Mathematical Methods for Social Sciences
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Course Introduction
This course provides an introduction to fundamental mathematical concepts and techniques commonly utilized in the social sciences, including linear algebra, calculus, probability, optimization, and introductory statistics. Emphasis is placed on applying these methods to model, analyze, and interpret data relevant to fields such as economics, political science, psychology, and sociology. Through theoretical instruction and practical examples, students gain proficiency in using mathematical tools to solve real-world problems and deepen their understanding of social phenomena.
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) Multiply and simplify: (3x + 1)(2x - 4)
Answer: 6 11ea83f6_629e_a9a2_b5a5_33f507342524_TB6578_11_TB6578_11 - 10x - 4
Q2) S = 2 rh is the formula for the surface area S of the curved surface of a cylinder having radius r and height h.You have a rectangular sheet of wrapping paper which has a length l and a width w.What is the radius of the largest-area cylinder having a height l that the wrapping paper will cover?
Answer: 11ea83f6_62b1_e306_b5a5_f9bf2ce7ff95_TB6578_11_TB6578_11
Q3) The volume of a rectangular prism with a square base and height which is 27 times as long as its width is 3 times its width.What are the dimensions of the rectangular prism?
Answer: 11ea83f6_62b6_ecab_b5a5_a9134b6f071c_TB6578_11_TB6578_11 × 11ea83f6_62b6_ecac_b5a5_d79d2629429b_TB6578_11_TB6578_11 × 9
Q4) Solve: 2(t - 1)- 3(t - 4)= 4t
Answer: t = 2
Q5) 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
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Chapter 2: Applications and More Algebra
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Sample Questions
Q1) A company produces and sells q units of its product.If the variable cost is $4/unit,fixed costs are $4800 and the selling price is $28/unit,find the number of units the company must produce to split even (i.e.,zero profit).
Answer: 200 units
Q2) A kitchen cabinet maker wants to establish a list price for a standard cabinet so that the selling price gives a trade discount of 10% of the list price.If he makes a markup of 20% of the selling price,and his cost is $144,what list price should he charge?
Answer: $200
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) Six minus four times x is a number that is at most 14 units from zero.What are the possible values for x?
Answer: -2 x 5
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Chapter 3: Functions and Graphs
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Sample Questions
Q1) To reduce inventory,a department store charges three rates.If you buy 0-5 pairs of socks,the price is $3.50 per pair of socks.If you buy 6-10 pairs of socks,the price is $3.00 per pair.If you buy more than 10,the price is $2.75 per pair.Graph the compound function that represents the cost of buying N pairs of socks.
Answer: 11ea83f6_62d0_67bd_b5a5_b12600ee3621_TB6578_00_TB6578_00
Q2) Find the inverse of the function: f(x)= 8x + 3
Answer: 11ea83f6_62ce_1d94_b5a5_d7ad9b3b631f_TB6578_11_TB6578_11 (x)= 11ea83f6_62ce_1d95_b5a5_ab055db6ee30_TB6578_11_TB6578_11 (x - 3)
Q3) If (f(x)= 3 - 2x,find f(f(x)).
Answer: 4x - 3
Q4) (a)Sketch the graph of y = 2x + 1.(b)Determine the intercepts.
(c)Based on your graph,is y a function of x? If so,state (d)the domain and (e)the range.
Answer: (a) 11ea83f6_62ce_e0fc_b5a5_55dc2c48aa79_TB6578_00_TB6578_00 (b) 11ea83f6_62cf_080d_b5a5_8fded677b738_TB6578_11_TB6578_11 ,(0,1)
(c)y is a function of x
(d)all real numbers
(e)all real numbers
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Chapter 4: Lines parabolas and Systems
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Sample Questions
Q1) Find the slope of the line passing through the points (3,9)and (2,-5).
Q2) A function describing the value of a house purchased for $180,000 after x years of appreciation is estimated to be f(x)= 12,000x + 180,000.Graph the function on your graphing calculator.
Q3) Suppose that a manufacturer will place 1000 units of a product on the market when the price is $10 per unit,and 1400 units when the price is $12 per unit.Find the supply equation for the product assuming the price p and quantity q are linearly related.
Q4) An accountant can complete a simple tax return in about 1.5 hours and a complicated return in about 4 hours.If she works 8 hours per day,find an equation that shows the possible ways that both types of returns can be completed in a 5-day work week.
Q5) Find the slope of the line 3x + 9y - 7 = 0.
Q6) In testing an experimental diet for goats,it was determined that the (average)live weight w (in kilograms)of a goat was statistically a linear function of the number of days d after the diet was started where 0 d 100.The weight of a goat starting the diet was 12 kg and 25 days later it was 20 kg.Determine w as a linear function of d and find the average weight of a goat when d = 80.
