# 微积分网课代修|微分学代写Differential calculus代考|MATH272 What is calculus about?

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|微分学代写Differential calculus代考|What is calculus about?

One of the main entry ways to calculus is the study of motion.
We present the idea of calculus in these two related pictures. First, we derive the speed from the distance that we have covered:

Beyond this conceivable sit uation, this formula is the definition of speed. On the flip side, we derive the distance we have covered from the known velocity:

The two problems are solved, respectively, with the help of these two versions of the same elementary school formula:
speed $=$ distance $/$ time $\quad$ and $\quad$ distance $=$ speed $\times$ time
We solve the equation for the distance or for the speed depending on what is known and what is unknown. What takes this idea bey ond elementary school is the possibility that velocity varies over time. The simplest case is when it varies incrementally.

Let’s be more specific. We will face the two situations above but with more data collected and more information derived from it.

First, imagine that our speedometer is broken. What do we do if we want to estimate how fast we are driving during our trip? We look at the odometer several times – say, every hour on the hour – during the trip and record the mileage on a piece of paper. The list of our consecutive locations might look like this:

• after the first hour: 10, 055 miles
• after the second hour: 10,095 miles
• after the third hour: 10,155 miles
• et c.

We formalize the way we represent sequences of numbers such as the ones we saw in the last section:
\begin{tabular}{l|l|llllllllllll}
time & minutes & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & $\ldots$ \
\hline location & miles & $0.00$ & $0.10$ & $0.20$ & $0.30$ & $0.39$ & $0.48$ & $0.56$ & $0.64$ & $0.72$ & $0.78$ & $0.84$ & $\ldots$
\end{tabular}
We first give a sequence a name, say, $a$, and then assign a specific variation of this name to each term of the sequence:
Indices of sequence
$$\begin{array}{ll|llllllll} \text { index: } & n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \ \hline \text { term: } & a_{n} & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} & a_{7} & \ldots \end{array}$$
The name of a sequence is a letter, while the subscript called the index indicates the place of the term within the sequence. It reads ” $a$ sub $1 “$ “. ” $a$ sub 2 “, etc.
This is what the notation means:
\begin{tabular}{|cc|}
\hline Index of a term \
& index \
$\uparrow$ & $\downarrow$ \
$\uparrow$ & \
name & \
\hline
\end{tabular}

## 微积分网课代修|微分学代写Differential calculus代考|Infinitesequences and their long-term trends

|begin{tabular}{||||IIIIIIIIII} 小时 \& 分钟 \& $0 \& 1 \& 2 \& 3 \& 4 \& 5 \& 6 \& 7 \& 8 \& 9 \& 10 \& \$ \mid$dots$\$\backslash \backslash$ hline 位置 \& 英里 \& \$C 我们首先给一个序列一个名字，比如说，$a$，然后将这个名称的特定变体分配给序列的 每个项: 序列 索引 \begin } { \text { array } } | | | I | I I I I I } \backslash \text { text } { \text { index: } } \& n \& 1 \& 2 \& 3 \& 4 \& 5 \& 6 \& 7 \& | \text { dots } \backslash \backslash \text { hlineltext } { \text { term: } } \& a _ { – } { n } \& a _ { – } { 1 } \& a _ { – } { 2 } \& a _ { – } { 3 } \& a _ { – } { 4 } \text { i } 序列的名称是一个字母，而称为索引的下标表示该术语在序列中的位置。上面写着“$a\$ 子 1 ““.” asub 2″等。