Introduction to Mathematical Analysis Exam Preparation Guide
Course Introduction
Introduction to Mathematical Analysis provides a rigorous foundation in the principles of real analysis, focusing on the structure and properties of the real number system. The course covers key topics such as sequences and series, limits, continuity, differentiation, and integration, with an emphasis on formal proofs and logical reasoning. Students will explore the underpinnings of calculus and develop the mathematical maturity necessary to rigorously analyze functions and their behaviors. The course is designed to deepen understanding of fundamental concepts and to prepare students for more advanced studies in mathematics and related fields.
Recommended Textbook
Introductory Mathematical Analysis for Business Economics and the Life and Social Sciences 14th
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Chapter 1: Review of Algebra
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Sample Questions
Q1) Sara and Jeff have agreed to pool their savings when they have saved the same amount of money.Sara can save $40 a week but she must give her first $65 to her mom.Four weeks ago Jeff started saving $25 a week.When will they pool their savings? How much will they each have saved?
Answer: 11 weeks: $375
Q2) C = 2 r is the formula for the circumference of a circle having a radius r.What is the radius of a bicycle wheel which has a circumference C?
Answer: 11ea83f6_62b2_0a18_b5a5_417e11e6f882_TB6578_11_TB6578_11
Q3) Joe and Sue have agreed to pool their savings when they have saved the same amount of money.Joe can save $125 a week but he must use his first $375 to pay off his car.Sue has been saving $75 a week for 13 weeks already.When will they pool their savings? How much will they each have saved?
Answer: 27 weeks; $3000
Q4) Multiply and simplify: (3x + 1)(2x - 4)
Answer: 6 11ea83f6_629e_a9a2_b5a5_33f507342524_TB6578_11_TB6578_11 - 10x - 4
Q5) Multiply and simplify: (t + 2s)(t - 2s)
Answer: 11ea83f6_62a4_4eb7_b5a5_6f5b9d944a66_TB6578_11_TB6578_1111ea83f6_62a4_4eb8_b5a5_d1790b23571b_TB6578_11_TB6578_11
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Chapter 2: Applications and More Algebra
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Sample Questions
Q1) Hooke's Law of springs states that the force F (in pounds)required to stretch a spring x inches beyond its natural length is F = kx,where k is the "spring constant" for a given spring.If a certain spring has a spring constant k of 6.5 and 13 F 26,what are the corresponding values for x?
Answer: 2 x 4
Q2) Rooms in a 130 unit student housing complex are fully rented at $300 per month.The university is raising rates to produce $1920 more in monthly income from these units.Each $10 per month increase will create two vacancies with no possibility of filling them.What should be charged for the new rent?
Answer: $330 or $620
Q3) The sum of two integers is 21.The second integer is three more than twice the first integer.Find both of the integers.
Answer: 6,15
Q4) Suppose a person has $23 in his pocket.What is the maximum number of pizzas that person can order if each pizza costs $4.39?
Answer: 5
Q5) Solve: Six plus three times x is a number 12 units from zero. Answer: x = 2 or x = -6
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Chapter 3: Functions and Graphs
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Sample Questions
Q1) Suppose the yearly supply function for paintings from an artist is p = 3000x.
(a)How many paintings per year will be supplied if the price is $21,000 per painting?
(b)How many paintings per year will be supplied if the price is $51,000 per painting?
(c)How does the amount supplied change as the price increases?
Answer: (a)7 paintings per year
(b)17 paintings per year
(c)Amount supplied increases as the price increases.
Q2) Find the domain of the function f(x)= 6.
Answer: all real numbers
Q3) A daily round trip train ticket to the city costs $4.25.Let x represent the passenger's income and y represent the cost of a daily round trip train ticket.Write an equation which represents the relationship between the cost of a daily round trip ticket and a passenger's income; describe the graph of this equation,and identify the intercepts. Answer: y = 4.25; horizontal line; no x-intercept; y-intercept (0,4.25)
Q4) Find an equation of the plane that is parallel to the y,z-plane and that passes through the point (1,2,3).
Answer: x = 1
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Chapter 4: Lines parabolas and Systems
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Sample Questions
Q1) Suppose the variables q and p are linearly related such that p = 3 when q = 20,and p = 5 when q = 15.Find p when q = 12.
Q2) The demand function for a manufacturer's product is p = f(q)= 600 - 2q,where p is the price (in dollars)per unit when q units are demanded (per week).Find the level of production that maximizes the manufacturer's total revenue and determine this revenue.
Q3) Tickets to an opera at the Masonic Auditorium cost $14 for main floor seats and $10 for the balcony seats.If $8600 must be collected to meet expenses,what is an equation for the possible combinations of ticket sales to cover costs?
Q4) The demand function for a manufacturer's product is p = f(q)= 800 - 2q,where p is the price (in dollars)per unit when q units are demanded (per week).Find the level of production that maximizes the manufacturer's total revenue.
A) 100
B) 125
C) 150
D) 175
E) 200
Q5) Suppose f is a linear function such that f(0)= 6 and f(3)= 4.Find f(x).
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Chapter 5: Exponential and Logarithmic Functions
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Sample Questions
Q1) log 0.001 =
A) -1
B) -2
C) -3
D) -4
E) -5
Q2) Assume your savings consist of a $5000 investment which is guaranteed to increase by 7% every year and $800 cash in a safe at your home.
(a)Make a table of the value of your investment at 0 to 3 years.
(b)Make a table of the value of total savings at 0 to 3 years.
(c)How could you use the graph of (a)to make a graph of (b)? Verify your answer using a graphing calculator.
Q3) What is the sum of the Richter Scale measurement of an earthquake which is 250,000 times the intensity of a zero-level earthquake and an earthquake with intensity twice the intensity of a zero-level earthquake? Write as an expression involving logarithms.Simplify by combining logarithms and then use a calculator to evaluate.
Q4) Find x: ln x = 1
Q5) Find x: logx = 0
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Chapter 6: Mathematics of Finance
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Sample Questions
Q1) Find the present value of an ordinary annuity with monthly payments of $125 for 10 years at 4.5% compounded monthly.
Q2) At an annual rate of 8% compounded continuously,in how many years would it take for a principal to double?
Q3) Suppose a truck costing $36,000 is to be replaced at the end of 10 years,at which time it will have a resale value of $12,000.In order to provide money at the time for a new truck costing $40,000,a sinking fund is set up into which equal payments are placed at the end of every six months.If the fund earns 6% compounded semiannually,what should each payment be?
Q4) $200 is invested at the rate of 3.5% compounded quarterly for six and a half years.How many terms are in the geometric sequence formed by the amounts at the end of each quarter?
Q5) A $5000 loan is to be repaid over three years by equal payments due at the end of every quarter.If interest is at the rate of 20% compounded quarterly,determine (a)the quarterly payment and (b)the total interest paid.
Q6) Find the future value of an annuity due with quarterly payments of $550 for 32 years at 3.75% compounded quarterly.
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Chapter 7: Matrix Algebra
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Sample Questions
Q1) What is the change in sales between 1994 and 1995?
Q2) EO
Q3) The spa staff determines that a client should take 190% of the RDA of vitamin A,150% of the RDA of vitamin C,and 120% of the RDA of vitamin D each day.How many tablets of each supplement should he take each day?
Q4) Use the matrix operations on a graphing calculator to find her income if increasing it by 20% would cause an increase in profit of 50%.(Round to the nearest thousand dollars.)
Q5) A boom in the economy causes an across the board increase of 10% in energy demands in the area.Calculate a new demand matrix which reflects this increase.
Q6) Give an example of 2 matrices A and B where A × B B × A.
Q7) The spa staff determines that a client should take 140% of the RDA of vitamin A,140% of the RDA of vitamin C,and 190% of the RDA of vitamin D each day.How many tablets of each supplement should he take each day?
Q8) A client wants 27 blocks of high-risk stock,27 blocks of moderate-risk stock,and 41 blocks of low-risk stock.Use the matrix operations on a graphing calculator to find how many of each portfolio should be suggested.
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Chapter 8: Linear Programming
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Sample Questions
Q1) To make some extra money,you make two types of picture frames,type A and type B,for sale.You have an initial start-up expense of $75.The production cost for type A is $3.60 per frame,and the production cost for type B is $5.20 per frame.The price for type A is $6.00 per frame and the price for type B is $10.00 per frame.Let x be the number of type A and y be the number of type B produced and sold.Write an inequality describing revenue less than cost.Solve the inequality and describe the region.Also,describe what this means in terms of frames.
Q2) Use the simplex method to solve the following problem: United Blimpo Co.produces two types of exercise devices,regular and heavy-duty,each of which requires in its manufacture the use of two machines,A and B.A regular model requires the use of machine A for 2 hours and machine B for 3 hours.A heavy-duty model requires the use of machine A for 3 hours and machine B for 3 hours.Machine A can be used at most 18 hours a day and machine B can be used at most 21 hours a day.If the profits on the regular and heavy-duty models are $20 and $15,respectively,and United Blimpo Co.can sell all it produces,how many of each model should be produced per day in order to realize maximum profit? What is the maximum profit per day?
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Chapter 9: Introduction to Probability and Statistics
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Sample Questions
Q1) In a certain state,all drivers are given a driver's license code consisting of 2 letters followed by 6 numerals.How many different drivers license codes are possible?
Q2) An urn contains three red and two green marbles,and a second urn contains two red and two green marbles.An urn is selected at random and a marble is randomly drawn from it.The marble is green.What is the probability that it came from the first urn?
Q3) If a fair die is rolled three times,find the probability that a 3 or 5 comes up each time.
Q4) An urn contains two red and three green marbles.Two marbles are randomly drawn in succession without replacement.Determine the probability that (a)the first marble is red and the second is green; (b)both marbles are red.
Q5) A college mathematics club of 30 students needs to elect new officers.If there are 3 positions available and no one can serve in more than one position,how many different slates of candidates are possible?
Q6) Ten tickets numbered from 1 to 10 are placed in a hat.If two tickets are randomly drawn with replacement,find the probability the sum is 15.
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Chapter 10: Additional Topics in Probability
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Sample Questions
Q1) Suppose that you pay $1 to play a game in which a pair of fair dice are rolled.For each 2 that shows on a die you receive $2.To the nearest cent,your expected loss on each play is
A) $0.67.
B) $0.33.
C) $0.25.
D) $0.15.
E) $0.07.
Q2) If a person watches a certain TV daily evening news program on one evening,then the probability that the person watches that program the next evening is 0.7.However,if the person does not watch the program one evening,then the probability that the person watches the program the next evening is 0.2.
(a)If the person watches the program on Monday,what is the probability that the person watches the program on Wednesday?
(b)If 20% of the population watches the program on Thursday,what percentage can be expected to watch on Friday?
Q3) A fair die is rolled four times.What is the probability that exactly three 2's appear?
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Chapter 11: Limits and Continuity
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Sample Questions
Q1) The solution of (x + 9)(x - 1)(x - 5) 0 is
A) x -9.
B) -9 x 1.
C) x 5.
D) x -9, 1 x 5.
E) -9 x 1, x 5.
Q2) Find: lim 15 x 8
Q3) Solve: (x - 1)(x + 2)(x - 3) 0
Q4) Find the value(s)of x for which f(x)= 10 is discontinuous.
Q5) Suppose that the rent for an apartment in a city with rent control is $740 per month.Show that the rent function R(t)= 740 is continuous at t = 3.
Q6) Use the definition of continuity to show f(x)= 2 is continuous at x = 5.
Q7) Use the definition of continuity to show f(x)= 2x + 3 is continuous at x = -1.
Q8) Suppose that the rent for an apartment in a city with rent control is $480 per month.Show that the rent function R(t)= 480 is continuous at t = 7.
Q9) Solve: (x + 6)(x - 1)(x - 4) 0
Q10) Find: \(\lim_{x \rightarrow 5}\) \(\frac{x^{2}}{h}\)
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Chapter 12: Differentiation
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Sample Questions
Q1) If a function is continuous at x = a,then it is differentiable at x = a.
A)True
B)False
Q2) Find the rate of change of the area of a square with respect to its one side.Evaluate it when side = 4 units.
Q3) If the revenue function is given by R(x)= 10x,find the derivative of the revenue function (this is called the marginal revenue).
Q4) The concentration of acid in a buffer solution is given by A = 0.8.Graph this function on your graphing calculator and describe its slope.
Q5) A catering company will plan a group dinner for groups of 20 or larger.If the group is exactly 20,the cost is $15 per person.The company offers to discount each person's ticket by $0.50 for each addition to the group above 20.The revenue for the catering company is given by: R(x)= (20 + x)(15 -0.5x)where x is the number of additional people above 20.Find the marginal revenue function using the product rule.
Q6) The total revenue r for selling x number of bicycles is given by r = 380x.Determine the relative and percentage rates of change of revenue with respect to x when 4 bicycles are sold.
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Chapter 13: Additional Differentiation Topics
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Q1) Find an equation of the tangent line to the curve y = ln(x + 3)when x = -2.
Q2) Find y' if ln(xy)+ y = 2.
Q3) Determine the point elasticity of the demand equation p = -3q + 120,where p > 0 and q > 0.
Q4) The total cost for a product is given by C(x)= 900 + 8 ln(3x + 7).Find the rate of change of C''(x).
Q5) The total cost for a product is given by C(x)= 36 + 5 ln(3x + 9).Find the rate of change of C''(x).
Q6) Determine the point elasticity of the demand equation pq + p + 50q = 10,000 when p = 200.
Q7) The total cost for a product is given by C(x)= 212 + 3 ln(2x + 1).Find the rate of change of C''(x).
Q8) Find the point elasticity of the demand equation p = 15 - 0.3q for the value q = 300,and determine whether demand is elastic,inelastic or has unit elasticity.
Q9) If the position of a particle moving along the x-axis at a given time is given by x(t)= 5t + 11,find all the higher-order derivatives of this function.
Q10) Determine the point elasticity of the demand equation pq = 81,where p > 0 and q > 0.
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Chapter 14: Curve Sketching
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Sample Questions
Q1) A company owns an apartment building containing 100 units.If the company charges $400 per month rent,then all units can be rented out,and for every increase of $20 per month in rent,the company will lose one customer.What rent should be charged to maximize revenue?
Q2) A cable TV company has 200 customers paying $10 each month.For each $1 reduction in price,the company attracts 50 more customers.Find the price that yields maximum revenue.
Q3) A rectangular plot adjacent to a stream is to be fenced in by using the stream as one side of the enclosed area.If 2000 ft of fencing are to be used,find the maximum area that can be enclosed.Assume the answer is in square feet.
A) 400,000
B) 500,000
C) 600,000
D) 700,000
E) 800,000
Q4) A cable TV company has 500 customers paying $20 each month.For each $1 reduction in price,the company attracts 50 more customers.Find the price that yields maximum revenue.
Q5) Find all the critical values of f(x)= x ln x.
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Chapter 15: Integration
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Sample Questions
Q1) By using differentials,an approximation of ln(1.03)is
A) -0.01.
B) 0.01.
C) 0.02.
D) 0.03.
E) 0.04.
Q2) Use differentials to approximate ln(0.99).
Q3) Use differentials and C = 250 + 0.30x to approximate the change in the cost C (in dollars)to produce x pounds of candy if the number of pounds of candy increases from 10 pounds to 10.6 pounds.
Q4) If y'' = 6x + 2 and y'(1)= 2 and y(1)= 2,find y.
Q5) Find y subject to the given conditions: y'' = 6x - 2; y'(1)= 2; y(1)= 2.
Q6) If y = x ln x,then dy =
A) 1 + ln x.
B) (1 + ln x) dx.
C) x + ln x.
D) (x + ln x) dx.
E) none of the above
Q7) Use differentials to approximate: ln(0.95).
Page 17
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Chapter 16: Methods and Applications of Integration
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Sample Questions
Q1) If 25% of the initial amount of a radioactive sample remains after 2.5 years,find the decay constant and the half-life of the element.
Q2) Find the present value of a continuous annuity at an annual rate of 9% for five years if the payment at time t is at the annual rate of f(t)= 3000 dollars.Express your answer in terms of e.
Q3) If 80% of the initial amount of a radioactive substance remains after 2 hours,find the decay constant and the half-life of the substance.Express your answers in terms of natural logarithms.
Q4) Suppose that the population of a city in 2000 was 25,400 and in 2005 was 27,300.If the exponential law of growth is assumed,what is the expected population in 2010?
Q5) The population of a certain city was 1,500,000 in the year 1580.The population grew by 5% each year of the next 15 years.The population declined by 10% each year for the rest of the century.What was the population of the city in the year 1600?
Q6) If 55% of the initial amount of a radioactive sample remains after 10 hours,find the decay constant and the half-life of the element.
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Chapter 17: Continuous Random Variables
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Sample Questions
Q1) The life (in hours)of light bulbs of a certain brand is normally distributed with mean 1000 and standard deviation 100.What percentage of such bulbs will burn more than 950 hours?
A) 40.95
B) 72.15
C) 68.34
D) 65.54
E) 69.15
Q2) If Z has a standard normal distribution,find P(Z > -1.81).
Q3) Suppose X is normally distributed with = 80 and = 15.Without using tables,approximate P(50 < X < 80).
Q4) If X is normally distributed with = 40 and = 6,find P(X 37).
Q5) If X is normally distributed with = 150 and = 30,find P(90 < X < 180).
A) 0.1359
B) 0.8185
C) 0.3185
D) 0.8243
E) 0.8173
Q6) If Z has a standard normal distribution,find P(-0.65 < Z < 1.92).
Q7) If X is normally distributed with = 18 and = 2,find P(11 X 16).
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Chapter 18: Multivariable Calculus
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Q1) An open rectangular cardboard box is to have a volume of 4 cubic feet.Find the dimensions of the box so that the amount of cardboard is minimized.
Q2) Use the method of Lagrange multipliers to determine the critical points of f(x,y)= x + 2y subject to the constraint xy = 8.
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