Mathematics for Economists
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Course Introduction
Mathematics for Economists provides students with the essential mathematical tools and techniques used in economic analysis. The course covers topics such as functions, limits, differentiation and integration, comparative statics, optimization, and matrix algebra, with a focus on their application to economic models and problems. Emphasis is placed on developing mathematical reasoning skills and applying quantitative methods to analyze consumer and producer behavior, equilibrium, and policy implications. By the end of the course, students will be equipped to formulate and solve a variety of economic models using mathematical methods.
Recommended Textbook
Introductory Mathematical Analysis for Business Economics and the Life and Social Sciences 14th
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Page 2
Chapter 1: Review of Algebra
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Sample Questions
Q1) Solve 2q - 1 = 2(1 - q)
Answer: q = 11ea83f6_62af_72be_b5a5_b706d3df5f66_TB6578_11_TB6578_11
Q2) One factor of x(y + 6)+ 2(y + 6)is
A) x + 6
B) y + 2
C) x + 2
D) x + y
E) y + 12
Answer: C
Q3) Solve for x: ax + 3 = b(x +2)
Answer: 11ea83f6_62b0_0e15_b5a5_6567eaa559c7_TB6578_11_TB6578_11
Q4) 10 L solution of water and acetic acid contains 10% of the acid.15 L of water was added to the solution.What is the percentage of the acetic acid in the new solution?
Answer: 4%
Q5) Solve the equation y = mx + b for x if m = -2,b = 3,and y = 4.
Answer: 11ea83f6_62b0_3529_b5a5_254837eca35e_TB6578_11_TB6578_11
Q6) A number squared is four more than the opposite of three times the number.What is the number?
Answer: The number is -4 or 1.
Page 3
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Chapter 2: Applications and More Algebra
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Sample Questions
Q1) Simplify: |3 - 8| - |8 - 3|
Answer: 0
Q2) Write the income range for line 3 as an inequality.Write the inequality that represents the tax owed.
Answer: 58,150 < I 121,300; 13,162 < T 32,738.50
Q3) A homeowner must decide whether to buy or rent a garden rototiller.If she rents the machine,the rental fee is $25 per day,and the daily cost to use it is $5 for gas.If she were to buy the machine,the purchase price is $650,and the daily cost is $7 for gas,oil,and maintenance.On which day of use would the rental costs become greater than the ownership costs?
Answer: 29th day
Q4) Ohm's Law in electrical theory states that V = IR,where R is the resistance (in ohms)of an object,V is the potential difference (in volts)across the object,and I is the current (in amperes)that flows through it.If the voltage is 220 volts,what values for the current will result in a resistance that is less than 20 ohms?
Answer: I > 11 amps
Q5) Solve: 3(4x - 1) 2(x + 4)
Answer: x 11ea83f6_62b9_83e4_b5a5_a7ce2a13296f_TB6578_11_TB6578_11
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Chapter 3: Functions and Graphs
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Sample Questions
Q1) An equation of the plane that is parallel to the x,y-plane and that passes through (2,7,3)is
A) x = 2.
B) y = 7.
C) z = 3.
D) x = 7.
E) y = 3.
Answer: C
Q2) Sketch the surface x + y = 3.
Answer: 11ea83f6_62d3_2730_b5a5_995ce730bf3d_TB6578_00_TB6578_00
Q3) To encourage large group sales,a theater charges two rates.If your group is fewer than 10,each ticket costs $7.50.If your group is 10 or more,each ticket costs $7.00.Graph the compound function that represents the cost of buying N tickets.
Answer: 11ea83f6_62d0_67be_b5a5_4593ee7e5c71_TB6578_00_TB6578_00
Q4) Suppose a committee of 6 people is to be selected from a group of 25.How many groups are possible? Represent as a factorial and give the solution.
Answer: 11ea83f6_62ca_9bb5_b5a5_17c514c62e2b_TB6578_11_TB6578_11 = 177,100
Q5) Find the domain of the function f(x)= 6.
Answer: all real numbers
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Chapter 4: Lines parabolas and Systems
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Sample Questions
Q1) 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?
