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Solution Manual for College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 14th Edition

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     L EARNING W ORKSHEETS C OLLEGE M ATHEMATICS FOR B USINESS , E CONOMICS , L IFE S CIENCES , AND S OCIAL S CIENCES F OURTEENTH E DITION Raymond A. Barnett Merritt College Michael R. Ziegler Marquette University Karl E. Byleen Marquette University Christopher J. Stocker Marquette University

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    iii Contents 1 Linear Equations and Graphs 1 1-1 Linear Equations and Inequalities 1 1-2 Graphs and Lines 5 1-3 Linear Regression 11 2 Functions and Graphs 15 2-1 Functions 15 2-2 Elementary Functions: Graphs and Transformations 19 2-3 Quadratic Functions 25 2-4 Polynomial and Rational Functions 39 2-5 Exponential Functions 53 2-6 Logarithmic Functions 59 3 Mathematics of Finance 69 3-1 Simple Interest 69 3-2 Compound and Continuous Compound Interest 75 3-3 FV of an Annuity; Sinking Funds 81 3-4 PV of an Annuity; Amortization 91 4 Systems of Linear Equations; Matrices 103 4-1 Review: Systems of Linear Equations in Two Variables 103 4-2 Systems of Linear Equations and Augmented Matrices 111 4-3 Gauss-Jordan Elimination 117 4-4 Matrices: Basic Operations 125 4-5 Inverse of a Square Matrix 129 4-6 Matrix Equations and Systems of Linear Equations 133 4-7 Leontief Input–Output Analysis 141 5 Linear Inequalities and Linear Programming 147 5-1 and 5-2 Systems of Linear Inequalities 147 5-3 Linear Programming in Two Dimensions: A Geometric Approach 155 6 Linear Programming: The Simplex Method 165 6-1 The Table Method: An Introduction to the Simplex Method 165 6-2 The Simplex Method: Maximization with Problem Constraints of the Form ≤ 169 6-3 The Dual Problem: Minimization with Problem Constraints of the Form ≥ 175 6-4 Maximization and Minimization with Mixed Problem Constraints 181 7 Logic, Sets, and Counting 185 7-1 Logic 185 7-2 Sets 189 7-3 Basic Counting Principles 193 7-4 Permutations and Combinations 197 8 Probability 201 8-1 Sample Spaces, Events, and Probability 201 8-2 Unions, Intersections, and Complement of Events; Odds 207 8-3 Conditional Probability, Intersection, and Independence 213 8-4 Bayes’ Formula 219 8-5 Random Variables, Probability Distribution, and Expected Value 225

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    iv 9 Limits and the Derivative 229 9-1 Introduction to Limits 229 9-2 Infinite Limits and Limits at Infinity 235 9-3 Continuity 241 9-4 The Derivative 245 9-5 Basic Differentiation Properties 251 9-6 Differentials 257 9-7 Marginal Analysis in Business and Economics 261 10 Additional Derivative Topics 269 10-1 The Constant e and Continuous Compound Interest 269 10-2 Derivatives of Exponential and Logarithmic Functions 273 10-3 Derivatives of Products and Quotients 277 10-4 The Chain Rule 283 10-5 Implicit Differentiation 289 10-6 Related Rates 293 10-7 Elasticity of Demand 297 11 Graphing and Optimization 303 11-1 First Derivatives and Graphs 303 11-2 Second Derivatives and Graphs 311 11-3 L’Hôpital’s Rule 323 11-4 Curve-Sketching Techniques 327 11-5 Absolute Maxima and Minima 333 11-6 Optimization 339 12 Integration 343 12-1 Antiderivatives and Indefinite Integrals 343 12-2 Integration by Substitution 351 12-3 Differential Equations; Growth and Decay 359 12-4 The Definite Integral 365 12-5 The Fundamental Theorem of Calculus 369 13 Additional Integration Topics 375 13-1 Area Between Curves 375 13-2 Applications in Business and Economics 381 13-3 Integration by Parts 387 13-4 Other Integration Methods 391 14 Multivariable Calculus 397 14-1 Functions of Several Variables 397 14-2 Partial Derivatives 401 14-3 Maxima and Minima 407 14-4 Maxima and Minima Using Lagrange Multipliers 413 14-5 Method of Least Squares 419 14-6 Double Integrals over Rectangular Regions 427 14-7 Double Integrals over More General Regions 431 15 Markov Chains 437 15-1 Properties of Markov Chains 437 15-2 Regular Markov Chains 443 15-3 Absorbing Markov Chains 447

