### 数学代写|微积分代写Calculus代写|MATH1051

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|微积分代写Calculus代写|Areas and tangents

The study of calculus begins with questions about change. What happens to the velocity of a swinging pendulum as its position changes? What happens to the position of a planet as time changes? What happens to a population of owls as its rate of reproduction changes? Mathematically, one is interested in learning to what extent changes in one quantity affect the value of another related quantity. Through the study of the way in which quantities change we are able to understand more deeply the relationships between the quantities themselves. For example, changing the angle of elevation of a projectile affects the distance it will travel; by considering the effect of a change in angle on distance, we are able to determine, for example, the angle which will maximize the distance.

Related to questions of change are problems of approximation. If we desire to approximate a quantity which cannot be computed directly (for example, the area of some planar region), we may develop a technique for approximating its value. The accuracy of our technique will depend on how many computations we are willing to make; calculus may then be used to answer questions about the relationship between the accuracy of the approximation and the number of calculations used. If we double the number of computations, how much do we gain in accuracy? As we increase the number of computations, do the approximations approach some limiting value? And if so, can we use our approximating method to arrive at an exact answer? Note that once again we are asking questions about the effects of change.

Two fundamental concepts for studying change are sequences and limits of sequences. For our purposes, a sequence is nothing more than a list of numbers. For example,
$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$$
might represent the beginning of a sequence, where the ellipsis indicates that the list is to continue on indefinitely in some pattern. For example, the 5 th term in this sequence might be
the 8th term
$$\frac{1}{16}=\frac{1}{2^4},$$
$$\frac{1}{128}=\frac{1}{2^7} \text {, }$$
and, in general, the $n$th term
$$\frac{1}{2^{n-1}},$$
where $n=1,2,3, \ldots$. Notice that the sequence is completely specified only when we have given the general form of a term in the sequence. Also note that this list of numbers is approaching 0 , which we would call the limit of the sequence. In the next section of this chapter we will consider in some detail the basic question of determining the limit of a sequence.

## 数学代写|微积分代写Calculus代写|Sequences

Ás we noted in Section 1.1, listing the first few terms of a sequence does not uniquely specify the remaining terms of the sequence. To fully specify a sequence, we need a formula that describes an arbitrary term in the sequence. For example, the first example above lists the first four terms of the sequence $\left{a_n\right}$ with
$$a_n=n$$
for $n=1,2,3, \ldots ;$ the second example lists the first four terms of $\left{b_n\right}$ with
$$b_n=2 n$$
for $n=1,2,3, \ldots ;$ the third example lists the first four terms of $\left{c_n\right}$ with
$$c_n=1-\frac{1}{n}$$
for $n=1,2,3, \ldots ;$ the fourth lists the first four terms of $\left{d_n\right}$ with
$$d_n=\frac{(-1)^n}{2^n}$$
for $n=0,1,2,3, \ldots ;$ and the fifth lists the first four terms of $\left{e_n\right}$ with
$$e_n=(-1)^n$$
for $n=0,1,2, \ldots$. Note, however, that although these are in some sense the natural formulas for these sequences, they are not the only possibilities.

As indicated in Section 1.1, we are often interested in the value, if one exists, which a sequence approaches. For example, the sequences $\left{a_n\right}$ and $\left{b_n\right}$ increase beyond any possible bound as $n$ increases, and hence they have no limiting value. To visualize what is happening here, you might plot the points of the sequence on the real line. For both of these sequences, the plotted points will march off to the right without any upper limit. Although a limit does not exist in these cases, we usually write
$$\lim {n \rightarrow \infty} a_n=\infty$$ and $$\lim {n \rightarrow \infty} b_n=\infty$$
to express the fact that the limits do not exist because the terms in the sequence are eventually always larger than any specified positive bound. On the other hand, if we plot the points of the sequence $\left{c_n\right}$, as in Figure 1.2.1, we see that although they are always increasing (that is, moving toward the right), nevertheless they never increase beyond 1. Moreover, even though no term in the sequence is ever equal to 1 , we can see that the points become arbitrarily close to 1 . Hence we say that the limit of the sequence is 1 and we write
$$\lim _{n \rightarrow \infty} c_n=1$$

## 数学代写|微积分代写Calculus代写|Areas and tangents

$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$$

$$\begin{gathered} \frac{1}{16}=\frac{1}{2^4}, \ \frac{1}{128}=\frac{1}{2^7}, \end{gathered}$$

$$\frac{1}{2^{n-1}}$$

## 数学代写|微积分代写Calculus代写|Sequences

$$a_n=n$$

$$b_n=2 n$$

$$c_n=1-\frac{1}{n}$$

$$d_n=\frac{(-1)^n}{2^n}$$

$$e_n=(-1)^n$$

$$\lim n \rightarrow \infty a_n=\infty$$

$$\lim n \rightarrow \infty b_n=\infty$$ 和
$$\lim n \rightarrow \infty b_n=\infty$$

$$\lim _{n \rightarrow \infty} c_n=1$$

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## MATLAB代写

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