Q2) Find the equation of a line having slope 0.5 and x-intercept = 1.3.
Q3) An investor has $12,000 to purchase stock in two companies.If the first stock sells for $45 per share,and the second stock sells for $62 per share,find an equation that shows the possible ways to purchase the stock.
Q4) A bank charges $10 for a money order.What is an equation for the relationship between the fee F charged and the amount of the money order?
Q5) Determine the equation of the line which is perpendicular to 2x - y + 3 = 0 and has y-intercept 6.
Q6) Suppose f is a linear function with slope -3 and f(2)= 5.Find f(x).
A) f(x) = -3x + 2
B) f(x) = -3x + 5
C) f(x) = -3x + 7
D) f(x) = -3x + 11
E) f(x) = -3x + 13
Q7) Suppose f is a linear function with slope 5 and such that f(1)= 4.Find f(x).
Page 6
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Chapter 5: Exponential and Logarithmic Functions
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Sample Questions
Q1) An earthquake which is 520,000 times as intense as a zero-level earthquake has a magnitude on the Richter Scale which is 2.7 more than the intensity of another earthquake.What is the intensity of the other earthquake?
Q2) Write log(x + 3)in terms of natural logarithms.
Q3) The number of bacteria in a culture is doubling every hour.Currently the culture has 128 bacteria.Make a table of values for the number of bacteria present each hour for 0 to 4 hours.For each hour write an expression for the number of bacteria as a product of 128 and a power of 2.Use the expressions to make an entry in your table for the number of bacteria after t hours.Write a function N for the number of bacteria after t hours.
Q4) Assume that log 4 = 0.6021.Determine the value of log 400.
Q5) Your friend started a savings plan with 3 pennies and each day she saved three times the amount she saved the previous day.Later you started a savings plan with 9 pennies and each day you saved 9 times the amount you saved the previous day.On the day that your friend had been saving for 8 less than 4 times as many days as you,you saved the same amount as your friend.How many days have you been saving?
Q6) Express 2 ln 3 - ln 4 as a single logarithm.
Q7) Find x: log 0.01 = x
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Chapter 6: Mathematics of Finance
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Sample Questions
Q1) At what nominal rate of interest,compounded quarterly,will an investment double in 15 years?
Q2) To purchase land for an industrial site,a company agrees to pay $20,000 down and $10,000 at the end of every six-month period for 10 years.If the interest rate is 10% compounded semiannually,what is the corresponding cash value of the land?
Q3) What is the present value of an annuity of $300 per quarter for five years at an interest rate of 4.5% compounded quarterly?
Q4) You have two investment opportunities.You can invest $6000 at 12% compounded monthly or you can invest $6100 at 12.1% compounded quarterly.Which has the better effective rate of interest? Use a graphing calculator to graph both amounts as a function of time in years.Which is the better investment over twenty years?
Q5) Find the future value of an annuity due with semiannual payments of $6,250 for 21 years at 4.25% compounded semiannually.
Q6) Suppose you deposit $200 at the end of every month into a bank account that pays 6% compounded monthly.After six years,how much will you have?
Q7) At what nominal rate of interest,compounded quarterly,will money double in 10 years?
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Chapter 7: Matrix Algebra
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Sample Questions
Q1) What would her income be if increasing it threefold would cause her profit to be five times as large? (Round to the nearest thousand dollars.)
Q2) A client wants 33 blocks of high-risk stock,27 blocks of moderate-risk stock,and 30 blocks of low-risk stock.How many of each portfolio should be suggested.
Q3) Use the matrix operations on a graphing calculator to show that A + (B + B) + C. C) = (A +
Q4) A group of investors has $500,000 to invest in the stocks of three companies.Company A sells for $50 a share and has an expected growth of 13% per year.Company B sells for $20 per share and has an expected growth of 15% per year.Company C sells for $80 a share and has an expected growth of 10% per year.The group decides to try a new investment strategy which entails buying equal amounts of shares in Company B and Company C,and having a goal of 11.4% growth per year.How many shares of each stock should they buy?
Q5) Find the income earned by the gas industry and its cost for the other forms of energy it uses.Then calculate its profit.
