x' = -6x - 3y + te^2t

y' = 4x + y

Find the general solution using undetermined coeffiecients

x' = -6x - 3y + te^2t

y' = 4x + y

Find the general solution using undetermined coeffiecients

Suppose that over a certain region of space the electrical potential V is given by the following equation.

V(x, y, z) = 4x² − 4xy + xyz

(a) Find the rate of change of the potential at P(3, 6, 6) in the direction of the vector v = i + j − k.
???
(b) In which direction does V change most rapidly at P?

???
(c) What is the maximum rate of change at P?

???

1.Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

ln(x) = 1/(x-3)

2. Use Newton's method to find all solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. (Enter your answers as a comma-separated list.)

6e^−x2 sin(x) = x² − x + 1

Hey.
Given C = {z | z = z(t) = 10*e^(it), 0 <= t <= 2 pi}.

How do I solve the following two integrals. Is there a way to do it with residues?

a) f(z) = (cos(z) -1)/ z^3
b) f(z) = (sin(pi*z))/(z^3-1)

Thank you!

Recall the following theorem, phrased in terms of least upper bounds.
Theorem (The Least Upper Bound Property of R). Every nonempty subset of R that
has an upper bound has a least upper bound.
A consequence of the Least Upper Bound Property of R is the Archimedean Property.
Theorem (Archimedean Property of R). For any x; y 2 R, if x > 0, then there exists
n 2 N so that nx > y.
Prove the following statements by using the above theorems.
(a) For any two real numbers a; b 2 R, if a < b, then there exists a real number r 2 R
such that a < r < b.
(b) Prove that for any two rational numbers a; b 2 Q, if a < b, then there exists an
irrational number r 2 R, r =2 Q, such that a < r < b.
(c) For any two real irrational numbers a; b 2 R, a; b =2 Q, if a < b, then there exists
a rational number q 2 Q such that a < q < b.
(d) Prove that the Least Upper Bound Property is equivalent to the Greatest Lower
Bound Property: \Every nonempty subset of R that has a lower bound has a
greatest lower bound."

Let T denote the counterclockwise rotation through 60 degrees, followed by reflection in the line y=x

(i) Show that T is a linear transformation.

(ii) Write it as a composition of two linear transformations.

(iii) Find the standard matrix of T.

The Verbrugge Publishing Company’s 2019 balance sheet and income statement are as follows (in millions of dollars):

Balance Sheet

Current assets Net fixed assets

Total assets

Income Statement

Net sales Operating expense

$300 200

$500

Current liabilities
Advance payments by customers

Noncallable preferred stock, $6 coupon, $110 par value (1,000,000 shares)
Callable preferred stock, $10 coupon, no par, $100 call price (200,000 shares)

Common stock, $2 par value (5,000,000 shares)

Retained earnings Total liabilities & equity

$ 40 80 110

200

10

60 $500

$540 516 $24 4 $28 7 $21 6 2 $13

Net operating income Other income
EBT
Taxes (25%)

Net income
Dividends on $6 preferred
Dividends on $10 preferred
Income available to common stockholders

(24-3)

Liquidation

Verbrugge and its creditors have agreed upon a voluntary reorganization plan. In this plan, each share of the noncallable preferred will be exchanged for 1 share of $2.40 preferred with a par value of $35 plus one 8% subordinated income debenture with a par value of $75. The callable preferred issue will be retired with cash generated by reducing current assets.

1. Assume that the reorganization takes place and construct the projected balance. Show the new preferred stock at its par value. What is the value for total assets? For debt? For preferred stock? 

2. Construct the projected income statement. What is the income available to common shareholders in the proposed recapitalization? 

3. What were the total cash flows received by the noncallable preferred stockholders prior to the reorganization? What were the total cash flows to the original noncall- able preferred stockholders after the reorganization? What was the net income to common stockholders before the reorganization? After the reorganization. 

4. Required pre-tax earnings are defined as the amount that is just large enough to meet fixed charges (debenture interest and/or preferred dividends). What are the required pre-tax earnings before and after the recapitalization? 

5. How is the debt ratio (i.e., liabilities/total assets) affected by the reorganization? Suppose you treated preferred stock as debt and calculated the resulting debt ratios. How are these ratios affected? If you were a holder of Verbrugge’s common stock, would you vote in favor of the reorganization? Why or why not? 


Consider again the Ohio Trust bank location problem discussed in Section 7.3. The file OhioTrustFull contains data for all of Ohio’s 88 counties. The file contains an 88 X 88 matrix with the rows and columns each being the 88 counties. The entries in the matrix are zeros and ones and indicate if the county of the row shares a border with the county of the column (1 = yes and 0 = no).

a. Create a model to find the location of required principal places of business (PPBs) to minimize the number of PPBs needed to open all counties to branches.

b. Solve the model constructed in part (a). What is the minimum number PPBs needed to open up the entire state to Ohio Trust branches?