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    College Mathematics: Learning Worksheets Chapter 1 1 Name ________________________________ Date ______________ Class ____________ Goal: To solve linear equation and linear inequalities In Problems 1–3, solve for the variable: 1. 7 8 2 28 7 8 2 2 28 2 5 8 28 5 8 8 28 8 5 20 5 20 5 5 4 x x x x x x x x x x x + = + + − = + − + = + − = − = = = Section 1-1 Linear Equations and Inequalities Equality Properties: 1. If x y  and a is any real number, then . x a y a    2. If x y  and a is any nonzero real number, then ax ay  and . x y a a  Inequality Properties: 1. If x y  and a is any real number, then . x a y a    2. If x y  and a is any positive real number, then ax ay  and . x y a a  3. If x y  and a is any negative real number, then ax ay  and . x y a a  Interval Notation: A bracket, ] or [, is used if the endpoint is included. A parentheses, ) or (, is used if the endpoint is not included. Infinity, either positive or negative, always uses a parentheses.

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    College Mathematics: Learning Worksheets Chapter 1 2 2. 7 3(6 11) 167 7 18 33 167 25 33 167 25 33 33 167 33 25 200 25 200 25 25 8 y y y y y y y y y                3. 8 10 6 3 6 6 8(6) 10(6) 6 3 48 2 60 48 60 12 m m m m m m m m + = + + = + + = + = + − = In Problems 4–6, solve for the variable and place the final answer in interval notation. 4. 9 4 22 9 18 2 (2, ) x x x + > > > ∞ 5. 5 2 1 13 6 2 12 3 6 6 3 [ 6,3) x x x x − < − + ≤ − < − ≤ > ≥ − − ≤ < − 6. 3 5 3 4 2 4 12 3(12) 12 5(12) 3 4 2 4 4 9 6 15 24 2 12 (12, ) u u u u u u u u            

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    College Mathematics: Learning Worksheets Chapter 1 3 7. Break-even Analysis . A publisher for a promising new novel figures fixed costs (overhead, advances, promotion, copyediting, typesetting, and so on) at $87,000 and variable costs (printing, paper, binding, shipping) at $4.50 for each book produced. If the book is sold to distributors for $28 each, how many must be produced and sold for the publisher to break even? Let x = the number of books produced. Since the break-even point is the point when cost is the same as the revenue: 28 87, 000 4.50 23.50 87, 000 3702.12766 x x x x = + = = Therefore, the publisher must produce 3703 books to break even.

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    College Mathematics: Learning Worksheets Chapter 1 5 Name ________________________________ Date ______________ Class ____________ Goal: To find the equations of lines, x- intercepts, and y- intercepts In Problems 1–12, write the equation of the line in slope-intercept form with the given characteristics: 1. Slope is 8 and y -intercept is (0, 3). Since the slope and y -intercept are given, 8 3. y x = + 2. Slope is –5 and y -intercept is (0, 6).  Since the slope and y -intercept are given, 5 6. y x    Section 1-2 Graphs and Lines Slope of a Line: 2 1 2 1 , y y m x x − = − where   1 1 1 : , P x y and   2 2 2 : , P x y Slope-Intercept Form of a Line: , y mx b   where m is the slope and (0, b ) is the y- intercept. Equation of a line in standard form: Ax By C   , where A and B are not both zero. Horizontal Line: y b  , slope is zero. Vertical Line: x a  , slope is undefined. y- intercept: (0 , b ) x- intercept: ( a, 0)

