## Exercises

1. For the example with , answer the following questions.
1. For which value(s) of t is the speed a maximum? What is the maximum speed?
2. For which value(s) of t is the speed a minimum? What is the minimum speed?
3. For which value(s) of t is the curvature a maximum? What is the maximum curvature?
4. For which value(s) of t is the curvature a minumum? What is the minimum curvature?

2. Consider the curve .
1. 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

2. Find the values of t, , for which the curvature of is a maximum or a minimum and identify which is which. What is the maximum and minimum curvature?
3. Find the values of t, , for which the speed for the trajectory is a maximum or a minimum and identify which is which. What is the maximum and minimum speed?

3. Consider . What is the normal vector at ? At ? What about at t=0? What goes wrong?

4. Consider the helix

where A, b, and are parameters.

1. 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);
```

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

5. A hairpin turn on a roadway can be approximated as half of an ellipse, including the major (longer) axis. If the major axis is 100 feet and the minor axis is 80 feet, what is the maximum (constant) speed a car can go through the turn while keeping the centripetal acceleration less than ?