### 数学代写|微积分代写Calculus代写|Basic trig graphs

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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|Basic trig graphs

Points on the unit circle have coordinates $(\cos \theta, \sin \theta)$, as in figure 25 (right). Points on the graphs of $y=\sin \theta$ and $y=\cos \theta$ have the form $(\theta, \sin \theta)$ and $(\theta, \cos \theta)$, respectively (figure 25 , left).

Tracing the points on the unit circle and the graphs as we allow $\theta$ to vary helps us see the relationship between the unit circle, values of $\sin \theta$ and $\cos \theta$, and their graphs. Start with $\theta=0$. The coordinates on the unit circle are $(1,0)$; therefore, $\cos 0=1$ and $\sin 0=0$. As $\theta$ gets larger, the point on the unit circle moves left and upward, meaning that the $x$-coordinate $(\cos \theta)$ decreases and the $y$-coordinate $(\sin \theta)$ increases. Eventually, as $\theta$ reaches $\frac{\pi}{2}$, the point on the unit circle reaches $(0,1)$, meaning $\cos \frac{\pi}{2}=0$ and $\sin \frac{\pi}{2}=1$. This is illustrated in figure 26 .
Figure 26 Points on the graphs of $y=\sin \theta$ (top left) and $y=\cos \theta$ (botlom left), for various values of $\theta$ from 0 to $\frac{\pi}{2}$
Continuing toward $\theta=\pi$ moves the point on the unit circle left and downward; therefore, $\cos \theta$ is becoming more negative and $\sin \theta$ is decreasing, eventually reaching $(-1,0)$, meaning that $\cos \pi=-1$ and $\sin \pi=0$. See figure 27 .

As we move toward $\theta=\frac{3 \pi}{2}$ and the point $(0,-1)$ on the unit circle, the $x$-coordinate moves right toward 0 and the $y$-coordinate moves more negative toward $-1$. Then, as $\theta$ moves toward $2 \pi$ and the point $(1,0)$ on the unit circle, the $x$-coordinate continues moving to the right toward 1 and the $y$-coordinate starts moving upward toward 0 . Then the entire process repeats, over and over, ad infinitum, every length of $2 \pi$. We may move clockwise into negative values of $\theta$, with the pattern still repeating. The final results are in figure 28 .

## 数学代写|微积分代写Calculus代写|Machine description of a function

The idea of a function is sometimes represented as a machine on a production line. A real-number input rolls along a conveyor belt into the function machine, the function does its job, and a real-number output comes out the other side. Imagine “The Squarer,” a function machine that squares all inputs. If the number 2 is the input, then the function spits out the number 4 on the other side, as pictured in figure 1 .

Use the input $-1$ and the machine outputs 1 . Use the input 7 and the machine outputs 49 . In general, if the input is $x$, then the output is $x^{2}$ (figure 2).

The domain of a function is the set of all inputs. Unless otherwise stated, we use the natural domain of a function, which is the set of all inputs that the function can actually handle without running into problems, such as division by zero or square roots of negative numbers. Because any real number can be squared, the domain of The Squarer is the set of all real numbers, written in interval notation as $(-\infty, \infty)$.
The range of a function is the set of all outputs as the input varies throughout the domain. Squares of numbers cannot be negative, so the range of The Squarer is the interval $[0, \infty)$.To be called a function, the machine is only allowed to have one output for a given input. When the input to a function has been chosen, its output is determined.

## 数学代写|微积分代写Calculus代写|Representations of functions

Functions can be represented in various ways. Before symbols for algebra were developed, words were used. A function may still be represented verbally. For instance, The Squarer can be described by the sentence “the output of function $f$ is the square of its input.”

A function can be represented numerically by giving a list of inputs and outputs. A numerical representation of The Squarer is in table 1. Notice that a numerical description is somewhat limited as it does not describe what happens to every input; inferences must be made if one wants to apply a numerical representation to other inputs. Numerical descriptions often arise in scientific and engineering settings from observing a process and measuring inputs and outputs. Values from an experiment might be the starting point for the exploration of a function.

An algebraic representation of a function is a complete description of the function using the power of symbolic notation. The Squarer can be represented algebraically in function notation as
$$f(x)=x^{2}$$
or as the equation
$$y=x^{2} .$$

In the equation $y=x^{2}, x$ is the independent variable and $y$ is the dependent variable; the inputs are represented by $x$ and the outputs, by $y$.
Functions may also be represented graphically. Using the horizontal axis for the inputs and the vertical axis for the outputs, points on the graph represent input-output pairs (figure 3).

F(X)=X2

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## MATLAB代写

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