Q6) What is the change in sales between 1995 and 1996?
Q7) Show that A + O = A.
Page 9
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Chapter 8: Linear Programming
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Sample Questions
Q1) Maximize
Z = 4x + y subject to -x + y 2
3x + y 18
x,y 0.
Q2) Maximize
Z = 5x - 3y subject to
2x - y 8
2x - 5y 0
x - y = -2
x,y 0.
Q3) Use the simplex method to maximize
Z = 30x + 50y subject to
2x + y 16
x + 2y 11
x + 3y 15
x,y 0
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Chapter 9: Introduction to Probability and Statistics
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Sample Questions
Q1) Two light bulbs are selected from a box of 20 bulbs.For this experiment,how many sample points are in the sample space?
Q2) An office worker spends each coffee break at the same deli.This deli has the same 8 varieties of coffee every day.To add variety and excitement to his or her day,the office worker decides to randomly choose a coffee variety in the morning and then randomly choose a variety in the afternoon.How many morning coffee - afternoon coffee choices are there?
Q3) If events E and F are independent with P(E)= 0.3 and P(F)= 0.5,find P(E' F).
Q4) If two cards are randomly drawn,without replacement,from a standard deck of 52 cards,find the probability that the second card is not a jack or queen,given that the first card is a jack or queen.
Q5) A certain disease is believed to affect 4% of the population.Results of a new blood test for the disease indicate that 95% of persons who have the disease will have a positive reaction to the test,whereas 5% of those who do not have the disease will also have a positive reaction.What is the probability that a randomly selected person who has a positive reaction will actually have the disease?
Q6) The plane arrives late in L.A.but left Denver as scheduled.
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Chapter 10: Additional Topics in Probability
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Q1) 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?
Q2) If a fair coin is tossed six times,determine the probability that at least two heads show.
Q3) n = 20; p = 0.2
A) = 4.0
B) = 4.3
C) = 4.7
D) = 3.5
Q4) An urn contains two red and eight green marbles.Four marbles are randomly drawn in succession with replacement.Determine the probability that exactly two marbles are red.
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Page 12
Chapter 11: Limits and Continuity
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Sample Questions
Q1) Solve: (x - 1)(x + 2)(x - 3) 0
Q2) Consumers will purchase q units of a certain product when the price is 340 - 2q dollars per unit.How many units must be sold in order that sales revenue be no less than $14,000?
Q3) 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.
Q4) Use the definition of continuity to show f(x)= 2 is continuous at x = 5.
Q5) Solve: (x + 6)(x - 1)(x - 4) 0
Q6) Suppose that the rent for an apartment in a city with rent control is $540 per month.Show that the rent function R(t)= 540 is continuous at t = 3.
Q7) 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.
Q8) Find: lim 15 x 8
Q9) Suppose that the rent for an apartment in a city with rent control is $540 per month.Show that the rent function R(t)= 540 is continuous at t = 5.
Q10) Find: \(\lim_{x \rightarrow 5}\) \(\frac{x^{2}}{h}\)
Page 13
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Chapter 12: Differentiation
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Sample Questions
Q1) Two children are selling cups of lemonade for $0.25 each.If they sell q cups of lemonade,the total revenue r is given by r = 0.25q.Find the marginal revenue.
Q2) A cone has a height of 10 inches.It is being filled with ice cream.Find the rate of change of its volume with respect to the radius of the base of the cone.Evaluate this rate of change when the radius is 12 inches.
Q3) 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.
Q4) 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.
Q5) The height of a boulder just sitting on a 30-foot cliff is given by y = 32.Graph this function on your graphing calculator and describe its slope.
Q6) Differentiate: g(x)= (x - 8)(2x - 7)(x + 5)
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14
Chapter 13: Additional Differentiation Topics
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Sample Questions
Q1) The total cost for a product is given by C(x)= 36 + 5 ln(3x + 9).Find the rate of change of C''(x).
Q2) The total cost for a product is given by C(x)= 900 + 8 ln(3x + 7).Find the rate of change of C''(x).
