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# 数学代写|微积分代写Calculus代考|The Fundamental Theorem of Calculus

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## 数学代写|微积分代写Calculus代考|The Fundamental Theorem of Calculus

In this section we present the Fundamental Theorem of Calculus, which is the central theorem of integral calculus. It connects integration and differentiation, enabling us to compute integrals by using an antiderivative of the integrand function rather than by taking limits of Riemann sums as we did in Section 5.3. Leibniz and Newton exploited this relationship and started mathematical developments that fueled the scientific revolution for the next 200 years.

Along the way, we will present an integral version of the Mean Value Theorem, which is another important theorem of integral calculus and is used to prove the Fundamental Theorem. We also find that the net change of a function over an interval is the integral of its rate of change, as suggested by Example 2 in Section 5.1.
Mean Value Theorem for Definite Integrals
In the previous section we defined the average value of a continuous function over a closed interval $[a, b]$ to be the definite integral $\int_a^b f(x) d x$ divided by the length or width $b-a$ of the interval. The Mean Value Theorem for Definite Integrals asserts that this average value is always taken on at least once by the function $f$ in the interval.

The graph in Figure 5.16 shows a positive continuous function $y=f(x)$ defined over the interval $[a, b]$. Geometrically, the Mean Value Theorem says that there is a number $c$ in $[a, b]$ such that the rectangle with height equal to the average value $f(c)$ of the function and base width $b-a$ has exactly the same area as the region beneath the graph of $f$ from $a$ to $b$.
THEOREM 3-The Mean Value Theorem for Definite Integrals If $f$ is continuous on $[a, b]$, then at some point $c$ in $[a, b]$,
$$f(c)=\frac{1}{b-a} \int_a^b f(x) d x .$$

## 数学代写|微积分代写Calculus代考|Fundamental Theorem, part 1

It can be very difficult to compute definite integrals by taking the limit of Riemann sums. We now develop a powerful new method for evaluating definite integrals, based on using antiderivatives. This method combines the two strands of calculus. One strand involves the idea of taking the limits of finite sums to obtain a definite integral, and the other strand contains derivatives and antiderivatives. They come together in the Fundamental Theorem of Calculus. We begin by considering how to differentiate a certain type of function that is described as an integral.

If $f(t)$ is an integrable function over a finite interval $I$, then the integral from any fixed number $a \in I$ to another number $x \in I$ defines a new function $F$ whose value at $x$ is
$$F(x)=\int_a^x f(t) d t .$$
For example, if $f$ is nonnegative and $x$ lies to the right of $a$, then $F(x)$ is the area under the graph from $a$ to $x$ (Figure 5.19). The variable $x$ is the upper limit of integration of an integral, but $F$ is just like any other real-valued function of a real variable. For each value of the input $x$, there is a single numerical output, in this case the definite integral of $f$ from $a$ to $x$.

Equation (1) gives a useful way to define new functions (as we will see in Section 7.1), but its key importance is the connection that it makes between integrals and derivatives. If $f$ is a continuous function, then the Fundamental Theorem asserts that $F$ is a differentiable function of $x$ whose derivative is $f$ itself. That is, at each $x$ in the interval $[a, b]$ we have
$$F^{\prime}(x)=f(x)$$
To gain some insight into why this holds, we look at the geometry behind it.
If $f \geq 0$ on $[a, b]$, then to compute $F^{\prime}(x)$ from the definition of the derivative we must take the limit as $h \rightarrow 0$ of the difference quotient
$$\frac{F(x+h)-F(x)}{h}$$

## 数学代写|微积分代写Calculus代考|The Fundamental Theorem of Calculus

$$f(c)=\frac{1}{b-a} \int_a^b f(x) d x .$$

## 数学代写|微积分代写Calculus代考|Fundamental Theorem, part 1

$$F(x)=\int_a^x f(t) d t .$$

$$F^{\prime}(x)=f(x)$$

$$\frac{F(x+h)-F(x)}{h}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。