We have seen curves defined using functions, such as *y* = *f* (*x*). We can define more complex curves that represent relationships between *x* and *y* that are not definable by a function using parametric equations. Parametric curves are defined using two separate functions, *x*(*t*) and *y*(*t*), each representing its respective coordinate and depending on a new parameter, *t*. As *t* varies, so do the *x* and *y* coordinates of points on the curve. A curve such as *y* = *x*² can be represented parametrically by *x*(*t*) = *t* and *y*(*t*) =* t*². More complex curves involve more complex functions for *x*(*t*).

To find the rate of change of *y* with respect to *x* for a parametric curve (i.e., the first derivative with respect to *x*), and to find the derivative of this (i.e., the second derivative), use the following formulas:

and
Note that both of these derivatives are defined in terms of the parameter, *t*.

## 1. A circle

## 2. Another circle

*x*(*t*) = cos(2*t*) and *y*(*t*) = sin(2*t*)
Move the *t* slider and notice what happens. What difference does that multiple of 2 inside the sine and cosine functions make? How does the graph change? How does the position of the point vary with *t*? How will the derivatives change? Parametric curves can retrace themselves, unlike curves defined using *y* = *f* (*x*).
## 3. A line

## 4. An ellipse

## 5. A Lissajous

## Explore

## Other 'Applications of Differentiation' topics

See About the calculus applets for operating instructions. |

The applet initially shows a graph of a circle defined parametrically (if the circle looks squished, click Equalize Axes). In this case, *x*(*t*) = cos(*t*) and *y*(*t*) = sin(*t*). Move the *t* slider, which changes the value of the *t* parameter. Changing *t* changes the values of *x*(*t*) and *y*(*t*), which are the coordinates of a point on the curve (the magenta dot). As you change *t*, the dot traces out the curve. The limit control panel now has some additional fields on it to specify the minimum and maximum values of *t* used in drawing the curve (tmin and tmax), plus 'tintervals' - the number of intervals into which this is divided. Making this smaller results in a coarser graph.

The graph also displays the value of *dy/dx*, which is just the slope of the tangent line at the point corresponding to the current value of *t*. This is computed using the formula above as:
The graph also shows the value of the second derivative, computed using the formula above as

Select the second example from the drop down menu. This is also a circle, with the definition

Select the third example from the drop down menu. This shows a straight line. Only part of the line is showing, due to setting tmin = 0 and tmax = 1. As you would expect, *dy/dx* is constant, based on using the formulas above:

Select the fourth example from the drop down menu. This shows an ellipse, which is just a slight modification of the equations for a circle. Can you compute the derivatives?

Select the fifth example, showing a Lissajous figure. In this case, the multiple of *t* inside the sine function is different from the one inside the cosine function. Move the *t* slider and see if you can understand why this curve comes out the way it does.

You can try your own examples by typing different functions of *t* for *x*(*t*) and *y*(*t*) and setting tmin and tmax to appropriate values.

- Curve Analysis: Basics
- Curve Analysis: Special Cases
- Curve Analysis: Global Extrema
- Optimization: Maximize Volume
- Extreme Value Theorem
- Related Rates
- L'Hopital's Rule
- Parametric Derivatives
- Polar Derivatives
- Motion on a Line
- Motion in the Plane

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