## 数学代写|微积分代写Calculus代写|MATH0220

2022年12月26日

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## 数学代写|微积分代写Calculus代写|Numerical approximations

Computing a definite integral of a function $f$ over an interval $[a, b]$ using upper and lower sums, or even as the limit of Riemann sums, is, for all but the simplest cases, a difficult task. As a result, definite integrals are almost never computed in that manner. For the most part, definite integrals are evaluated either using the Fundamental Theorem of Calculus or using numerical approximation techniques. We will take up the Fundamental Theorem of Calculus approach in the next section; in this section wc consider several methods for numerical approximation.

4.2.1 The left-hand and right-hand rules. Recall that for an integrable function $f$ on an interval $[a, b]$, the left-hand rule approximation for $\int_a^b f(x) d x$, using $n$ intervals, is given by
$$A_L=h \sum_{i=0}^{n-1} f(a+i h)$$
and the right-hand rule approximation by
$$A_R=h \sum_{i=1}^n f(a+i h),$$
where
$$h=\frac{b-a}{n} .$$
We now look at the accuracy of these approximations. Let $x_i=a+i h, i=0,1,2, \ldots, n$, the endpoints for a partition of $[a, b]$ using $n$ intervals of equal length $h$. Assume $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ and that $x$ is a point in the $i$ th interval, that is, $x_{i-1} \leq x \leq x_i$. Then the Mean Value Theorem tells us that there exists a point $c_i$ in the interval $\left(x_{i-1}, x_i\right)$ such that
$$f^{\prime}\left(c_i\right)=\frac{f(x)-f\left(x_{i-1}\right)}{x-x_{i-1}} .$$
Solving for $f(x)$ in (4.2.3), we have
$$f(x)=f\left(x_{i-1}\right)+f^{\prime}\left(c_i\right)\left(x-x_{i-1}\right) .$$

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

We are now ready to make the long-promised connection between differentiation and integration, between areas and tangent lines. We will look at two closely related theorems, both of which are known as the Fundamental Theorem of Calculus. We will call the first of these the Fundamental Theorem of Integral Calculus.

Suppose $f$ is integrable on $[a, b]$ and $F$ is an antiderivative of $f$ on $(a, b)$ which is continuous on $[a, b]$. In particular, $F^{\prime}(x)=f(x)$ for all $x$ in $(a, b)$. Let $P=$ $\left{x_0, x_1, x_2, \ldots, x_n\right}$ be a partition of $[a, b]$ and, as usual, let $\Delta x_i=x_i-x_{i-1}, i=1,2,3, \ldots, n$. Now
\begin{aligned} F(b)-F(a)= & F\left(x_n\right)-F\left(x_0\right) \ = & F\left(x_n\right)+\left(F\left(x_{n-1}\right)-F\left(x_{n-1}\right)\right)+\left(F\left(x_{n-2}\right)-F\left(x_{n-2}\right)\right) \ & \quad+\cdots+\left(F\left(x_1\right)-F\left(x_1\right)\right)-F\left(x_0\right) \ = & \left(F\left(x_n\right)-F\left(x_{n-1}\right)\right)+\left(F\left(x_{n-1}\right)-F\left(x_{n-2}\right)\right) \ & \quad+\cdots+\left(F\left(x_1\right)-F\left(x_0\right)\right) \ & \quad+\sum_{i=1}^n\left(F\left(x_i\right)-F\left(x_{i-1}\right)\right) \end{aligned}
By the Mean Value Theorem, for every $i=1,2,3, \ldots, n$, there exists a point $c_i$ in the interval $\left[x_{i-1}, x_i\right]$ such that
$$F^{\prime}\left(c_i\right)=\frac{F\left(x_i\right)-F\left(x_{\mathrm{i}-1}\right)}{x_i-x_{i-1}}$$
Since $F^{\prime}\left(c_i\right)=f\left(c_i\right)$ and $x_i-x_{i-1}=\Delta x_i$, it follows that
$$F\left(x_i\right)-F\left(x_{i-1}\right)=f\left(c_i\right) \Delta x_i .$$
Hence, putting (4.3.3) into (4.3.1),
$$F(b)-F(a)=\sum_{i=1}^n f\left(c_i\right) \Delta x_i .$$
Thus $F(b)-F(a)$ is equal to the value of a Riemann sum using the partition $P$, and so must lie between the upper and lower sums for $P$. That is, we have shown that for any partition $P$,
$$L(f, P) \leq F(b)-F(a) \leq U(f, P)$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Numerical approximations

4.2.1 左手和右手规则。回想一下可积函数F在一个时间间隔[一种,b], 的左手法则近似∫一种bF(X)dX， 使用n间隔，由下式给出

H=b−一种n.

F′(C一世)=F(X)−F(X一世−1)X−X一世−1.

F(X)=F(X一世−1)+F′(C一世)(X−X一世−1).

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

Veft $\left{x_{-} 0, x_{-} 1, x_{-} 2\right.$, Vdots, $\left.\mathrm{x}{-} n \backslash r i g h t\right}$ 是一个分区 $[a, b]$ 和往常一样，让 $\Delta x_i=x_i-x{i-1}, i=1,2,3, \ldots, n$. 现在
$$F(b)-F(a)=F\left(x_n\right)-F\left(x_0\right)=F\left(x_n\right)+\left(F\left(x_{n-1}\right)\right.$$

$$F^{\prime}\left(c_i\right)=\frac{F\left(x_i\right)-F\left(x_{\mathrm{i}-1}\right)}{x_i-x_{i-1}}$$

$$F\left(x_i\right)-F\left(x_{i-1}\right)=f\left(c_i\right) \Delta x_i .$$

$$F(b)-F(a)=\sum_{i=1}^n f\left(c_i\right) \Delta x_i$$

$$L(f, P) \leq F(b)-F(a) \leq U(f, P)$$

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## MATLAB代写

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