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# 数学代写|微积分代写Calculus代考|MATH1051 Parametric Equations and the Cycloid

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## 数学代写|微积分代写Calculus代考|Parametric Equations and the Cycloid

Topics

Parametric equations.

Slope in parametric equations.

The cycloid.

Arc length in parametric equations.
Definitions and Theorems

Definition of parametric curve: Let $f$ and $g$ be continuous functions of $t$ on an interval $I$. The equations $x=f(t), y=g(t)$ are parametric equations of the parameter $t$. The curve $C$ given by these equations is smooth if the derivatives of $f$ and $g$ are continuous and not both zero on $I$.

Given the parametric equations $x=f(t), y=g(t)$ for the smooth curve $C$, the slope at the point $(x, y)$ on $C$ is $\frac{d y}{d x}=\frac{d y / d t}{d x / d t}, \frac{d x}{d t} \neq 0$.

A cycloid is defined as the curve traced out by a point on the circumference of a circle rolling along a line. The parametric equations of a cycloid are $x=a(\theta-\sin \theta), y=a(1-\cos \theta)$.

The arc length of a parametric curve is given by
$$s=\int_a^b \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t=\int_a^b \sqrt{\left[f^{\prime}(t)\right]^2+\left[g^{\prime}(t)\right]^2} d t$$
Summary
In this lesson, we introduce parametric equations. The idea is to consider $x$ and $y$ as functions of a third variable (“parameter”) $t$. This allows us to add an orientation to the graph of the parametric curve. After a brief example of how to analyze a parametric curve, we look at the calculus concept of slope in parametric equations. We will look closely at the equation of the cycloid, its derivative, and its arc length.

## 数学代写|微积分代写Calculus代考|Polar Coordinates and the Cardioid

Topics

Definition of polar coordinates.

Conversion formulas.

Graphs of polar equations.

Graph of the cardioid.

The slope of a polar graph.

Horizontal and vertical tangents.
Definitions and Theorems

Let $P=(x, y)$ be a point in the plane. Let $r$ be the distance from $P$ to the origin, and let $\theta$ be the angle that the segment $\overline{O P}$ makes with the positive $x$-axis. Then, $(r, \theta)$ is a set of polar coordinates for the point $P$.

conversion formulas: $x=r \cos \theta, y=r \sin \theta ; \tan \theta=\frac{y}{x}, r^2=x^2+y^2$.

Let $r=f(\theta)$ be a polar curve. The slope of the graph of the curve is given by
$$\frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{f(\theta) \cos \theta+f^{\prime}(\theta) \sin \theta}{-f(\theta) \sin \theta+f^{\prime}(\theta) \cos \theta} .$$
Summary
You are probably familiar with polar coordinates. Even so, this lesson reviews their main properties and graphs. One example we consider is the cardioid. We then look at the derivative of a function in polar coordinates and study where the graph has horizontal and vertical tangents.

## 数学代写|微积分代写Calculus代考|Parametric Equations and the Cycloid

$$s=\int_a^b \sqrt{\left(\frac{d x}{d t}\right)^2+\left(\frac{d y}{d t}\right)^2} d t=\int_a^b \sqrt{\left[f^{\prime}(t)\right]^2+\left[g^{\prime}(t)\right]^2} d t$$

## 数学代写|微积分代写Calculus代考|Polar Coordinates and the Cardioid

$$\frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{f(\theta) \cos \theta+f^{\prime}(\theta) \sin \theta}{-f(\theta) \sin \theta+f^{\prime}(\theta) \cos \theta} .$$

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