parametric equation...Think of a curve being traced out over time, sometimes doubling back on itself or crossing itself. Such a curve cannot be described by a function y=f(x). Instead, we will describe our position along the curve at time t by
x y = = x(t) y(t)
Then x and y are related to each other through their dependence on the parameter t.
Example
Suppose we trace out a curve according to
x y = = t2−4t 3t
where t0. The arrow on the curve indicates the direction of increasing time or orientation of the curve. Drag the box along the curve and notice how x and y vary with t.
The parameter does not always represent time:
Example
Consider the parametric equation
x y = = 3cos 3sin
Here, the parameter represents the polar angle of the position on a circle of radius 3 centered at the origin and oriented counterclockwise.
Differentiating Parametric Equations
Let x=x(t) and y=y(t). Suppose for the moment that we are able to re-write this as y(t)=f(x(t)). Then dtdy=dxdydtdx by the Chain Rule. Solving for dxdy and assuming dtdx=0,
dxdy= dtdx dtdy
a formula that holds in general.
Example
If x=t2−3 and y=t8, then dtdx=2t and dtdy=8t7. So
dxdy dx2d2y = = dtdxdtdy=2t8t7=4t6 ddxdxdy= dtdx dtd[dxdy] =2t24t5=12t4
Notes
* It is often possible to re-write the parametric equations without the parameter. In the second example, x3=cos, y3=sin. Since cos2+sin2=1, x32+x32=1 . Then x2+y2=9, which is the equation of a circle as expected. When you do eliminate the parameter, always check that you have not introduced extraneous portions of the curve.
* Every curve has infinitely many parametrizations, amounting to different scales for the parameter. For example,
x y = = 3cos2 3sin2
traces out the circle from the second example twice as "quickly," completing a full revolution in rather than 2 units of .
* Every equation y=f(x) may be re-written in parametric form by letting x=t, y=f(t).
Key Concepts
A curve in the xy-plane may be described by a pair of parametric equations
x=x(t)
y=y(t)
where x and y are related through their dependence on t. This is particularly useful when neither x nor y is a function of the other.
The derivative of y with respect to x (in terms of the parameter t) is given by
dxdy=dtdxdtdy