College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)
5th Edition
ISBN: 9780134179025
Author: Ronald J. Harshbarger, Lisa S. Yocco
Publisher: PEARSON
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Chapter 6.3, Problem 16E
To determine
To solve: The polynomial equation,
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5. (a) State the Residue Theorem. Your answer should include all the conditions required
for the theorem to hold.
(4 marks)
(b) Let y be the square contour with vertices at -3, -3i, 3 and 3i, described in the
anti-clockwise direction. Evaluate
に
dz.
You must check all of the conditions of any results that you use.
(5 marks)
(c) Evaluate
L
You must check all of the conditions of any results that you use.
ཙ
x sin(Tx)
x²+2x+5
da.
(11 marks)
3. (a) Lety: [a, b] C be a contour. Let L(y) denote the length of y. Give a formula
for L(y).
(1 mark)
(b) Let UCC be open. Let f: U→C be continuous. Let y: [a,b] → U be a
contour. Suppose there exists a finite real number M such that |f(z)| < M for
all z in the image of y. Prove that
<
||, f(z)dz| ≤ ML(y).
(3 marks)
(c) State and prove Liouville's theorem. You may use Cauchy's integral formula without
proof.
(d) Let R0. Let w € C. Let
(10 marks)
U = { z Є C : | z − w| < R} .
Let f UC be a holomorphic function such that
0 < |ƒ(w)| < |f(z)|
for all z Є U. Show, using the local maximum modulus principle, that f is constant.
(6 marks)
3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
Chapter 6 Solutions
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences (5th Edition)
Ch. 6.1 - Graph the function h(x) = 3x3 + 5x2 x 10 on the...Ch. 6.1 - Graph the function f(x) = 2x3 3x2 6x on the...Ch. 6.1 - Prob. 3ECh. 6.1 - Prob. 4ECh. 6.1 - Prob. 5ECh. 6.1 - Prob. 6ECh. 6.1 - Prob. 7ECh. 6.1 - Prob. 8ECh. 6.1 - Prob. 9ECh. 6.1 - Prob. 10E
Ch. 6.1 - Prob. 11ECh. 6.1 - Prob. 12ECh. 6.1 - Prob. 13ECh. 6.1 - Prob. 14ECh. 6.1 - Prob. 15ECh. 6.1 - For Exercises 1116, match the polynomial function...Ch. 6.1 - For Exercises 1720, use the equation of the...Ch. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Prob. 21ECh. 6.1 - Prob. 22ECh. 6.1 - Prob. 23ECh. 6.1 - Prob. 24ECh. 6.1 - Prob. 25ECh. 6.1 - Prob. 26ECh. 6.1 - Prob. 27ECh. 6.1 - Prob. 28ECh. 6.1 - Prob. 29ECh. 6.1 - Prob. 30ECh. 6.1 - Prob. 31ECh. 6.1 - Sketch a graph of any cubic polynomial function...Ch. 6.1 - Prob. 33ECh. 6.1 - Prob. 34ECh. 6.1 - Prob. 35ECh. 6.1 - Prob. 36ECh. 6.1 - Prob. 37ECh. 6.1 - Prob. 38ECh. 6.1 - Daily Revenue The daily revenue in dollars from...Ch. 6.1 - Prob. 40ECh. 6.1 - Prob. 41ECh. 6.1 - Prob. 42ECh. 6.1 - Starbucks Using data from 2005 and projected to...Ch. 6.1 - Prob. 44ECh. 6.1 - Prob. 45ECh. 6.1 - Prob. 46ECh. 6.1 - Prob. 47ECh. 6.1 - Salaries The median salary for male workers, in...Ch. 6.1 - Prob. 49ECh. 6.1 - Prob. 50ECh. 6.2 - Find the cubic function that models the data in...Ch. 6.2 - Find the cubic function that is the best fit for...Ch. 6.2 - Prob. 3ECh. 6.2 - Find the quartic function that is the best fit for...Ch. 6.2 - Prob. 5ECh. 6.2 - a. Make a scatter plot of the data in the table...Ch. 6.2 - Prob. 7ECh. 6.2 - a. Find a cubic function that models the data in...Ch. 6.2 - Prob. 9ECh. 6.2 - a. Graph each of the functions found in Exercise 9...Ch. