Show that the map Q[X] → Q[X],

sum^{n}{i=0}(a_nX^i) → sum^{n}{i=0}(a_n(2X + 3)^i) , is an automorphism of Q[X],

Un tanque inicialmente tiene 220 galones de agua limpia, pero una solución de sal de concentración desconocida se vierte a un ritmo de 6 galones por minuto. Si a la vez que se vierte se extrae solución a la misma velocidad y si al cabo de 40 minutos la concentración en el tanque fue de 0.2 libras de sal por galón, determine la concentración de la solución vertida (en libras por galón).

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows.

For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory is not to exceed 215,000 hr; and the amount of labor for on-site work is to be less than or equal to 234,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture.

Use the simplex method to solve the linear programming problem.

The maximum is P = ________

at

(x, y, z) = (_______)

.

Exercise 2.1.39 Let A be a 2×2 invertible matrix, with

A =
[a b
c d]
Find a formula for A−1 in terms of a,b, c,d by using elementary row operations

4) In this problem, we will explore how the cardinality of a subset S ⊆ X relates to the cardinality of a finite set X.

(i) Explain why |S| ≤ |X| for every subset S ⊆ X when |X| = 1.

(ii) Assume we know that if S ⊆ <n>, then |S| ≤ n. Explain why we can show that if T ⊆ <n+ 1>, then |T| ≤ n + 1.

(iii) Explain why parts (i) and (ii) imply that for every n ∈ N, every subset of <n> is finite and has cardinality less than n + 1.

A spring with a spring constant 4 N/m is loaded with a 2 kgmass and allowed to reach equilibrium. It is then displaced 1 meter downward and released. Suppose the mass experiences a damping force in Newtons equal to 1 times the velocity at every point and an external force of F(t)=4sin(3t) driving the system. Set up a differential equation that describes this system and find a particular solution to this non-homogeneous differential equation:

Let D8 be the group of symmetries of the square.

(a) Show that D8 can be generated by the rotation through 90◦ and any one of the four reflections.

(b) Show that D8 can be generated by two reflections.
(c) Is it true that any choice of a pair of (distinct) reflections is a generating set of D8?

Note: What is mainly required here is patience. The first important step is to set up your notation in a clear way, so that you (and your reader) can see what you are doing. You might find it useful to write out the whole group table for D8, which is a useful exercise anyway. Then for part (a), choose one of the four reflections, think about how it composes with the rotation through 90◦, and how you can use this to obtain the remaining reflections. Try to explain why your argument would work for any of the four reflections. For parts (b) and (c), think about the geometry of the different pairs of reflections that you could choose. The composition of two reflections is always a rotation, but how does the angle of rotation depend on the two reflections that you choose?

Let [x]_B be the coordinate vector of a vector x ∈ V with respect to the basis B for V . Show
that x is nonzero if and only if [x]_B is nonzero.

10. SHALL WE SCREAM?

What mathematics is involved in the design of roller coasters? How does one make them safe but still scary?

in the group R* find elements a and b such that |a|=inf, |b|=inf and |ab|=2

4) Find the volume of the solid formed by the region bounded by the graphs of y= x3 , y=x for x=0 and x=1

-Sketch the region bounded by the graphs of the functions and find the area of the region bounded by the graphs of y=x-1 and y= (x − 1)3

-calculate the arc length of the graph y= x=1 to x=2 14x7 + 101x5 from

-Use the washer method to find the volume of the solid formed by revolving the region bounded by the graphs by y=-2x+1 and y= x2 +1

Problem for submission: For which positive integers k can a simple graph G = (V, E) be constructed such that: G has k vertexes, that is, |V | = k, G is bipartite, and its complement G is bipartite? Prove your answer is correct

Please show and explain your full proof.