## Section4.1Introduction

Use the following definitions and properties to solve the problems contained in this Chapter.

Parametric Curves - Vocabulary

Let $I$ be an interval and let $f$ and $g$ be continuous on $I\text{.}$

1. The set of points $C=\{ (f(t),g(t)):t\in I\}$ is called a parametric curve.

2. The variable $t$ is called a parameter.

3. We say that the curve $C$ is defined by parametric equations $x=f(t)\text{,}$ $y=g(t)\text{.}$

4. We say that $x=f(t)\text{,}$ $y=g(t)$ is a parametrization of $C\text{.}$

5. If $I=[a,b]$ then $(f(a),g(a))$ is called the initial point of $C$ and $(f(b),g(b))$ is called the terminal point of $C\text{.}$

Derivative of Parametric Curves

The derivative to the parametric curve $x=f(t)\text{,}$ $y=g(t)$ is given by $\ds \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{g'(t)}{f'(t)}\text{.}$

Polar Coordinate System
1. Choose a point in the plane. Call it $O\text{,}$ the pole.

2. Choose a ray starting at $O\text{.}$ Call it the polar axis.

3. Take any point $P\text{,}$ except $O\text{,}$ in the plane. Measure the distance $d(O,P)$ and call this distance $r\text{.}$

4. Measure the angle between the polar axis and the ray starting at $O$ and passing through $P$ going from the polar axis in counterclockwise direction. Let $\theta$ be this measure in radians.

5. There is a bijection between the plane and the set

\begin{equation*} \mathbb{R}^+\times [0,2\pi )=\{ (r,\theta ):r\in \mathbb{R}^+ \mbox{ and } \theta \in [0,2\pi )\}\text{.} \end{equation*}

This means that each point $P\text{,}$ except $O\text{,}$ in the plane is uniquely determined by a pair $(r,\theta )\in \mathbb{R}^+\times [0,2\pi )\text{.}$

6. $r$ and $\theta$ are called polar coordinates of $P\text{.}$

Polar Curves

The graph of a polar equation $r=f(\theta)$ consists of all points $P$ whose polar coordinates satisfy the equation.

Derivative of Polar Curves

Suppose that $y$ is a differentiable function of $x$ and that $r=f(\theta )$ is a differentiable function of $\theta\text{.}$ Then from the parametric equations $x=r\cos \theta\text{,}$ $y=r\sin \theta$ it follows that

\begin{equation*} \ds \frac{dy}{dx}= \frac{\ds~~ \frac{dy}{d\theta}~~}{\ds \frac{dx}{d\theta}}=\frac{\ds \frac{dr}{d\theta }\sin \theta +r\cos \theta }{\ds \frac{dr}{d\theta }\cos \theta -r\sin \theta }\text{.} \end{equation*}
Parabola

A parabola is a set of points in the plane that are equidistant from a fixed point $F$ (called the focus) and a fixed line called the directrix). An equation of the parabola with focus $(0,p)$ and directrix $y=-p$ is $x^2=4py\text{.}$

Ellipse

An ellipse is a set of point in plane the sum of whose distances from two fixed points $F_1$ and $F_2$ is constant. The fixed points are called foci. The ellipse $\ds \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\text{,}$ $a\geq b>0\text{,}$ has foci $(\pm c,0)\text{,}$ where $c=\sqrt{a^2-b^2}\text{,}$ and vertices $(\pm a,0)\text{.}$

Hyperbola

A hyperbola is a set of points in plane the difference of whose distances from two fixed points $F_1$ and $F_2$ (the foci) is constant. The hyperbola $\ds \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has foci $(\pm c,0)\text{,}$ where $c=\sqrt{a^2+b^2}\text{,}$ vertices $(\pm a,0)\text{,}$ and asymptotes $\ds y=\pm \frac{bx}{a}\text{.}$

Eccentricity

Let $F$ be a fixed point in the plane, let $l$ be a fixed line in the plane, and let $e$ be a fixed positive number (called the eccentricity). The set of all points $P$ in the plane such that $\ds \frac{|PF|}{|Pl|}=e$ is a conic section. The conic is an ellipse if $e\lt 1\text{,}$ a parabola if $e=1\text{,}$ and a hyperbola if $e>1\text{.}$