1. In class we saw that the solution to the traffic flow problem
and initial condition  is given by the formula

(a) Sketch a graph of this solution which shows a representative sample of the characteristics. Make sure you label the boundaries of the rarefaction fan in the solution with their equations, and label each region of the solution with the value that u takes in that region. 

(b) The speed of an individual car which is at position x at time t is given by


Determine the function f(t) that gives the position at time t of a car which begins at a position x0 < 0 at time 0. Add a representative few of these trajectories for different values of x0 to your graph in part (a).
2. Sketch a graph of the solution to the traffic flow problem
and initial condition

Include some trajectories of individual cars in your graph, similar to problem 1(b).
3. Consider the vector field
F~ (x; y) = hx; yi in the region Ω = fx2 +y2 < 1g. Verify the divergence theorem
for this example by checking that the two sides of the equation agree:

4. Consider the scalar function u(x; y) = x4 + x2y2 in the region Ω = fx2 + y2 < 1g. Verify the following identity (a special case of the divergence theorem) for this example by checking that the two sides of the equation agree: