FINITE MATH.F/BUS,ECON,LIFE..-ACCESS
14th Edition
ISBN: 9780134984209
Author: Barnett
Publisher: PEARSON
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Chapter 1.1, Problem 44E
In Problems 43 and 44, solve and graph.
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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?
(a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's
integral formula for derivatives. Your answer should include all the conditions
required for the results to hold.
(8 marks)
(b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at
0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate
dz.
You must check the conditions of any results you use.
(d) Let U C. Calculate
Liz-1ym dz,
(z - 1) 10
(5 marks)
where 2 is the same as the previous part. You must check the conditions of any
results you use.
(4 marks)
(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it
means for this singularity to be a pole of order k.
(2 marks)
(b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given
by
1
res (f, w):
=
Z
dk
(k-1)! >wdzk−1
lim
-
[(z — w)* f(z)] .
(5 marks)
(c) Using the previous part, find the singularity of the function
9(z) =
COS(πZ)
e² (z - 1)²'
classify it and calculate its residue.
(5 marks)
(d) Let g(x)=sin(211). Find the residue of g at z = 1.
(3 marks)
(e) Classify the singularity of
cot(z)
h(z) =
Z
at the origin.
(5 marks)
Chapter 1 Solutions
FINITE MATH.F/BUS,ECON,LIFE..-ACCESS
Ch. 1.1 - According to equality property 2, multiplying both...Ch. 1.1 - Replace ? with < or > in each of the following:...Ch. 1.1 - The solution to Example 5B shows the graph of the...Ch. 1.1 - Solve and check: 3x2(2x5)=2(x+3)8Ch. 1.1 - Solve and check: x+13x4=12Ch. 1.1 - Matched Problem 3 If a cardboard box has length L,...Ch. 1.1 - Replace each question mark with either < or >....Ch. 1.1 - (A) Write 7,4 as a double inequality and graph....Ch. 1.1 - Solve and graph: 3(x1)5(x+2)5Ch. 1.1 - Solve and graph: 83x57
Ch. 1.1 - Mary paid 8.5 sales tax and a 190 title and...Ch. 1.1 - How many bike computers would a company have to...Ch. 1.1 - What net annual salary in 1973 would have had the...Ch. 1.1 - In Problems 1-6, solve for x. 5x+3=x+23Ch. 1.1 - In Problems 1-6, solve for x. 7x6=5x24Ch. 1.1 - In Problems 1-6, solve for x. 9(4x)=2(x+7)Ch. 1.1 - In Problems 1-6, solve for x. 3(x+6)=52(x+1)Ch. 1.1 - In Problems 1-6, solve for x. x+14=x2+5Ch. 1.1 - In Problems 1-6, solve for x. 2x+135x2=4Ch. 1.1 - In Problems 7-12, write the interval as an...Ch. 1.1 - In Problems 7-12, write the interval as an...Ch. 1.1 - In Problems 7-12, write the interval as an...Ch. 1.1 - In Problems 7-12, write the interval as an...Ch. 1.1 - In Problems 7-12, write the interval as an...Ch. 1.1 - In Problems 7-12, write the interval as an...Ch. 1.1 - In Problems 13-18, write the solution set using...Ch. 1.1 - In Problems 13-18, write the solution set using...Ch. 1.1 - In Problems 13-18, write the solution set using...Ch. 1.1 - In Problems 13-18, write the solution set using...Ch. 1.1 - In Problems 13-18, write the solution set using...Ch. 1.1 - In Problems 13-18, write the solution set using...Ch. 1.1 - In Problems 19-32, find the solution set. x4+12=18Ch. 1.1 - In Problems 19-32, find the solution set. m34=23Ch. 1.1 - In Problems 19-32, find the solution set. y532Ch. 1.1 - In Problems 19-32, find the solution set. x456Ch. 1.1 - In Problems 19-32, find the solution set....Ch. 1.1 - In Problems 19-32, find the solution set....Ch. 1.1 - In Problems 19-32, find the solution set....Ch. 1.1 - In Problems 19-32, find the solution set. 34x=5x+1Ch. 1.1 - In Problems 19-32, find the solution set. 3y4y3Ch. 1.1 - In Problems 19-32, find the solution set. x22x5Ch. 1.1 - In Problems 19-32, find the solution set. x5x6=65Ch. 1.1 - In Problems 19-32, find the solution set. y4y3=12Ch. 1.1 - In Problems 19-32, find the solution set. m5335m2Ch. 1.1 - In Problems 19-32, find the solution set. u223u3+2Ch. 1.1 - In Problems 33-36, solve and graph. 23x714Ch. 1.1 - In Problems 33-36, solve and graph. 45x+621Ch. 1.1 - In Problems 33-36, solve and graph. 495C+3268Ch. 1.1 - In Problems 33-36, solve and graph. 123t+511Ch. 1.1 - In Problems 37-42, solve for the indicated...Ch. 1.1 - In Problems 37-42, solve for the indicated...Ch. 1.1 - In Problems 37-42, solve for the indicated...Ch. 1.1 - In Problems 37-42, solve for the indicated...Ch. 1.1 - In Problems 37-42, solve for the indicated...Ch. 1.1 - In Problems 37-42, solve for the indicated...Ch. 1.1 - In Problems 43 and 44, solve and graph. 347x18Ch. 1.1 - In Problems 43 and 44, solve and graph. 1083u6Ch. 1.1 - If both a and b are positive numbers and b/a. is...Ch. 1.1 - If both a and b are negative numbers and b/a is...Ch. 1.1 - Ticket sales. A rock concert brought in $432,500...Ch. 1.1 - Parking meter coins. An all-day parking meter...Ch. 1.1 - IRA. You have $500,000 in an IRA (Individual...Ch. 1.1 - IRA. Refer to Problem 49. How should you divide...Ch. 1.1 - Car prices. If the price change of cars parallels...Ch. 1.1 - Home values. If the price change in houses...Ch. 1.1 - Retail and wholesale prices. Retail prices in a...Ch. 1.1 - Retail and sale prices. Sale prices in a...Ch. 1.1 - Equipment rental. A golf course charges $52 for a...Ch. 1.1 - Equipment rental. The local supermarket rents...Ch. 1.1 - Sales commissions. One employee of a computer...Ch. 1.1 - Sales commissions. A second employee of the...Ch. 1.1 - Break-even analysis. A publisher for a promising...Ch. 1.1 - Break-even analysis. The publisher of a new book...Ch. 1.1 - Break-even analysis. The publisher in Problem 59...Ch. 1.1 - Break-even analysis. The publisher in Problem 60...Ch. 1.1 - Wildlife management. A naturalist estimated the...Ch. 1.1 - Temperature conversion. If the temperature for a...Ch. 1.1 - Psychology. The IQ (intelligence quotient) is...Ch. 1.1 - Psychology. Refer to Problem 65. If the IQ of a...Ch. 1.2 - (A) As noted earlier, (4,3) is a solution of the...Ch. 1.2 - (A) Graph y=x+b for b=5,3,0,3 and 5 simultaneously...Ch. 1.2 - Graph:4x3y=12Ch. 1.2 - Graph 4x-3y=12 on a graphing calculator and find...Ch. 1.2 - (A) Graph x=5andy=3 simultaneously in the same...Ch. 1.2 - Find the slope of the line through each pair of...Ch. 1.2 - Write the equation of the line with slope 12 and y...Ch. 1.2 - (A) Find an equation for the line that has slope...Ch. 1.2 - Answer parts (A) and (B) in Example 7 for fixed...Ch. 1.2 - At a price of $12.59 per box of grapefruit, the...Ch. 1.2 - \ Problems 1-4 refer to graphs (A)-(D). Identify...Ch. 1.2 - Problems 1-4 refer to graphs (A)-(D). Identify the...Ch. 1.2 - Problems 1-4 refer to graphs (A)-(D). Identify the...Ch. 1.2 - Problems 1-4 refer to graphs (A)-(D). Identify the...Ch. 1.2 - In Problems 5-8, sketch a graph of each equation...Ch. 1.2 - In Problems 5-8, sketch a graph of each equation...Ch. 1.2 - In Problems 5-8, sketch a graph of each equation...Ch. 1.2 - In Problems 5-8, sketch a graph of each equation...Ch. 1.2 - In Problems 9-14, find the slope and y intercept...Ch. 1.2 - In Problems 9-14, find the slope and y intercept...Ch. 1.2 - In Problems 9-14, find the slope and y intercept...Ch. 1.2 - In Problems 9-14, find the slope and y intercept...Ch. 1.2 - In Problems 9-14, find the slope and y intercept...Ch. 1.2 - In Problems 9-14, find the slope and y intercept...Ch. 1.2 - In Problems 15-20, find the slope and x intercept...Ch. 1.2 - In Problems 15-20, find the slope and x intercept...Ch. 1.2 - In Problems 15-20, find the slope and x intercept...Ch. 1.2 - In Problems 15-20, find the slope and x intercept...Ch. 1.2 - In Problems 15-20, find the slope and x intercept...Ch. 1.2 - In Problems 15-20, find the slope and x intercept...Ch. 1.2 - In Problems 21-24, write an equation of the line...Ch. 1.2 - In Problems 21-24, write an equation of the line...Ch. 1.2 - In Problems 21-24, write an equation of the line...Ch. 1.2 - In Problems 21-24, write an equation of the line...Ch. 1.2 - In Problems 25-28, use the graph of each line to...Ch. 1.2 - In Problems 25-28, use the graph of each line to...Ch. 1.2 - In Problems 25-28, use the graph of each line to...Ch. 1.2 - In Problems 25-28, use the graph of each line to...Ch. 1.2 - In Problems 29-34, sketch a graph of each equation...Ch. 1.2 - In Problems 29-34, sketch a graph of each equation...Ch. 1.2 - In Problems 29-34, sketch a graph of each equation...Ch. 1.2 - In Problems 29-34, sketch a graph of each equation...Ch. 1.2 - In Problems 29-34, sketch a graph of each equation...Ch. 1.2 - In Problems 29-34, sketch a graph of each equation...Ch. 1.2 - In Problems 35-40, find the slope of the graph of...Ch. 1.2 - In Problems 35-40, find the slope of the graph of...Ch. 1.2 - In Problems 35-40, find the slope of the graph of...Ch. 1.2 - In Problems 35-40, find the slope of the graph of...Ch. 1.2 - In Problems 35-40, find the slope of the graph of...Ch. 1.2 - In Problems 35-40, find the slope of the graph of...Ch. 1.2 - Given Ax+By=12, graph each of the following three...Ch. 1.2 - Given Ax+By=24, graph each of the following three...Ch. 1.2 - Graph y=25x+200,x0.Ch. 1.2 - Graph y=40x+160,x0.Ch. 1.2 - (A) Graph y=1.2x4.2 in a rectangular coordinate...Ch. 1.2 - (A) Graph y=0.8x+5.2 in a rectangular coordinate...Ch. 1.2 - In Problems 47-50, write the equations of the...Ch. 1.2 - In Problems 47-50, write the equations of the...Ch. 1.2 - In Problems 47-50, write the equations of the...Ch. 1.2 - In Problems 47-50, write the equations of the...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 51-58, write the slope-intercept form...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - In Problems 59-66, (A) Find the slope of the line...Ch. 1.2 - Discuss the relationship among the graphs of the...Ch. 1.2 - Discuss the relationship among the graphs of the...Ch. 1.2 - Cost analysis. A donut shop has a fixed cost of...Ch. 1.2 - Cost analysis. A small company manufactures picnic...Ch. 1.2 - Cost analysis. A plant can manufacture 80 golf...Ch. 1.2 - Cost analysis. A plant can manufacture 50 tennis...Ch. 1.2 - Business-Markup policy. A drugstore sells a drug...Ch. 1.2 - Business-Markup policy. A clothing store sells a...Ch. 1.2 - Business-Depreciation. A farmer buys a new tractor...Ch. 1.2 - Business-Depreciation. A charter fishing company...Ch. 1.2 - Boiling point. The temperature at which water...Ch. 1.2 - Boiling point. The temperature at which water...Ch. 1.2 - Flight conditions. In stable air, the air...Ch. 1.2 - Flight navigation. The airspeed indicator on some...Ch. 1.2 - Demographics. The average number of persons per...Ch. 1.2 - Demographics. The median household income divides...Ch. 1.2 - Cigarette smoking. The percentage of female...Ch. 1.2 - Cigarette smoking. The percentage of male...Ch. 1.2 - Supply and demand. At a price of $9.00 per bushel,...Ch. 1.2 - Supply and demand. At a price of $3.20 per bushel,...Ch. 1.2 - Physics. Hooke’s law states that the relationship...Ch. 1.2 - Physics. The distance d between a fixed spring and...Ch. 1.3 - As illustrated in Example 1A, the slope m of a...Ch. 1.3 - As stated previously, we used linear regression to...Ch. 1.3 - The equation a=28.55w+118.7 expresses BSA for...Ch. 1.3 - A 400-pound load of grain is dropped from an...Ch. 1.3 - Prices for emerald-shaped diamonds from an online...Ch. 1.3 - Using the model of Example 4, estimate the...Ch. 1.3 - Figure 5 shows the scatter plot for white spruce...Ch. 1.3 - Ideal weight. Dr. J. D. Robinson published the...Ch. 1.3 - Ideal weight. Dr. J. D. Robinson also published...Ch. 1.3 - Underwater