5. Find the generating function for the number of ways to create a bunch of n balloons selected from white, gold, and blue balloons so that the bunch contains at least one white balloon, at least one gold balloon, and at most two blue balloons. How many ways are there to create a bunch of 10 balloons subject to these requirements?

You wish to retire in 18 years , at which time , you want to have accumulated enough money to receive an annuity of $500 monthly for 20 years of retirement. During the period before retirement you can earn 4% annually,while after retirement you can earn 6 percent on your money. What monthly contributions to the retirement fund will allow you to receive the 500 dollars annuity?

X and Y are subsets of a universal set U.

Is the following statement true or false? Support your answer with a venn diagram.

X - (Y u Z) = (X - Y) n (A - C)

Answer each one, with (brief) justification. Throughout, V denotes a vector space, and bold-face letters like u, v, etc denote vectors in V.

A1. Suppose there are three linearly independent vectors in V. Is V necessarily finite-dimensional?

A2. Suppose there are three vectors in V that span V. Is V necessarily finite-dimensional?

A3. What is the dimension of the vector space of all polynomials of degree 4 or less?

A4. Can an infinite-dimensional vector space contain a finite-dimensional subspace?

A5, In a finite-dimensional vector space, is every spanning set necessarily finite?

A6. Can R³ contain a four-dimensional subspace?

Solve the following equations in complex numbers and write your answer in polar and rectangular form.

(a) z² − i = 0 in C

(b) z⁷ + z⁶ + z⁵ + z⁴ + z³ + z² + z = 0

(c) e^2z + 2e^z = −2

(d) z + 1 / z−1 = e^iπ/3

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements.

3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements.

4. Define a relation R as follows:

R = {(a, a),(b, b),(c, c),(d, d),(c, a),(a, d),(c, d),(b, c),(b, d),(b, a)}

Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements.

5. How many different partial ordering relations are there on the set {a, b, c}?

6. Which of the following relations are partial orderings? Which are total orderings? Which are well-orderings?

            a. The relation described in Problem 2

            b. The relation described in Problem 3

            c. The relation described in Problem 4

            d. The "less-than" relation on the integers

            e. The "less-than-or-equal" relation on the integers

            f. The "less-than-or-equal" relation on the natural numbers

            g. The "less-than-or-equal" relation on the real numbers

            h. The "less-than-or-equal" relation on the non-negative real numbers

            i. The relations in Problem 5

What happens when the signs of the function in IVT are the same? Is it true that they don’t have any roots? Show an example

Diminishing Returns. Spending money on advertising for a product can increase the amount of revenue generated by selling that product. Eventually, however, the amount by which revenue increases is offset by the cost of the advertising. The revenue generated (in thousands of dollars) by spending x thousands of dollars advertising a certain product is measured and found to be

R(x) = x3e−0.3x,where x is at most 10 (that is, $10,000).

6. If R′(x) is negative, it means that spending more money will actually reduce sales. IsR′(x) ever negative on the interval 0 ≤ x ≤ 10?

7. We would like to find the point at which R′(x) stops increasing. This is called the point of diminishing returns.

   (a) Calculate R′′(x).
   (b) On what intervals is R′(x) increasing and on what intervals is R′(x) decreasing?

   (c) What is the point of diminishing returns for this particular product?

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a) Prove that Z[ √ 2] is a subring of R. (b) Find a unit in Z[ √ 2] that is different than 1 or −1.

Find the line y = b + mx of best fit through the data {(.1, .2),(.2, .3),(.3, .7),(.5, .2),(.75, .8)}, using the least squares criterion.

(Use one of the software tools: Excel, SPSS, Mathematica, or MATLAB to answer the following items, and print out your results directly from the software)

A mass of 1 kg is attached to a spring with constant

k = 16 kg/ft.

Initially at equilibrium, a periodic external force of

f(t) = cos(3t)

begins to affect the mass for

t > 0.

Find the resultant motion x(t).

5. [§ 3.1,3.3] — Consider the plane, P , described by the equation(s) ?

(a) (10 pts) Write the system of equations in matrix-vector form (A⃗x = ⃗b).

(b) (10 pts) Find all solutions to the system of equations. x1+x3+x4 = 0 x1+x2+3x3+5x4 = 0

(c) (10 pts) Identify a basis for P .

(d) Next, consider the orthogonal projection onto P, defined by T(⃗x) = M⃗x, M ∈ R4×4. { We do not have the tools to derive the matrix M yet. }

i. (10 pts) What is the image of T [im(T ) = im(M )]?

ii. (10 pts) What is the dimension of the image of T [dim(im(T ))]?

iii. (10 pts) What is the dimension of the kernel of T [dim(ker(T ))]?

iv. (10 pts) Identify a basis for ker(T )?

Let f be a group homomorphism from a group G to a group H

If the order of g equals the order of f(g) for every g in G must f be one to one.

for the matrix, A= [1 2 -1; 2 3 1; -1 -1 -2; 3 5 0]

a. calculate the transpose of A multiplied by A

b. find the eigenvectors and eigenvalues of the answer to a

c. Find the SVD of matrix A

Use the Laplace transform to solve the given initial value problem.

y′′−12y′−13y=0; y(0)=5, y′(0)= 23

Enclose arguments of functions in parentheses. For example, sin(2x).

Differential Equations