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) If $1000 is invested for 2 years at 6% compounded quarterly,then the compound amount at the end of the period is
A) $1120.00. B) $1123.60.
C) $1126.49.
D) $1141.23. E) $1593.84.
Q2) 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?
Q3) How much more on the Richter Scale is an earthquake with intensity 300,000 times the intensity of a zero-level earthquake than an earthquake with intensity 150,000 times 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) If $2000 is invested for 3 years at 8% compounded quarterly,find (a)the compound amount and (b)the compound interest.
Q5) What power of 3 is 36?
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Chapter 6: Mathematics of Finance
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Sample Questions
Q1) You wish to be a millionaire when you retire.To accomplish this you decide to open up an IRA and make equal monthly payments at the end of each month.If the IRA earns 7.5% compounded monthly and you know you will retire in 40 years,what must be the monthly payment to obtain your $1,000,000?
Q2) Find the future value of an ordinary annuity with semiannual payments of $475 for 30 years at 6.15% compounded semiannually.
Q3) If $3,100 is invested at a nominal rate of 6% compounded quarterly for 7 years,find a)the compound amount,and b)the compound interest.
Q4) Find the future value of an ordinary annuity with quarterly payments of $300 for 30 years at 7.5% compounded quarterly.
Q5) After putting $10,000 down on a piece of property,a woman began paying $2500 a quarter for nine years.Given an interest rate of 7.75% compounded quarterly,how much would the property cost if the woman had paid for it in cash?
Q6) A person amortizes a loan of $180,000 for a new home by obtaining a 30-year mortgage at the rate of 8.7% compounded monthly.Find (a)the monthly payment,(b)the total interest charges,and (c)the principal remaining after 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) Use the matrix operations on a graphing calculator to find the total demand for the four types of energy from only the consumers.
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) An employer offers medical,dental,and life insurance to both salaried and unsalaried employees.She describes her annual costs for these benefits in a matrix,and finds that,after employee contributions,only medical insurance requires an employer contribution of $20,000 for salaried and $15,000 for unsalaried employees.Construct a matrix which shows this result.
Q5) What is the change in sales between 1995 and 1996?
Q6) Use the matrix operations on a graphing calculator to find the profit made by the manufacturer on these orders.
Q7) 4(E + 3I)
Q8) Use the matrix operations on a graphing calculator to show that A + C = C + A.
Q9) EF
Page 9
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Chapter 8: Linear Programming
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Sample Questions
Q1) Maximize
Z = 5x - 3y subject to 2x - y 8
2x - 5y 0
x - y = -2 x,y 0.
Q2) The XYZ Corporation produces two models of home computers,the Alpha model and the Beta model.Let x be the number of Alpha models and y the number of Beta models produced at the San Antonio factory per week.If the factory can produce at the most 100 Beta models in a week,write an inequality to describe this situation.Describe the region for the inequality.What additional inequalities can you add to the situation?
Q3) A store sells two types of calculators.In order to cover overhead,it must sell at least 40 calculators total per week,and in order to satisfy distribution requirements,it must sell at least twice as many of type II as type I.Write a system of inequalities to describe the situation.Let x be the number of type I that it sells in a week and y be the number of type II that it sells in a week.Find the region described by the system of linear inequalities.
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Chapter 9: Introduction to Probability and Statistics
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Sample Questions
Q1) A math instructor has twelve students in his class.It so happens that he has exactly 12 desks in his classroom.How many ways can the 12 students be arranged in the 12 desks?
Q2) Five different books are to be arranged horizontally on a bookshelf.(a)In how many ways can this be done? (b)If two are mathematics books and three are accounting books,in how many ways can all the books be arranged if the first two books are to be in mathematics?
Q3) An urn contains five chips numbered from 1 to 5.A chip is randomly drawn.Let E be the event of drawing a 3 and F be the event of drawing a 5.Are E and F independent?
Q4) An urn contains ten marbles numbered 1 through 10.If two marbles are randomly drawn in succession without replacement,determine the probability that (a)the first marble drawn shows 1 and the second shows 2; (b)both show an odd number; (c)at least one marble shows a number greater than 5.
Q5) How many distinguishable arrangements of all letters in the word MISSISSIPPI are possible?
Q6) In how many ways can a four-member committee be selected from a group of seven people?
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Chapter 10: Additional Topics in Probability
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Sample Questions
Q1) n = 42; p = 0.2
A) = 2.59
B) = 5.86
C) = 0.18
D) = 6.71
Q2) A random variable X has a distribution given by f(1)= 0.6,f(2)= 0.2,f(3)= 0.2.The mean of X is
A) 0.33.
B) 0.4.
C) 1.6.
D) 1.8.
E) 3.
