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Handout 4

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:

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

Plot the length of the acceleration . From the picture estimate the magnitudes of the maximal and minimal acceleration between t=0 and .

Problem 2

Let be defined by

Calculate the derivative of f at the point (1,-1).
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).
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 .
Discuss how the two pictures compare.