微积分网课代修|预备微积分代写precalculus辅导|MAC1140 Functions and Function Notation

微积分网课代修|预备微积分代写precalculus辅导|MAC1140 Functions and Function Notation

简单的说,学好微积分(数学分析)是一个毁灭自己的先天直觉然后重新塑造一个后天直觉。

转变思维永远不是简单,但是不转变,贪图一时的捷径只是饮鸩止渴罢了。高中的时候,我一个同学很背单词的时候喜欢用汉字去拼那些单词的发音,还喜欢学各种解题技巧,这个时候我和他的成绩是一样的。

国外的老师较为看重学生homework的完成情况,对于同学们来说,完成一门科目作业并获得不错的成绩是尤为重要的事情。但对于不少同学来说,在自身英语说存在局限的情况下,当数学基础较为薄弱时,微积分作业的难度一下子就提升了,很难独立完成微积分作业。Calculus-do™提供的专业微积分代写能为大家解决所有的学术困扰,我们不仅会帮大家完成作业,还提供相应的数学知识辅导课程,以此来提高同学们学习能力。

我们提供的econ代写服务范围广, 其中包括但不限于:

  • 单变量微积分
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  • 黎曼积分
  • ODE
  • 微分学
微积分网课代修|预备微积分代写precalculus辅导|MAC1140 Functions and Function Notation

微积分网课代修|预备微积分代写precalculus辅导|Functions and Function Notation

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
$$
{(1,2),(2,4),(3,6),(4,8),(5,10)}
$$
The domain is ${1,2,3,4,5}$. The range is ${2,4,6,8,10}$.
Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter $x$. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter $y$.

A function $f$ is a relation that assigns a single value in the range to each value in the domain. In other words, no $x$-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain, ${1,2,3,4,5}$, is paired with exactly one element in the range, ${2,4,6,8,10}$.

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as
$$
\text { {(odd, 1), (even, 2), (odd, 3), (even, 4), (odd, 5) }}
$$
Notice that each element in the domain, {even,odd $}$ is not paired with exactly one element in the range, ${1,2,3,4,5}$. For example, the term “odd” corresponds to three values from the domain, ${1,3,5}$ and the term “even” corresponds to two values from the range, ${2,4}$. This violates the definition of a function, so this relation is not a function. Figure $\mathbf{1}$ compares relations that are functions and not functions.

微积分网课代修|预备微积分代写precalculus辅导|Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.

To represent “height is a function of age,” we start by identifying the descriptive variables $h$ for height and $a$ for age. The letters $f, g$, and $h$ are often used to represent functions just as we use $x, y$, and $z$ to represent numbers and $A, B$, and $C$ to represent sets.
$\begin{array}{ll}h \text { is } f \text { of } a & \text { We name the function } f \text {; height is a function of age. } \ h=f(a) & \text { We use parentheses to indicate the function input. } \ f(a) & \text { We name the function } f \text {; the expression is read as ” } f \text { of } a \text {.” }\end{array}$
Remember, we can use any letter to name the function; the notation $h(a)$ shows us that $h$ depends on $a$. The value $a$ must be put into the function $h$ to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.

We can also give an algebraic expression as the input to a function. For example $f(a+b)$ means “first add $a$ and $b$, and the result is the input for the function $f .$. The operations must be performed in this order to obtain the correct result.

微积分网课代修|预备微积分代写precalculus辅导|MAC1140 Functions and Function Notation

微积分网课代修|预备微积分代写precalculus辅导|Functions and
Function Notation


关系是一组有序对。每个有序对的第一个分量的集合称为域,每个有序对的第二个分量
的集合称为范围。考虑以下一组有序对。每对中的第一个数字是前五个自然数。每对中
的第二个数字是第一个数字的两倍。
$$
(1,2),(2,4),(3,6),(4,8),(5,10)
$$
域是 $1,2,3,4,5$. 范围是 $2,4,6,8,10$.
请注意,域中的每个值也称为输入值或自变量,通常用小写字母标记 $x$. 范围内的每个值
也称为输出值或因变量,通常标记为小写字母 $y$.
一个函数 $f$ 是一种将范围中的单个值分配给域中的每个值的关系。换句话说,没有 $x$-值
重复。对于我们将前五个自然数与其值加倍的数字相关的示例,这种关系是一个函数,
因为域中的每个元素, $1,2,3,4,5$ ,范围内的一个元素配对, $2,4,6,8,10$.
现在让我们考虑将术语“偶数”和“奇数”与前五个自然数相关联的有序对集合。它会显示

$$
{(\text { odd }, 1),(\text { even, } 2),(\text { odd }, 3),(\text { even, } 4),(\text { odd }, 5)}
$$
请注意,域中的每个元素 ${e v e n, o d d}$ } 不与范围内的一个元素配对, $1,2,3,4,5$. 例
如,术语“奇数”对应于域中的三个值, $1,3,5$ 并且术语“偶数”对应于范围中的两个值,
2,4 . 这违反了函数的定义,所以这个关系不是函数。数字 $\mathbf{1}$ 比较是函数而不是函数的关
系。


微积分网课代修|预备微积分代写precalculus辅导|Using Function
Notation


一旦我们确定一个关系是一个函数,我们就需要显示和定义函数关系,以便我们能够理
解和使用它们,有时还可以将它们编程到计算机中。有多种表示函数的方法。标准函数
表示法是一种便于使用函数的表示。
为了表示“身高是年龄的函数”,我们首先确定描述性变量 $h$ 对于身高和 $a$ 年龄。这些信
$f, g$ ,和 $h$ 经常被用来表示函数,就像我们使用的一样 $x, y$ ,和 $z$ 表示数字和 $A, B$ ,
和 $C$ 来表示集合。
$h$ is $f$ of $a \quad$ We name the function $f$; height is a function of age. $h=f(a) \quad$ We use parentheses to indicat 请记住,我们可以使用任何字母来命名函数;符号 $h(a)$ 向我们展示了 $h$ 取决于 $a$. 价值 $a$
必须放入函数中 $h$ 得到一个结果。括号表示年龄被输入到函数中;它们不表示乘法。
我们还可以将代数表达式作为函数的输入。例如 $f(a+b)$ 意思是“首先添加 $a$ 和 $b$ ,结果
是函数的输入 $f . .$ 必须按此顺序执行操作才能获得正确的结果。

微积分网课代修|预备微积分代写precalculus辅导|MAC1140 Functions and Function Notation
微积分网课代修|预备微积分代写precalculus辅导

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