Q3) At the time of purchase of its product,a company offers a contract to its customers that guarantees the replacement of the product if it malfunctions within a one-year period.The cost of the contract is $10.The probability that a unit of the product malfunctions within one year of purchase is 0.06.If the cost to the company for replacement of a unit is $160,determine the company's expected gain or loss per contract.
Q4) A random variable X has a distribution given by f(0)= 0.3,f(1)= 0.2,f(2)= 0.5.(a)Find .(b)Find Var(X).
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Chapter 11: Limits and Continuity
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Sample Questions
Q1) 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.
Q2) Use the definition of continuity to show f(x)= 2 is continuous at x = 5.
Q3) 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.
Q4) 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.
Q5) Solve: (x - 1)(x + 2)(x - 3) 0
Q6) Find the value(s)of x for which f(x)= 10 is discontinuous.
Q7) 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.
Q8) 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?
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Chapter 12: Differentiation
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Sample Questions
Q1) Find the rate of change of the area of a square with respect to its one side.Evaluate it when side = 4 units.
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 stereo company sells 300 car stereos per month at a price of $280 per car stereo.Market research indicates that they can sell one additional stereo for each $1 they reduce the price.In this case the total revenue is given by: R(x)= (300 + x)(280 - x)where x is the number of additional stereos above 300.Find the marginal revenue function using the product rule.
Q4) 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.
Q5) If the revenue function is given by R(x)= 11x + 9,find the derivative of the revenue function (this is called the marginal revenue).
Q6) 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.
Q7) Differentiate: g(x)= (x - 8)(2x - 7)(x + 5)
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Chapter 13: Additional Differentiation Topics
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Sample Questions
Q1) Find y' if ln(xy)+ y = 2.
Q2) Determine the point elasticity of the demand equation pq = 81,where p > 0 and q > 0.
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) Determine the point elasticity of the demand equation pq + p + 50q = 10,000 when p = 200.
Q6) The total cost for a product is given by C(x)= 36 + 5 ln(3x + 9).Find the rate of change of C''(x).
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) If the point elasticity of demand for a product is -2.3 and the price of the product increases 3%,what is the approximate percentage decrease in demand?
Q9) 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.
Q10) Find an equation of the tangent line to the curve y = ln(x + 3)when x = -2.
Page 15
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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) 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.
Q4) Find two positive numbers whose product is 25 and whose sum is a minimum.
Q5) 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
Q6) Find all the critical values of f(x)= x ln x.
Page 16
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Chapter 15: Integration
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Sample Questions
Q1) 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
Q2) If y'' = 6x + 2 and y'(1)= 2 and y(1)= 2,find y.
Q3) 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.
Q4) If the marginal cost (in dollars)for a company is C'(x)= 200,and the cost for producing 8 units is $2250,that is C(8)= 2250,then find C(x).
Q5) Use differentials to approximate ln(0.99).
Q6) Use differentials to approximate: ln(0.95).
Q7) Find y subject to the given conditions: y'' = 6x - 2; y'(1)= 2; y(1)= 2.
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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) An ecologist has determined that a local metro-park has food to support up to 200 rabbits at any given time.Now,from observations,he estimates that there are 20 rabbits.One week from now,he estimates that there are 30 rabbits.Assuming logistic growth,find the logistic function that will model this rabbit population.
Q3) The demand equation for a certain product is p = 30 - 0.1q,where p is the price per unit (in dollars)for q units.If its supply equation is p = 0.15q + 5,find the consumers' surplus and the producers' surplus when market equilibrium has been established.
Q4) Health officials are studying the spread of a disease in a city of 50,000 people.At the beginning of the study,1000 people were infected.One week later,2000 people are infected.Assuming logistic growth,determine the number N of people that are infected t weeks after the beginning of the study.
Q5) 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.
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Chapter 17: Continuous Random Variables
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Sample Questions
Q1) If X is normally distributed with = 100 and = 10,find P(X < 118).
Q2) The scores on an examination are normally distributed with = 500 and = 100.What percentage of the scores are less than 360?
Q3) If X is normally distributed with mean and standard deviation ,then what is the approximate probability that - 2 X + 2 ?
A) 0.5
B) 0.68
C) 0.95
D) 0.997
E) 0.999
Q4) If X is normally distributed with = 12 and = 2,find P(12 X 15).
Q5) If Z has a standard normal distribution,find P(-0.65 < Z < 1.92).
Q6) 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.
Q7) If X is normally distributed with = 8 and = 2,find P(3 X 8).
Q8) If Z has a standard normal distribution,find P(Z > -1.81).
Page 19
Q9) If Z has a standard normal distribution,find P(0 < Z < 1.48).
Q10) If Z has a standard normal distribution,find P(-2 Z -1).
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Chapter 18: Multivariable Calculus
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Sample Questions
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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