Due on ** Friday, October 13**.

** Problem 1**

A model for the motion of a moon around a planet is given by the curve

where is the angle . Plot a picture of the curve between 0 and . Plot a picture of the speed s(t) against t. From the picture estimate:

- a)
- The maximum and minimum speed of the particle between
**t=0**and . - b)
- All times between
**t=1**and**t=2**when the particle will have a speed of 10.

** Problem 2**

Let be defined by

- a)
- Calculate the derivative of f at the point (1,-1).
- b)
- Write down the linear approximation g(x,y) to f(x,y) at the point (1,-1). (So is an affine linear function whose value and derivative agree with f at (1,-1).
- c)
- Now view f and g as vector fields. Use the Diffeq,Phase Plane program in MacMath to plot a number of integral curves for each of these vector fields in the region and .
- d)
- Discuss how the two pictures compare.