pressure. At sea level, the weight of...Ch. 1.3 - Underwater pressure. Refer to Problem 3. In fresh...Ch. 1.3 - Rate of descent Parachutes. At low altitudes, the...Ch. 1.3 - Rate of descent Parachutes. The U.S Army is...Ch. 1.3 - Speed of sound. The speed of sound through air is...Ch. 1.3 - Speed of sound. The speed of sound through sea...Ch. 1.3 - Energy production. Table 5 lists U.S. fossil fuel...Ch. 1.3 - Energy consumption. Table 6 lists U.S. fossil fuel...Ch. 1.3 - Cigarette smoking. The data in Table 7 shows that...Ch. 1.3 - Cigarette smoking. The data in Table 7 shows that...Ch. 1.3 - Licensed drivers. Table 8 contains the state...Ch. 1.3 - Licensed drivers. Table 9 contains the state...Ch. 1.3 - Net sales. A linear regression model for the net...Ch. 1.3 - Operating income. A linear regression model for...Ch. 1.3 - Freezing temperature. Ethylene glycol and...Ch. 1.3 - Freezing temperature. A linear regression model...Ch. 1.3 - Forestry. The figure contains a scatter plot of...Ch. 1.3 - Forestry. The figure contains a scatter plot of...Ch. 1.3 - Undergraduate enrollment. Table 12 lists fall...Ch. 1.3 - Graduate enrollment. Table 13 lists fall graduate...Ch. 1.3 - Climate. Find a linear regression model for the...Ch. 1.3 - Climate. Find a linear regression model for the...Ch. 1.3 - Olympic Games. Find a linear regression model for...Ch. 1.3 - Olympic Games. Find a linear regression model for...Ch. 1.3 - Supply and demand. Table 16 contains price supply...Ch. 1.3 - Supply and demand. Table 17 contains price supply...Ch. 1 - Solve2x+3=7x11.Ch. 1 - Solvex12x33=12.Ch. 1 - Solve2x+5y=9fory.Ch. 1 - Solve3x4y=7forx.Ch. 1 - Solve Problems 5-7 and graph on a real number...Ch. 1 - Solve Problems 5-7 and graph on a real number...Ch. 1 - Solve Problems 5-7 and graph on a real number...Ch. 1 - Sketch a graph of 3x+2y=9.Ch. 1 - Write an equation of a line with x intercept 6 and...Ch. 1 - Sketch a graph of 2x3y=18. What are the intercepts...Ch. 1 - Write an equation in the form y=mx+b for a line...Ch. 1 - Write the equations of the vertical line and the...Ch. 1 - Write the equation of a line through each...Ch. 1 - Write the equation of the line through the two...Ch. 1 - Solve Problems 15-19. 3x+25=5xCh. 1 - Solve Problems 15-19. u5=u6+65Ch. 1 - Solve Problems 15-19. 5x34+x2=x24+1Ch. 1 - Solve Problems 15-19. 0.05x+0.2530x=3.3Ch. 1 - Solve Problems 15-19. 0.2x3+0.05x=0.4Ch. 1 - Solve Problems 20-24 and graph on a real number...Ch. 1 - Solve Problems 20-24 and graph on a real number...Ch. 1 - Solve Problems 20-24 and graph on a real number...Ch. 1 - Solve Problems 20-24 and graph on a real number...Ch. 1 - Solve Problems 20-24 and graph on a real number...Ch. 1 - Given Ax+By=30, graph each of the following cases...Ch. 1 - Describe the graphs of x=3 and y=2 Graph both...Ch. 1 - Describe the lines defined by the following...Ch. 1 - Solve Problems 28 and 29 for the indicated...Ch. 1 - Solve Problems 28 and 29 for the indicated...Ch. 1 - For what values of a and b is the inequality a+bba...Ch. 1 - If a and b are negative numbers and ab, then is...Ch. 1 - Graph y=mx+bandy=1mx+b simultaneously in the same...Ch. 1 - Investing. An investor has $300,000 to invest. If...Ch. 1 - Break-even analysis. A producer of educational...Ch. 1 - Sports medicine. A simple rule of thumb for...Ch. 1 - Linear depreciation. A bulldozer was purchased by...Ch. 1 - Business Pricing. A sporting goods store sells...Ch. 1 - Income. A salesperson receives a base salary of...Ch. 1 - Price demand. The weekly demand for mouthwash in a...Ch. 1 - Freezing temperature. Methanol, also known as wood...Ch. 1 - High school dropout rates. Table 2 gives U.S. high...Ch. 1 - Consumer Price Index. The U.S. Consumer Price...Ch. 1 - Forestry. The figure contains a scatter plot of 20...
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