- For the example with , answer the following questions.
- For which value(s) of
**t**is the speed a maximum? - For which value(s) of
**t**is the speed a minimum? - For which value(s) of
**t**is the curvature a maximum? - For which value(s) of
**t**is the curvature a minumum?

- For which value(s) of
- Consider the curve .
- Show that the image curve of is an ellipse. That
is, find values of
**a**and**b**such that the component functions of satisfy the equation - Find the values of
**t**, , for which the curvature of is a maximum or a minimum and identify which is which. - Find the values of
**t**, , for which the speed for the trajectory is a maximum or a minimum and identify which is which.

- Show that the image curve of is an ellipse. That
is, find values of
- Consider . What is the
normal vector at ? At ? What about at
**t=0**? What goes wrong? - Consider the helix
where

**A**,**b**, and are parameters.- Plot for a few values of the parameters. Try to
identify how each parameter affects the curve. A good way to do this
is with the
`subs`command, like the following example.> VPlot(subs({A=1,b=1,omega=1},r(t)),t=0..2*Pi);

- Show, using the
`Speed`command that the speed is constant and describe how it depends on the parameters. - Compute the curvature and describe how it depends on the parameters.

- Plot for a few values of the parameters. Try to
identify how each parameter affects the curve. A good way to do this
is with the

Wed Apr 5 15:41:18 EDT 1995