172 MATH Spring, 2004 Exam #2 Name: There are 100 points. For full credit you must show your work. You may use a calculator, but this does not exempt you from explaining your answers by giving results of computations or sketches of graphs, etc. 1. (36 points) For each model equation, and initial condition, rst give the solution equation. Then answer the other questions. a. Model equation un = (1.07)un1 with initial condition u0 = 350 ; then . u5 =
b.
Model equation P (t) = 0.07P (t) with initial contion P (0) = 350 ; then P (5) = . The population is double the initial population when t= .
c.
Model equation zn = zn1 + (3/2) with initial condition z0 = 9 .
d.
Model equation vn = 0.96vn1 + 5 with v0 = 200 . Does the equilibrium . Briey explain. appear to be stable (yes or no)?
e.
In (d) compute the ratio how rapidly vn
vn E and explain its signicance by saying vn1 E goes towards or away from the equilibrium.
f.
(7 bonus points) Model equation P = 0.04P + 5 with P (0) = 200 .
2. (10 points) Verify that Q(x) = 2x2 + cx + d , where c and d are constants, satises the model equation Q (x) = 4 . Compute the values of c and d so that Q(0) = 5 and Q (0) = 3 .
3. (7 points) Convert r = 2 , = 5/6 (radians) to (x, y) coordinates. Also give the equivalent measure of = 5/6 in degrees.
. Find A and B so that 4. (10 points) The period of sin(3x) x is = A cos(Bx) has an amplitude of 5 and a period of 4.
5. (12 points) Determine the model equation satised by R(t) = 3 cos 5t + 2 sin 5t .
6. (15 points) Let A = a.
2 2 1.3 0.2 , w= . , v= 1 3 0.15 0.9 Compute Av and Aw .
b.
The eigenvalues for this matrix are 1 = 1.2 and 2 = 1 , and eigenvectors are v and w . Which goes with which? Briey explain.
c.
Find a vector that lines up with the eigenvector belonging to eigenvalue 1, but whose column total is 1.
7. (10 points) Compute the sum of each series; or state that no sum exists. k a. k=0 (3/10)(1/5)
b.
j j=0 (1/4)(5/2)
8. (6 bonus points) In a 2-variable system you nd that eventually un 1.09un1 and vn 1.02vn1 . Is the total population Tn = un + vn growing or decling? Can you say at what rate? Is there eventually a stable distributio...