explicit function

简明释义

显函数,显示函数

英英释义

例句

1.An explicit function can be easily graphed on a coordinate plane.

显式函数可以很容易地在坐标平面上绘制。

2.Using an explicit function helps avoid ambiguity in calculations.

使用显式函数有助于避免计算中的模糊性。

3.The explicit function in the mathematical model allows for direct calculation of results.

数学模型中的显式函数允许直接计算结果。

4.When you create an explicit function, you directly specify how the output is derived.

当你创建一个显式函数时，你直接指定输出是如何推导的。

5.In programming, an explicit function is defined with a specific output for given inputs.

在编程中，显式函数是针对给定输入定义了特定输出的函数。

作文

In mathematics, the concept of an explicit function is fundamental to understanding relationships between variables. An explicit function is defined as a function where the dependent variable is expressed directly in terms of the independent variable(s). This means that for every input value, there is a clear output value that can be calculated without any ambiguity. For example, in the equation y = 2x + 3, y is an explicit function of x. Here, you can easily determine the value of y by substituting any value of x into the equation. The importance of explicit functions cannot be overstated, especially in fields such as physics, engineering, and economics, where precise calculations are essential. When dealing with explicit functions, one can quickly analyze how changes in the independent variable affect the dependent variable. This characteristic makes explicit functions particularly useful for graphing and visualization. For instance, if we were to graph the function y = 2x + 3, we would see a straight line where the slope is 2 and the y-intercept is 3. On the other hand, there are also implicit functions, which are defined differently. An implicit function is one where the relationship between variables is given in a form that does not isolate the dependent variable. For example, in the equation x^2 + y^2 = 1, y is not expressed explicitly in terms of x. Instead, it defines a circle, and while we can find y for a given x, it requires solving for y, which can lead to multiple values (both positive and negative roots). Understanding the difference between explicit and implicit functions is crucial for students and professionals alike. In calculus, for instance, we often differentiate explicit functions more easily than implicit functions. The derivative of an explicit function can be found directly using standard rules of differentiation. However, when dealing with implicit functions, one may need to use implicit differentiation, which involves applying the chain rule and considering both variables simultaneously. Moreover, explicit functions are typically easier to work with in optimization problems. When trying to find maximum or minimum values of a function, having the function in an explicit form allows for straightforward application of techniques such as taking derivatives and setting them to zero. In contrast, optimizing implicit functions may require additional steps to express one variable in terms of the other first. In conclusion, mastering the concept of explicit functions is vital for anyone studying mathematics or related fields. They provide clarity and simplicity in expressing relationships between variables, making them indispensable tools in analysis and problem-solving. As we continue to explore more complex mathematical concepts, the foundation laid by understanding explicit functions will undoubtedly enhance our ability to tackle various challenges in science, engineering, and beyond. Ultimately, recognizing the utility of explicit functions will empower individuals to apply mathematical reasoning effectively in real-world situations.

在数学中，显式函数的概念对理解变量之间的关系至关重要。显式函数被定义为依赖变量以独立变量的形式直接表达的函数。这意味着对于每个输入值，都有一个明确的输出值可以计算，而不会产生任何歧义。例如，在方程y = 2x + 3中，y是x的显式函数。在这里，你可以通过将任何x值代入方程轻松确定y的值。显式函数的重要性不容小觑，尤其是在物理、工程和经济等领域，精确计算至关重要。在处理显式函数时，人们可以迅速分析独立变量的变化如何影响依赖变量。这一特性使得显式函数在图形化和可视化方面特别有用。例如，如果我们要绘制函数y = 2x + 3的图形，我们会看到一条直线，其中斜率为2，y截距为3。另一方面，还有隐式函数，其定义方式不同。隐式函数是指变量之间的关系以一种不孤立依赖变量的形式给出。例如，在方程x^2 + y^2 = 1中，y并没有以x的形式显式表达。相反，它定义了一个圆，而虽然我们可以为给定的x找到y，但这需要求解y，这可能导致多个值（正根和负根）。理解显式函数和隐式函数之间的区别对学生和专业人士来说至关重要。例如，在微积分中，我们通常比处理隐式函数更容易对显式函数进行微分。显式函数的导数可以直接使用标准微分法则找到。然而，当处理隐式函数时，可能需要使用隐式微分，这涉及应用链式法则并同时考虑两个变量。此外，在优化问题中，显式函数通常更易于处理。当试图找到函数的最大值或最小值时，拥有显式形式的函数允许直接应用诸如取导数并将其设为零等技术。相比之下，优化隐式函数可能需要额外的步骤，首先将一个变量表示为另一个变量。总之，掌握显式函数的概念对于任何学习数学或相关领域的人来说都是至关重要的。它们提供了清晰和简单的方式来表达变量之间的关系，使其成为分析和解决问题不可或缺的工具。随着我们继续探索更复杂的数学概念，理解显式函数所奠定的基础无疑将增强我们在科学、工程等领域应对各种挑战的能力。最终，认识到显式函数的实用性将使个人能够有效地在现实世界中应用数学推理。