### 计算机代写|计算机图形学作业代写computer graphics代考|Explicit and Implicit Equations

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 计算机代写|计算机图形学作业代写computer graphics代考|Explicit and Implicit Equations

The equation
$$y=3 x^{2}+2 x+4$$
associates the value of $y$ with different values of $x$. The directness of the equation: ‘ $y=$ ‘, is why it is called an explicit equation, and their explicit nature is extremely useful. However, simply by rearranging the terms, creates an implicit equation:
$$4=y-3 x^{2}-2 x$$
which implies that certain values of $x$ and $y$ combine to produce the result 4 . Another implicit form is
$$0=y-3 x^{2}-2 x-4$$
which means the same thing, but expresses the relationship in a slightly different way.

An implicit equation can be turned into an explicit equation using algebra. For example, the implicit equation
$$4 x+2 y=12$$
has the explicit form:
$$y=6-2 x$$
where it is clear what $y$ equals.

## 计算机代写|计算机图形学作业代写computer graphics代考|Function Notation

The explicit equation
$$y=3 x^{2}+2 x+4$$
tells us that the value of $y$ depends on the value of $x$, and not the other way around. For example, when $x=1, y=9$; and when $x=2, y=20$. As $y$ depends upon the value of $x$, it is called the dependent variable; and as $x$ is independent of $y$, it is called the independent variable.
We can also say that $y$ is a function of $x$, which can be written as
$$y=f(x)$$
where the letter ‘ $f$ ‘ is the name of the function, and the independent variable is enclosed in brackets. We could have also written $y=g(x), y=h(x)$, etc.

Eventually, we have to identify the nature of the function, which in this case is
$$f(x)=3 x^{2}+2 x+4$$
Nothing prevents us from writing
$$y=f(x)=3 x^{2}+2 x+4$$
which means: $y$ equals the value of the function $f(x)$, which is determined by the independent variable $x$ using the expression $3 x^{2}+2 x+4$.

An equation may involve more than one independent variable, such as the volume of a cylinder:
$$V=\pi r^{2} h$$
where $r$ is the radius, and $h$, the height, and is written:
$$V(r, h)=\pi r^{2} h .$$

## 计算机代写|计算机图形学作业代写computer graphics代考|Intervals

An interval is a continuous range of numerical values associated with a variable, which can include or exclude the upper and lower values. For example, a variable such as $x$ is often subject to inequalities like $x \geq a$ and $x \leq b$, which can also be written as
$$a \leq x \leq b$$
and implies that $x$ is located in the closed interval $[a, b]$, where the square brackets indicate that the interval includes $a$ and $b$. For example,
$$1 \leq x \leq 10$$
means that $x$ is located in the closed interval [1, 10], which includes 1 and 10 .
When the boundaries of the interval are not included, then we would state $x>a$ and $x<b$, which is written
$$a<x<b$$
and means that $x$ is located in the open interval ]a, $b[$, where the reverse square brackets indicate that the interval excludes $a$ and $b$. For example,
$$1<x<10$$
means that $x$ is located in the open interval ]1, 10[, which excludes 1 and 10 .

4=是−3X2−2X

0=是−3X2−2X−4

4X+2是=12

F(X)=3X2+2X+4

1≤X≤10

1<X<10

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## MATLAB代写

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