Parametric Curves and Their Derivatives in Calculus III

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Parametric Curves

Introduction to Parametric Curves

In calculus, curves in the plane are often represented by functions such as y = f(x). However, not all curves can be described this way. Parametric equations provide a more complete method for representing curves, including those that cannot be written as a function of x or y alone.

Definition: A parametric curve is defined by two continuous functions of a parameter t, typically written as x = f(t) and y = g(t) for t in an interval [a, b].
The set of all points (x, y) = (f(t), g(t)) for all t in [a, b] forms the parametric curve.
This approach allows for the representation of curves that are not functions in the traditional sense.

Examples of Parametric Equations

Parametric equations can describe a wide variety of curves. Here are some examples:

x = t, y = t2 - 2 for 0 < t < 2
x = \ln t, y = t2 for 1 < t < 4
x = 3 \cos t, y = 2 \sin t for 0 < t < 2\pi

Graphing Parametric Equations

To graph a parametric curve, follow these steps:

Construct a table of values for t, x, and y.
Plot the points (x, y) in the xy-plane.
Indicate the orientation of the curve, which is the direction in which the curve is traced as t increases.

Example: Graph the curve described by x = t + 1, y = t2 for -1 < t < 3.

Table of values:
- t = -1: (x, y) = (0, 1)
- t = 0: (x, y) = (1, 0)
- t = 1: (x, y) = (2, 1)
- t = 2: (x, y) = (3, 4)
- t = 3: (x, y) = (4, 9)
Orientation: As t increases, the curve moves from (0, 1) to (4, 9).

Eliminating the Parameter

Sometimes, it is useful to eliminate the parameter t to find a Cartesian equation for the curve.

Given x = t + 1 and y = t2, solve for t: t = x - 1.
Substitute into y: y = (x - 1)2.
This is a parabola shifted one unit to the right.

Example: Circle Representation

Parametric equations can represent circles easily:

x = 4 \cos t, y = 4 \sin t for 0 < t < 2\pi
Eliminate t:
- \( x^2 + y^2 = 16 \)
This is a circle of radius 4 centered at the origin.
Orientation: As t increases from 0 to 2\pi, the curve is traced counterclockwise.

Derivatives of Parametric Equations

Finding the Slope of the Tangent Line

For a parametric curve x = f(t), y = g(t), the slope of the tangent line at a point is given by:

\( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \)
This formula is derived using the chain rule.

Example: Derivative Calculation

Given x = a \sin t, y = a (1 - \cos t):

\( \frac{dx}{dt} = a \cos t \)
\( \frac{dy}{dt} = a \sin t \)
\( \frac{dy}{dx} = \frac{a \sin t}{a \cos t} = \tan t \)

Derivative formula for parametric equations

Special Parametric Curves: Cycloid and Astroid

Certain curves have special names and properties:

Cycloid: The path traced by a point on the circumference of a rolling circle.
Astroid: The path traced by a point on a circle of radius r/4 rolling inside a larger circle of radius r.

Cycloid curve illustration

Example: Derivative for Astroid

Given x = 4 \cos t, y = 1 \sin t for 0 < t < 2\pi:

\( \frac{dx}{dt} = -4 \sin t \)
\( \frac{dy}{dt} = \cos t \)
\( \frac{dy}{dx} = \frac{\cos t}{-4 \sin t} \)
To find points where the curve has a horizontal or vertical tangent, set \( \frac{dy}{dt} = 0 \) (horizontal) or \( \frac{dx}{dt} = 0 \) (vertical).

Astroid curve illustration

Summary Table: Parametric Curve Properties

Curve Type	Parametric Equations	Cartesian Equation	Orientation
Parabola	x = t + 1, y = t2	y = (x - 1)2	As t increases, curve moves right and up
Circle	x = r \cos t, y = r \sin t	x2 + y2 = r2	Counterclockwise as t increases
Cycloid	x = a(t - \sin t), y = a(1 - \cos t)	None (complex)	As t increases, curve moves right
Astroid	x = a \cos3 t, y = a \sin3 t	\( x^{2/3} + y^{2/3} = a^{2/3} \)	As t increases, curve traces four cusps

Additional info: The notes also briefly mention how to find parametric equations for a given Cartesian curve, and how to analyze tangent lines using derivatives.