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Prob. 15ECh. 6.2 - Find the cubic function that is the best fit for...Ch. 6.2 - Use unrounded models for graphing and calculations...Ch. 6.2 - Use unrounded models for graphing and calculations...Ch. 6.2 - Prob. 19ECh. 6.2 - Prob. 20ECh. 6.2 - Median Age at First Marriage (Source: U.S. Census...Ch. 6.2 - Prob. 22ECh. 6.2 - Prob. 23ECh. 6.2 - Inflation Rate The annual changes in the consumer...Ch. 6.2 - Elderly Men in the Workforce The table below gives...Ch. 6.2 - Prob. 26ECh. 6.2 - Prob. 27ECh. 6.2 - Consumer Prices The data in the table give the...Ch. 6.2 - Federal Tax Per Capita The table gives the amount...Ch. 6.2 - Japanese Economy The economy of Japan is the third...Ch. 6.2 - Women in the Workforce The following table gives...Ch. 6.2 - Homicide Rates The table gives the U.S. homicide...Ch. 6.2 - Prob. 33ECh. 6.2 - Prob. 34ECh. 6.2 - Prob. 35ECh. 6.2 - Union Membership The table on the next page gives...Ch. 6.2 - Prob. 37ECh. 6.2 - Medicaid The numbers of thousands of people...Ch. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - Prob. 8ECh. 6.3 - Prob. 9ECh. 6.3 - Prob. 10ECh. 6.3 - Prob. 11ECh. 6.3 - Prob. 12ECh. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Prob. 20ECh. 6.3 - Prob. 21ECh. 6.3 - Prob. 22ECh. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Prob. 27ECh. 6.3 - Prob. 28ECh. 6.3 - Prob. 29ECh. 6.3 - Prob. 30ECh. 6.3 - Prob. 31ECh. 6.3 - Prob. 32ECh. 6.3 - Prob. 33ECh. 6.3 - Prob. 34ECh. 6.3 - Prob. 35ECh. 6.3 - Prob. 36ECh. 6.3 - Prob. 37ECh. 6.3 - Prob. 38ECh. 6.3 - Constructing a Box A box can be formed by cutting...Ch. 6.3 - Prob. 40ECh. 6.3 - Prob. 41ECh. 6.3 - Prob. 42ECh. 6.3 - Prob. 43ECh. 6.3 - Prob. 44ECh. 6.3 - Prob. 45ECh. 6.3 - Prob. 46ECh. 6.3 - Prob. 47ECh. 6.3 - Prob. 48ECh. 6.3 - Prob. 49ECh. 6.3 - Prob. 50ECh. 6.4 - In Exercises 14, use synthetic division to find...Ch. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 9ECh. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - Prob. 12ECh. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - In Exercises 1316, find one solution graphically...Ch. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.4 - Prob. 21ECh. 6.4 - Prob. 22ECh. 6.4 - Prob. 23ECh. 6.4 - In Exercises 2124, find all rational zeros of the...Ch. 6.4 - Prob. 25ECh. 6.4 - Prob. 26ECh. 6.4 - Prob. 27ECh. 6.4 - Solve each of the equations in Exercises 2530...Ch. 6.4 - Prob. 29ECh. 6.4 - Prob. 30ECh. 6.4 - Prob. 31ECh. 6.4 - In Exercises 3136, use synthetic division and...Ch. 6.4 - Prob. 33ECh. 6.4 - Prob. 34ECh. 6.4 - Prob. 35ECh. 6.4 - Prob. 36ECh. 6.4 - Drunk Driving Crashes Suppose the total number of...Ch. 6.4 - Prob. 38ECh. 6.4 - Prob. 39ECh. 6.4 - Births The number of births to females in the...Ch. 6.4 - Prob. 41ECh. 6.4 - Personal Savings The model y = 0.000447x3 ...Ch. 6.5 - Give the equations of any (a) vertical and (b)...Ch. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6.5 - Prob. 10ECh. 6.5 - Prob. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Prob. 13ECh. 6.5 - Prob. 14ECh. 6.5 - Prob. 15ECh. 6.5 - Prob. 16ECh. 6.5 - Prob. 17ECh. 6.5 - Prob. 18ECh. 6.5 - Prob. 19ECh. 6.5 - Prob. 20ECh. 6.5 - Prob. 21ECh. 6.5 - Prob. 22ECh. 6.5 - Prob. 23ECh. 6.5 - Prob. 24ECh. 6.5 - Prob. 25ECh. 6.5 - Prob. 26ECh. 6.5 - Prob. 27ECh. 6.5 - Prob. 28ECh. 6.5 - Prob. 29ECh. 6.5 - Prob. 30ECh. 6.5 - Prob. 31ECh. 6.5 - Prob. 32ECh. 6.5 - Prob. 33ECh. 6.5 - Prob. 34ECh. 6.5 - Prob. 35ECh. 6.5 - Prob. 36ECh. 6.5 - Prob. 37ECh. 6.5 - Prob. 38ECh. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - Average Cost The average cost per unit for the...Ch. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - Prob. 45ECh. 6.5 - Prob. 46ECh. 6.5 - Prob. 47ECh. 6.5 - Prob. 48ECh. 6.5 - Prob. 49ECh. 6.5 - Prob. 50ECh. 6.5 - Prob. 51ECh. 6.5 - Prob. 52ECh. 6.5 - Prob. 53ECh. 6.5 - Prob. 54ECh. 6.5 - Age at First Marriage The table that follows shows...Ch. 6.5 - Fences Suppose that a rectangular field is to have...Ch. 6.5 - Prob. 57ECh. 6.5 - Prob. 58ECh. 6.6 - Prob. 1ECh. 6.6 - Prob. 2ECh. 6.6 - Prob. 3ECh. 6.6 - Prob. 4ECh. 6.6 - In Exercises 116, use algebraic and/or graphical...Ch. 6.6 - Prob. 6ECh. 6.6 - Prob. 7ECh. 6.6 - Prob. 8ECh. 6.6 - Prob. 9ECh. 6.6 - Prob. 10ECh. 6.6 - Prob. 11ECh. 6.6 - Prob. 12ECh. 6.6 - Prob. 13ECh. 6.6 - Prob. 14ECh. 6.6 - Prob. 15ECh. 6.6 - Prob. 16ECh. 6.6 - For Exercises 1720, use the graph of y = f(x) to...Ch. 6.6 - Prob. 18ECh. 6.6 - Prob. 19ECh. 6.6 - Prob. 20ECh. 6.6 - Prob. 21ECh. 6.6 - Prob. 22ECh. 6.6 - Prob. 23ECh. 6.6 - Prob. 24ECh. 6.6 - Prob. 25ECh. 6.6 - Combine factoring with graphical and/or numerical...Ch. 6.6 - Prob. 27ECh. 6.6 - 28. Average Cost The average cost per set for the...Ch. 6.6 - Prob. 29ECh. 6.6 - Prob. 30ECh. 6.6 - Prob. 31ECh. 6.6 - Prob. 32ECh. 6.6 - Prob. 33ECh. 6.6 - Prob. 34ECh. 6.6 - Population Suppose the number of employees of a...Ch. 6 - For Exercises 14, (a) give the degree of the...Ch. 6 - For Exercises 14, (a) give the degree of the...Ch. 6 - Prob. 3TECh. 6 - Prob. 4TECh. 6 - Prob. 5TECh. 6 - Prob. 6TECh. 6 - Prob. 7TECh. 6 - Prob. 8TECh. 6 - Prob. 9TECh. 6 - Prob. 10TECh. 6 - Prob. 11TECh. 6 - Prob. 12TECh. 6 - Prob. 13TECh. 6 - Prob. 14TECh. 6 - In Exercises 1116, simplify each rational...Ch. 6 - Prob. 16TECh. 6 - Prob. 17TECh. 6 - Prob. 18TECh. 6 - Prob. 19TECh. 6 - Prob. 20TECh. 6 - Prob. 21TECh. 6 - Prob. 22TECh. 6 - Prob. 23TECh. 6 - Prob. 24TECh. 6 - Prob. 25TECh. 6 - Prob. 26TECh. 6 - Prob. 27TECh. 6 - Prob. 28TECh. 6 - Prob. 29TECh. 6 - Prob. 30TECh. 6 - Prob. 31TECh. 6 - Prob. 32TECh. 6 - Prob. 33TECh. 6 - Prob. 34TECh. 6 - Prob. 35TECh. 6 - Prob. 36TECh. 6 - Prob. 37TECh. 6 - Prob. 38TECh. 6 - Prob. 39TECh. 6 - Prob. 40TECh. 6 - In Exercises 3841, perform the long division. 41....Ch. 6 - Prob. 1RECh. 6 - Prob. 2RECh. 6 - Prob. 3RECh. 6 - Prob. 4RECh. 6 - Prob. 5RECh. 6 - Prob. 6RECh. 6 - Prob. 7RECh. 6 - Prob. 8RECh. 6 - Solve x4 x3 20x2 = 0.Ch. 6 - Prob. 10RECh. 6 - Prob. 11RECh. 6 - Prob. 12RECh. 6 - Prob. 13RECh. 6 - Prob. 14RECh. 6 - Prob. 15RECh. 6 - Prob. 16RECh. 6 - Prob. 17RECh. 6 - Prob. 18RECh. 6 - Find one solution of 3x3 x2 12x + 4 = 0...Ch. 6 - Find one solution of 2x3 + 5x2 4x 3 = 0...Ch. 6 - Prob. 21RECh. 6 - Prob. 22RECh. 6 - Prob. 23RECh. 6 - Prob. 24RECh. 6 - Prob. 25RECh. 6 - Prob. 26RECh. 6 - Prob. 27RECh. 6 - Prob. 28RECh. 6 - Prob. 29RECh. 6 - Prob. 30RECh. 6 - Prob. 31RECh. 6 - Prob. 32RECh. 6 - Prob. 33RECh. 6 - Revenue The monthly revenue for a product is given...Ch. 6 - Prob. 35RECh. 6 - Prob. 36RECh. 6 - Prob. 37RECh. 6 - Prob. 38RECh. 6 - Foreign Born Population The table on the following...Ch. 6 - Prob. 40RECh. 6 - Prob. 41RECh. 6 - Break-Even The daily profit in dollars for a...Ch. 6 - Starbucks The table gives the total number of...Ch. 6 - Drugs in the Bloodstream The concentration of a...Ch. 6 - Prob. 45RECh. 6 - Prob. 46RECh. 6 - Prob. 47RECh. 6 - Prob. 48RECh. 6 - Prob. 49RECh. 6 - Prob. 50RECh. 6 - Prob. 51RECh. 6 - Prob. 52RECh. 6 - Prob. 53RECh. 6 - Cost-Benefit The percent p of particulate...
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