Q3) Determine the point elasticity of the demand equation p = -3q + 120,where p > 0 and q > 0.
Q4) 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.
Q5) The total cost for a product is given by C(x)= 1750 + 300 ln(x + 2).Find the rate of change of C''(x).
Q6) Determine the point elasticity of the demand equation pq = 81,where p > 0 and q > 0.
Q7) An experiment was set up to find a relationship between weight and systolic blood pressure in children.It was found that p(x),the systolic blood pressure,was approximately p(x)= 17.5(1 + ln x),where x is the child's weight in pounds.Find p'(x).
Q8) The total cost for a product is given by C(x)= 212 + 3 ln(2x + 1).Find the rate of change of C''(x).
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Chapter 14: Curve Sketching
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Sample Questions
Q1) Find two numbers whose sum is 50 and such that the product of one of them and five more than the other is a maximum.
Q2) Find all the critical values of f(x)= x ln x.
Q3) A real estate firm owns 70 apartments.At $200 per month each apartment can be rented.However,for each $10 per month increase,there will be two vacancies with no possibility of filling them.What rent per apartment will maximize monthly revenue?
Q4) An open box is to be made by cutting equal squares from each corner of a 20-in.square piece of cardboard and then folding up the sides.Find the length of the side of the square that must be cut out if the volume of the box is to be maximized.
Q5) Find two positive numbers whose product is 25 and whose sum is a minimum.
Q6) A rectangular plot of ground is to be fenced in so that it has an area of 15,000 square feet.The plot is to be divided in half with a fence parallel to one pair of sides.Find the dimensions of the plot if the least amount of fencing is to be used.
Q7) A company owns an apartment building containing 80 units.If the company charges $300 per month rent,then all units can be rented out,and for every increase of $10 per month in rent,the company will lose one customer.What rent should be charged to maximize revenue?
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Chapter 15: Integration
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Sample Questions
Q1) 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.
Q2) 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.
Q3) 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
Q4) If y'' = 6x + 2 and y'(1)= 2 and y(1)= 2,find y.
Q5) Use differentials to approximate: ln(0.95).
Q6) Find y subject to the given conditions: y'' = 6x - 2; y'(1)= 2; y(1)= 2.
Q7) Use differentials to approximate ln(0.99).
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Chapter 16: Methods and Applications of Integration
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Sample Questions
Q1) 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?
Q2) 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.
Q3) Find the exact area of the region bounded by the graphs of y = -x,y = 2x,y = 1,and y = 2.Also sketch the region.
Q4) 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.
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.
Q7) If 65% of the initial amount of a radioactive sample has decayed after 5 hours,find the decay constant and the half life of the element.
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Page 18
Chapter 17: Continuous Random Variables
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Sample Questions
Q1) If Z has a standard normal distribution,find P(Z > 2 ).
Q2) If Z has a standard normal distribution,find P(-2 Z -1).
Q3) If Z has a standard normal distribution,find P(Z > -1.81).
Q4) The life expectancy (in years)of patients after they have contracted a certain disease is exponentially distributed with k = 0.4.Find the mean life expectancy and the standard deviation.
Q5) If X is normally distributed with = 40 and = 6,find P(X 37).
Q6) If X is normally distributed with = 80 and = 8,find P(X < 74).
A) 0.2734
B) 0.7734
C) 0.2266
D) 0.7732
E) 0.7266
Q7) If Z has a standard normal distribution,find P(|Z| > 1.75).
Q8) If Z has a standard normal distribution,find P(-3 Z -0.5).
Q9) If X is normally distributed with = 50 and = 8,find P(X 44).
Q10) If X is normally distributed with = 100 and = 10,find P(X < 115).
Q11) If Z has a standard normal distribution,find P(0 < Z < 1.48).
Q12) If Z has a standard normal distribution,find P(Z > 0.12). Page 19
Q13) If Z has a standard normal distribution,find P(2.5 Z 3).
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Page 20
Chapter 18: Multivariable Calculus
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Sample Questions
Q1) Use the method of Lagrange multipliers to determine the critical points of f(x,y)= x + 2y subject to the constraint xy = 8.
Q2) 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.
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