Math 142 Test3 S21
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Test 3 but should know for Test 2 with beta=0 Know proof as in class including MVT for integrals.
1 . suppose for all −∞ < a < b < ∞ that
b ρ(x, t) dx = q(a, t) − q(b, t) + lab β(x, t) dx.
(a) (10 points) Assume ρ, q, β are all continuous with continuous partial derivatives.
prove that
+ 
= β(x, t).
(b) (10 points) Now assume that ρ(x, t)
ρ− < ρ+ , where ρ− = lim ρ(x, t)，
x→xs(−)(t)
=
is discontinuous along a curve xs(t)
ρ+ = lim ρ(x, t). prove that
x→xs(+)(t)
q(ρ+) − q(ρ−)
ρ+ − ρ− ,
Test 2
2 . suppose a road satisfes the velocity/density relationship u(ρ) = 60 − .2ρ. The initial density is
（ 200 x < 0
ρ(x, 0) =〈 200 − 30x 0 ≤ x ≤ 5
（ 50 x > 5
(a) (14 points) sketch the characteristics in the xt-plane, and solve for ρ(x, t) in all regions of the plane.
(b) (6 points) using your formula for ρ(x, t) in the rarifed (central) region, show that ρ is continuous by showing that ρ(x, t) = 200 on the left-hand boundary, and ρ(x, t) = 50 on the right-hand boundary
(c) (5 points) sketch the trafc after 30 minutes, ρ(x, .5) 
nclude any important points on the x-axis.
3 . suppose we are again on a road with velocity/density relationship u(ρ) = 60 − .2ρ 
but with initial density now being ρ(x, 0) = 100 + 50 arctan(x). Now a shock will form.
(a) (10 points) Find the point (xs, ts) where the shock forms in the xt-plane.
ρ(x, 0)
150
100
x
−2 −1 1 2
b) (5 points) Draw a sketch of ρ(x, ts). (Include xs on the x-axis · )
(c) (5 points) Draw a sketch of ρ(x, 2ts). (Include xs on the x-axis. )
4. Assume a road has velocity-density relationship u(ρ) = 10 − ρ , with initial density
（ 8 x < 0 6 /
−6 −4 −2 2 4 6 8 10 12 14
(a) (6 points) sketch the characteristics in the xt-plane. (A Shock will occur, but it will not be a straight line. we will solve for it later. )
(b) (5 points) Find the point (xs, ts) where the shock forms , and include it on your sketch.
(c) (5 points) sketch 
s)，as a function of x. (LooK at your characteristics from part a) to see what it should be) .
(d) (5 points) Find ρ(x, t) in the region on the left side of the shock. (ρ is constant on the right side. )
(e) (4 points) Find a formula for 
. You 
ll need to simplify it for the next part.
(f) (10 points) You should get a linear ODE for xs(t). solve for the exact path of the shock by integrating factors.
2023-07-26