- 10.3 | Parametric Equations and Calculus
- 10.4 | Polar Coordinates and Polar Graphs
- 10.5 | Area and Arc Length in Polar Coordinates

**parametric curve**, or

**plane curve**, while the equations which define it are called

**parametric equations**. The following are equivalent:

- \(x = f(t) \), \(y = g(t)\)
- \((f(t), g(t))\)
- \( \langle f(t), g(t) \rangle \)
- \( f(t)\mathbf{i} + g(t)\mathbf{j} \)

**Definition of a Smooth Curve**

A curve \(C\) represented by \(x = f(t)\) and \(y = g(t)\) on an interval \(I\) is called

**smooth**when \(f’\) and \(g’\) are continuous on \(I\) and not simultaneously \(0\), except possibly at the endpoints of \(I\). The curve \(C\) is called

**piecewise smooth**when it is smooth on each subinterval of some partition of \(I\).

Match the parametric equations with the graphs on the left labled I-VI.

- \( x = \frac{ \sin 2t}{4+t^2}\), \(y = \frac{ \cos 2t}{4+t^2} \)
- \( x= \cos 5t\), \( y = \sin 2t\)
- \(x = \sin 2t \), \( y = \sin(t + \sin 2t) \)
- \( x= t^2-2t\), \( y = \sqrt{t}\)
- \( x= t + \sin 4t \), \( y = t^2 + \cos 3t\)
- \(x = t^4 -t+1\), \(y =t^2\)

Graph the curve with parametric equations

$$

x= t \text{ and } y = t^2

$$

$$

\begin{array}{c|c}

t & (x,y) \\ \hline

-2 & (-2,4) \\

-1 & (-1,1) \\

0 & (0,0) \\

1 & (1,1)

\end{array}

$$

The ordered pairs in the second column of the table all seem to lie on the parabola with equation \(y = x^2\). To see this, note that

$$

y = t^2 = x^2

$$

since \(x = t\).

Graph the curve with parametric equations

$$

x= 2\cos \theta \text{ and } y = 2 \sin \theta; \; \; 0 \leq \theta \leq 2 \pi

$$

$$

\begin{array}{c|c}

\theta & (x,y) \\ \hline

0 & (2,0) \\

\pi/4 & (\sqrt{2},\sqrt{2}) \\

\pi/2 & (0,2) \\

3\pi/4 & (-\sqrt{2},\sqrt{2})\\

\pi & (-2,0)\\

\end{array}

$$

The ordered pairs in the second column of the table all seem to lie on the circle with equation \(x^2+y^2=4\). To see this, note that

the parametric equations satisfy the rectangular equation:

$$

x^2+y^2=4.

$$

That is,

$$

\begin{aligned}

x^2 + y^2 & \overset{?}{=} 4 \\

(2 \cos \theta)^2 + (2 \sin \theta)^2 & \overset{?}{=} 4 \\

4(\cos^2 \theta + \sin^2 \theta) & \overset{?}{=} 4 \\

4 &= 4

\end{aligned}

$$

Since \(\theta\) ranges from \(0\) to \(2 \pi\), the graph will be the graph of a complete circle of radius \(2\) centered at the origin. If, instead, \(\theta\) ranged from \(0\) to \(\pi/2\), then the graph would be the portion of the full circle in the first quadrant.

Graph the curve with parametric equations

$$

x= \cos(2\theta) \text{ and } y = \sin \theta; \; \; 0 \leq \theta \leq 2 \pi.

$$

$$

\begin{array}{c|c}

\theta & (x,y) \\ \hline

0 & (1,0) \\

\pi/4 & (0,\sqrt{2}/2) \\

\pi/2 & (-1,1) \\

3\pi/4 & (0,\sqrt{2}/2)\\

\pi & (1,0)\\

5\pi/4 & (0,-\sqrt{2}/2)\\

3\pi/2 & (-1,1)\\

\end{array}

$$

The ordered pairs in the second column of the table all seem to lie on a parabola, opening sideways, with equation \(x=1-2y^2\). To see this, note that the parametric equations satisfy the rectangular equation:

$$

x=1-2y^2.

$$

That is,

$$

\begin{aligned}

x & \overset{?}{=} 1-2y^2 \\

\cos(2 \theta) & \overset{?}{=} 1-2(\sin \theta)^2 \\

2- 2 \sin^2 \theta & \overset{?}{=} 1-2(\sin \theta)^2 \\

2- 2 \sin^2 \theta & = 1-2\sin^2 \theta \\

\end{aligned}

$$

Since \(\theta\) ranges from \(0\) to \(2 \pi\), the graph will only be a portion of the parabola. In fact, as \(\theta \) ranges from \(0\) to \(2 \pi\) portions of the graph are retraced.