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    College Mathematics: Learning Worksheets Chapter 1 6 3. Slope is 3 7 and passes through the point (–14, 2). 1 1 ( ) 3 2 ( ( 14)) 7 3 2 6 7 3 8 7 y y m x x y x y x y x − = − − = − − − = + = + 4. Slope is 4 5  and passes through the point (2, –3). 1 1 ( ) 4 ( 3) ( 2) 5 4 8 3 5 5 4 7 5 5 y y m x x y x y x y x                5. Passes through the points (4, 8) and (8, 4). 2 1 2 1 4 8 4 1 8 4 4 y y m x x − − − = = = = − − − 1 1 ( ) 8 1( 4) 8 4 12 y y m x x y x y x y x − = − − = − − − = − + = − + 6. Passes through the points (–1, 4) and (2, –2). 2 1 2 1 2 4 6 2 2 ( 1) 3 y y m x x             1 1 ( ) 4 2( ( 1)) 4 2 2 2 2 y y m x x y x y x y x               

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    College Mathematics: Learning Worksheets Chapter 1 7 7. Passes through the points (0, 6) and (5, 0). 2 1 2 1 0 6 6 5 0 5 y y m x x         1 1 ( ) 6 0 ( 5) 5 6 6 5 y y m x x y x y x           8. Passes through the points (0, –6) and (1, 0). 2 1 2 1 6 0 6 6 0 1 1 y y m x x − − − − = = = = − − − 1 1 ( ) 0 6( 1) 6 6 y y m x x y x y x − = − − = − = − 9. A horizontal line that passes through the point (–2, 8). A horizontal line is parallel to the x -axis and in the form , y b  therefore, the equation is 8. y = 10. A horizontal line that passes through the point (2, –5). A horizontal line is parallel to the x -axis and in the form , y b  therefore, the equation is 5. y   11. A vertical line that passes through the point (–2, 7). A vertical line is parallel to the y -axis and in the form , x a  therefore, the equation is 2. x = − 12. A vertical line that passes through the point (2, –8). A vertical line is parallel to the y -axis and in the form , x a  therefore, the equation is 2. x 

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    College Mathematics: Learning Worksheets Chapter 1 8 In Problems 13–17, find the x -intercept and the y -intercept. Solutions for the x -intercept will be found by setting y = 0, and the y -intercepts will be found by setting x = 0. 13. 3 3 y x    14. 2 2 y x    15. 1 1 2 y x   3(0) 3 3 y y     2(0) 2 2 y y      1 (0) 1 2 1 y y     (0, 3) y -intercept (0, –2) y -intercept (0, –1) y -intercept 0 3 3 3 3 1 x x x      0 2 2 2 2 1 x x x        1 0 1 2 1 1 2 2 x x x       (1, 0) x -intercept (–1, 0) x -intercept (2, 0) x -intercept 16. 4 4 3 y x   17. 3 3 x y    4 (0) 4 3 4 y y     0 3 3 1 y y     (0, –4) y -intercept (0, 1) y -intercept 4 0 4 3 4 4 3 3 x x x       3(0) 3 3 x x      (3, 0) x -intercept (–3, 0) x -intercept

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    College Mathematics: Learning Worksheets Chapter 1 9 18. Graph each line in Problems 13–17. Graph for 13 Graph for 14 Graph for 15 Graph for 16 Graph for 17

