Soundex produces x Model A radios and y Model B radios. Model A requires 15 min of work on Assembly Line I and 10 min of work on Assembly Line II. Model B requires 10 min of work on Assembly Line I and 12 min of work on Assembly Line II. At most, 25 labor-hours of assembly time on Line I and 22 labor-hours of assembly time on Line II are available each day. It is anticipated that Soundex will realize a profit of $15 on model A and $12 on model B. How many clock radios of each model should be produced each day in order to maximize Soundex's profit?

Prove by induction:

1 + 1/4 + 1/9 +⋯+ 1/?^2 < 2 − 1/?,

for all integers ?>1

The table shows the estimated percentage P of the population of a certain country that are mobile-phone subscribers. (End of year estimates are given.)

(c) Estimate the instantaneous rate of growth in 2003 by sketching a graph of P and measuring the slope of a tangent. (Sketch your graph so that it is a smooth curve through the points, and so that the tangent line has an x-intercept of 1999.3 and passing through the point

(2006, 46.6).  Round your answer to two decimal places.)

For part, c use the two points that are on the tangent line to determine the slope, which is the instantaneous rate of change.
.........................................  percentage points per year

y"+y'-6y=1

1. general solution of corresponding homogenous equation

2. particular solution

3.solution of initial value problem with initial conditions y(0)=y'(0)=0

Express the equation of the plane in explicit form
a. d = [1, -3, -5]; e = [-2, 1, -1] and f = [5, -1, -3] Express the equation of the plane through d, e, and f explicitly
b. d = [-1, 7, -6]; e = [2, -1, 3] and f = [4, -2, 9] Express the equation of the plane through d, e, and f explicitly

Let A =

and eigenvalue λ1 = 3 and associated eigenvector v(1) = (1, 0, 1)t . Find the second dominant eigenvalue λ2 (or the approximation to λ2) by the Wielandt Dflation method

1.

Use Euler's method with step size 0.50.5 to compute the approximate yy-values y1≈y(0.5), y2≈y(1),y3≈y(1.5), and y4≈y(2) of the solution of the initial-value problem

y′=1+3x−2y,   y(0)=2.

y1=

y2=

y3=

y4=

2.  

Consider the differential equation dy/dx=6x, with initial condition y(0)=3

A. Use Euler's method with two steps to estimate y when x=1:

y(1)≈ (Be sure not to round your calculations at each step!)

Now use four steps:
y(1)≈

B. What is the solution to this differential equation (with the given initial condition)?
y=

C. What is the magnitude of the error in the two Euler approximations you found?
Magnitude of error in Euler with 2 steps =

Magnitude of error in Euler with 4 steps =

D. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)?
factor =

(How close to this is the result you obtained above?)

Let a < c < b, and let f be defined on [a,b]. Show that f ∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover, Integral a,b f = integral a,c f + integral c,b f .

The weighted voting systems for the voters A, B, C, ... are given in the form q: w1, w2, w3, w4, ..., wn . The weight of voter A is w1, the weight of voter B is w2, the weight of voter C is w3, and so on. Calculate, if possible, the Banzhaf power index for each voter. Round to the nearest hundredth. (If not possible, enter IMPOSSIBLE.) {82: 53, 36, 24, 18} BPI(A) = BPI(B) = BPI(C) = BPI(D) =

Graph Theory: Let S be a set of three pairwise-nonadjacent edges in a 3-connected graph G. Show that there is a cycle in G containing all three edges of S unless S is an edge-cut of G

Let En be the subspace of V (n, 2) consisting of all vectros of even weight.

(a) What are the parameters [n, k, d] of En.

(b) Write down a generator matrix for En in standard form