the forces acting on the object are balanced, allowing it to remain stationary.

在切点，作用于物体的力量是平衡的，使其保持静止。

2.The curve of the wheel meets the road at the point of tangency, where the tire makes contact with the surface.

轮子的曲线在与地面接触的地方形成了切点，这是轮胎与表面接触的地方。

3.The architect used the point of tangency to align the circular window with the rectangular wall.

建筑师使用切点将圆形窗户与矩形墙壁对齐。

4.During the design process, we identified the point of tangency between the two curves to ensure a smooth transition.

在设计过程中，我们确定了两条曲线之间的切点，以确保平滑过渡。

5.In calculus, the derivative of a function at a given point of tangency represents the slope of the tangent line.

在微积分中，一个函数在给定的切点处的导数代表切线的斜率。

作文

In the realm of mathematics and geometry, the term point of tangency refers to the specific point where a tangent line touches a curve without crossing it. This concept is fundamental in understanding the behavior of curves and their derivatives. For instance, when we analyze the graph of a function, the point of tangency indicates where the slope of the tangent line equals the slope of the curve at that very point. This relationship is not only crucial for calculus but also for various applications in physics and engineering.To comprehend the significance of the point of tangency, let us consider a practical example involving a circle and a line. Imagine a circle drawn on a coordinate plane. A line can be drawn such that it just touches the circle at one distinct point. This unique interaction between the line and the circle defines the point of tangency. At this point, if we were to draw a radius from the center of the circle to the point of tangency, we would notice that the radius is perpendicular to the tangent line. This geometric property highlights the importance of the point of tangency in determining the relationship between different shapes.Moreover, the point of tangency plays a pivotal role in optimization problems. In calculus, finding maximum or minimum values of functions often involves identifying critical points. When a function reaches its extremum, the derivative at that point of tangency equals zero, indicating a horizontal tangent line. This is particularly useful in fields such as economics, where businesses strive to maximize profit or minimize costs by analyzing these critical points of tangency.Additionally, the concept extends beyond pure mathematics into real-world applications. Engineers utilize the point of tangency when designing roads and highways. For instance, when a road curves, the design must ensure that vehicles can transition smoothly from a straight path to a curved one. The point of tangency here is crucial for maintaining safety and comfort for drivers, as it dictates how sharply a vehicle can turn without losing control.Furthermore, in the field of physics, the point of tangency is significant in understanding motion. Consider the trajectory of a projectile; the point of tangency on its path can reveal the instantaneous velocity at that moment. By analyzing these points, physicists can predict the future position of the projectile based on its current state.In conclusion, the point of tangency serves as a vital concept across various disciplines. Whether in mathematics, engineering, or physics, understanding this point enhances our ability to analyze and interpret complex systems. It allows us to draw connections between different mathematical principles and their practical applications. As we continue to explore the depths of these subjects, the point of tangency will undoubtedly remain an essential element in our quest for knowledge and understanding.

在数学和几何的领域中，术语切点指的是切线与曲线相接触而不相交的特定点。这个概念对于理解曲线及其导数的行为是基础性的。例如，当我们分析一个函数的图形时，切点表明切线的斜率在该点等于曲线的斜率。这种关系不仅对微积分至关重要，而且对物理和工程等各种应用也同样重要。为了理解切点的重要性，让我们考虑一个涉及圆和直线的实际例子。想象一下在坐标平面上画出的一个圆。可以画一条直线，使其仅在一个独特的点上触及圆。这条直线与圆之间的这种独特交互定义了切点。在这一点上，如果我们从圆心向切点画一条半径，我们会注意到这条半径与切线是垂直的。这一几何特性突显了切点在确定不同形状之间关系中的重要性。此外，切点在优化问题中起着关键作用。在微积分中，寻找函数的最大值或最小值通常涉及识别临界点。当一个函数达到其极值时，在该切点处的导数等于零，表明切线水平。这在经济学等领域尤其有用，因为企业通过分析这些关键的切点来努力最大化利润或最小化成本。此外，这个概念超越了纯数学，延伸到现实世界的应用中。工程师在设计道路和高速公路时利用切点。例如，当道路弯曲时，设计必须确保车辆能够平稳地从直线路径过渡到弯曲路径。在这里，切点对于保持驾驶者的安全和舒适至关重要，因为它决定了车辆可以多么急剧地转弯而不会失去控制。此外，在物理学领域，切点在理解运动中也是重要的。考虑一个抛射物的轨迹；其路径上的切点可以揭示该时刻的瞬时速度。通过分析这些点，物理学家可以根据当前状态预测抛射物的未来位置。总之，切点在各个学科中都是一个至关重要的概念。无论是在数学、工程还是物理学中，理解这一点增强了我们分析和解释复杂系统的能力。它使我们能够在不同的数学原理及其实际应用之间建立联系。在我们继续探索这些学科的深度时，切点无疑将继续成为我们追求知识和理解的重要元素。