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    College Mathematics: Learning Worksheets Chapter 1 10 19. A piece of equipment used in a landfill has an original value of $200,000. After two years of use, the piece of equipment is valued at $150,000. a) If the depreciation of the equipment is assumed to be linear, find an equation to relate the value ( V ) of the equipment over time ( t ). b) What would the value of the piece of equipment be after 6 years? c) In how many years would the value of the piece of equipment be $0? Solution: a) Since the value started at $200,000 and after two (2) years it was worth $150,000, the equipment depreciated as follows: 150, 000 200, 000 50, 000 25, 000 2 2 m − − = = = − Since the slope is –25,000 and the equipment had a starting value of $200,000, the equation is 25, 000 200, 000. V t = − + b) Substitute 6 in for t : 25, 000 200, 000 25, 000(6) 200, 000 150, 000 200, 000 50, 000 V t V V V = − + = − + = − + = Therefore, the equipment will be worth $50,000 after 6 years. c) Find the value of t when V = 0. 25, 000 200, 000 0 25, 000 200, 000 25, 000 200, 000 8 V t t t t = − + = − + = = Therefore, the value will be $0 after 8 years.

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    College Mathematics: Learning Worksheets Chapter 1 11 Name ________________________________ Date ______________ Class ____________ Goal: To interpret slopes and find linear regression equations In Problems 1–3, use the given information to answer the questions. 1. Depreciation . A new car worth $45,000 is depreciating in value by $5000 per year. a) Find the linear model for the current value of the car, v , and the number of years, y , after it was purchased. b) Interpret the slope of the model. c) If the car is 3 years old, what does the model predict for its value? d) After how many years will the car be worth nothing? Solution: a) 5000 45, 000 v y = − + b) The value of the car decreases $5000 for every year after it was purchased. c) 5000 45, 000 5000(3) 45, 000 15, 000 45, 000 30, 000 v y v v v = − + = − + = − + = The car is worth $30,000 after it is 3 years old. Section 1-3 Linear Regression Solving Real-World Problems 1. Construct a mathematical model. 2. Solve the mathematical model. 3. Interpret the solution. Linear Regression on a Graphing Calculator 1. Enter the data in columns L 1 and L 2 . 2. In the “STAT” mode, find the “LinReg” function. 3. Read the display to find the values of the slope and the y -intercept.

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    College Mathematics: Learning Worksheets Chapter 1 12 d) 5000 45, 000 0 5000 45, 000 5000 45, 000 9 v y y y y = − + = − + = = The car will be worth nothing after 9 years. 2. Health Club Membership . A health club offers membership for a fee of $59 plus a monthly fee of $15 per month. a) Find the linear model for the membership fee, f , and the number of months, m , since you have been a member. b) Interpret the slope of the model. c) If you have been a member for 24 months, what does the model predict for the fee you have paid so far? d) After how many months will you have paid the health club $329? Solution: a) 15 59 f m = + b) The fee will increase by $15 for every additional month of membership. c) 15 59 15(24) 59 360 59 419 f m f f f = + = + = + = The fee after 24 months will be $419. d) 15 59 329 15 59 270 15 18 f m m m m = + = + = = After 18 months, you will have paid $329 for your membership.

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    College Mathematics: Learning Worksheets Chapter 1 13 3. Stress. The table below shows the relationship between a stress test score and the diastolic blood pressure for 8 patients. A linear regression model for this data is 0.56 41.71, y x = + where x represents the stress test score and y represents the blood pressure. Stress Test Score, x 55 62 58 78 92 88 75 80 Blood Pressure, y 70 85 72 85 96 90 82 85 a) Interpret the slope of the model. b) Use the model to predict the blood pressure for a person with a stress test score of 75 c) Use the model to estimate the stress test score for if the diastolic blood pressure was 90. Solution: a) For every 1 point increase in the stress test score, the diastolic blood pressure will increase by 0.56 points. b) 0.56 41.71 0.56(75) 41.71 42 41.71 83.71 y x y y y        A person with a stress test score of 75 will have an approximate diastolic blood pressure of 84. c) 0.56 41.71 90 0.56 41.71 48.29 0.56 86.23 y x x x x       A person with a diastolic blood pressure of 90 will have a stress test score of approximately 86.

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