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Section 4.1 Introduction

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 \(e>1\text{.}\)