2. The following data represent Salary, years of experience, and rank (A, B, or C). Salary is the dependent variable. For class and the interaction effect between class and years of experience, do the following: a. Indicate a scheme (method) of coding “Rank”. How many variables are needed? b. Using the coding method given in Part a, give the data (the numbers) that you would input for the following cases including the interaction term between rank and experience: Absenteeism (Hrs/Yr) Y Experience (Years) X1 Rank 36.5 4 B 27.7 1 A 32.2 5 C In the heading, give the variable. In the body, give the numbers. You might not need all columns.

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V ,

what is the relationship between S and the basis of V ?

Determine whether each of these proposed definitions is
a valid recursive definition of a function f from the set
of nonnegative integers to the set of integers. If f is well
defined, find a formula for f (n) when n is a nonnegative
integer and prove that your formula is valid.

e) f (0) = 2, f (n) = f (n − 1) if n is odd and n ≥ 1 and
f (n) = 2f (n − 2) if n ≥ 2

Let f : G → G′ be a surjective homomorphism between two groups, G and G′, and let N be a normal subgroup of G. Prove that f (N) is a normal subgroup of G′.

Question:

1. Prove that the intersection of two compact sets is compact, using criterion (2).
2. Prove that the intersection of two compact sets is compact, using criterion (1).
3. Prove that the intersection of two compact sets is compact, using criterion (3).

Probably the most important new idea you'll encounter in real analysis is

the concept of compactness. It's the compactness of [a, b] that makes a

continuous function reach its maximum and that makes the Riemann in-

tegral exist. For subsets of R"h, there are three equivalent definitions of

compactness. The first, 9.2(1), promises convergent subsequences. The sec-

ond, 9.2(2) brings together two apparently unrelated adjectives, closed and

bounded. The third, 9.2(3), is the elegant, modern definition in terms of

open sets; it is very powerful, but it takes a while to get used to.

9.1. Definitions. Let S be a set in Rn. S is bounded if it is contained

in some ball B(0, R) about 0 (or equivalently in a ball about any point). A

collection of open sets {U} is an open cover of S if S is contained in U U(.

A finite subcover is finitely many of the Ua which still cover S. Following

Heine and Borel, S is compact if every open cover has a finite subcover.

9.2. Theorem. Compactness. The following are all equivalent conditions

on a set S in ]Rn.

(1) Every sequence in S has a subsequence converging to a point of S.

(2) S is closed and bounded.

(3) S is compact: every open cover has a finite subcover.

Criterion

(1)is the Bolzano-Weierstrass condition for compactness,

which you met for R in Theorem 8.3. The more modern Heine-Borel crite-

rion (3) will take some time to get used to. A nonclosed set such as (0, 1] is

not compact because the open cover {(1/n, oo)} has no finite subcover. An

unbounded set such as R is not compact because the open cover {(-n, n)}

has no finite subcover. This is the main idea of the first part of the proof.

Proof. We will prove that (3)->(2) -> (1) -> (3).

(3) ->(2). Suppose that S is not closed. Let a be an accumulation point

not in S. Then the open cover {{Ix - aI > 1/n}} has no finite subcover.

Suppose that S is not bounded. Then the open cover {{JxI < n}} has no

finite subcover.

(2) -> (1). Take any sequence of points in S C ]Rn. First look at just the

first of the n components of each point. Since S is bounded, the sequence

of first components is bounded. By Theorem 8.3, for some subsequence,

the first components converge. Similarly, for some further subsequence, the

second components also converge. Eventually, for some subsequence, all of

the components converge. Since S is closed, the limit is in S.

(1) =>. (3). Given an open cover {Ua}, first we find a countable subcover.

Indeed, every point x of S lies in a ball of rational radius about a rational

point, contained in some Ua. Each of these countably many balls lies in

some U,,. Let {Vi} be that countable subcover.

Suppose that {V} has no finite subcover. Choose xl in S but not in

V1. Choose X2 in S but not in V1 U V2. Continue, choosing xn in S but

not in U{V : 1 < i < n}, which is always possible because there is no finite

subcover. Note that for each i, only finitely many xn (for which n < i)

lie in V. By (1), the sequence xn has a subsequence converging to some

x in S, contained in some Vi. Hence infinitely xn are contained in Vi, a

contradiction.

9.3. Proposition. A nonempty compact set S of real numbers has a largest

element (called the maximum) and a smallest element (called the minimum).

Proof. We may assume that S has some positive numbers, by translating it

to the right if necessary. Since S is bounded, there is a largest integer part D

before the decimal place. Among the elements of S that start with D, there

is a largest first decimal place d1. Among the elements of S that start with

D.d1, there is a largest second decimal place d2. Keep going to construct

a = D.dld2d3.... By construction, a is in the closure of S.

Since S is closed, a lies in S and provides the desired maximum.

A minimum is provided by - max(-S).

Let P^N denote the vector space of all polynomials of degree N or less, with real coefficients. Let the linear transformation: T: P³ --> P¹ be the second derivative. Is T onto? Explain. Is T one-to-one? What is the Kernel of T? Find the standard matrix A for the linear transformation T. Let B= {x+1 , x-1 , x²+x , x³+x² } be a basis for P³ ; and

F={ x+2 , x-3 } be a basis for P¹ . Find A _F<--B ( the matrix for T relative to the bases B and F).