Student’s Manual
Essential Mathematics for
Economic Analysis
3rd edition
Knut Sydsæter
Arne Strøm
Peter Hammond
, Preface
This student’s solutions manual accompanies Essential Mathematics for Economic Analysis (3rd edition, FT
Prentice Hall, 2008). Its main purpose is to provide more detailed solutions to the problems marked⊂ ⊃SM in the
text. This Manual should be used in conjunction with the answers in the text. In some few cases only a part
of the problem is done in detail, because the rest follows the same pattern. We are grateful to Carren Pindiriri
for help with the proofreading. We would appreciate suggestions for improvements from our readers as well
as help in weeding out inaccuracies and errors.
Oslo and Coventry, July 2008
Knut Sydsæter ()
Arne Strøm ()
Peter Hammond ()
Version: 9 July 2008
Contents
1 Introductory Topics I: Algebra ................................................................................................................ 1
2 Introductory Topics II: Equations............................................................................................................ 4
3 Introductory Topics III: Miscellaneous ................................................................................................... 6
4 Functions of One Variable ....................................................................................................................... 8
5 Properties of Functions .......................................................................................................................... 12
6 Differentiation ....................................................................................................................................... 14
7 Derivatives in Use ................................................................................................................................. 18
8 Single-Variable Optimization................................................................................................................ 22
9 Integration ............................................................................................................................................. 26
10 Interest Rates and Present Values .......................................................................................................... 33
11 Functions of Many Variables................................................................................................................. 35
12 Tools for Comparative Statics ............................................................................................................... 37
13 Multivariable Optimization ................................................................................................................... 42
14 Constrained Optimization...................................................................................................................... 48
15 Matrix and Vector Algebra .................................................................................................................... 57
16 Determinants and Inverse Matrices ....................................................................................................... 61
17 Linear Programming.............................................................................................................................. 67
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
, CHAPTER 1 I N T R O D U C T O R Y T O P I C S I: A L G E B R A 1
Chapter 1 Introductory Topics I: Algebra
1.1
1. (a) True (b) False. −5 is smaller than −3, so on the number line it is to the left of −3. (See Fig. 1.1.1
in the book.) (c) False. −13 is an integer, but not a natural number. (d) True. Any natural number is
rational. For example 5 = 5√
/1. (e) √ False, since 3.1415 = 31415/10000, the quotient of two integers.
(f) False. Counterexample: 2 + (− 2) = 0. (g) True. (h) True.
1.3
9. (a) (2t −1)(t2 −2t +1) = 2t (t 2 −2t +1)−(t 2 −2t +1) = 2t 3 −4t 2 +2t −t 2 +2t −1 = 2t 3 −5t 2 +4t −1
(b) (a + 1)2 + (a − 1)2 − 2(a + 1)(a − 1) = a2 + 2a + 1 + a2 − 2a + 1 − 2a2 + 2 = 4
(c) (x + y + z)2 = (x + y + z)(x + y + z) = x(x + y + z) + y(x + y + z) + z(x + y + z) =
x2 + xy + xz + yx + y2 + yz + zx + zy + z2 = x2 + y2 + z2 + 2xy + 2xz + 2yz (d) (x − y − z)2 =
(x−y−z)(x−y−z) = x 2 −xy−xz−xy+y 2 −yz−xz−yz+z 2 , so (x+y+z)2−(x−y−z)2 = 4xy+4xz
13. (a) a2 +4ab+4b2 = (a +2b)2 using the first quadratic identity. (d) 9z2 −16w2 = (3z−4w)(3z+4w),
according to the difference-of-squares formula. (e) − 1 x2 + 2xy − 5y2 = − 1 (x2 − 10xy + 25y2) =
5 5
− 15 (x − 5y) 2 (f) a — b = (a — b )(a + b ), using the difference-of-squares formula. Since
4 4 2 2 2 2
a2 − b2 = (a − b)(a + b), the answer in the book follows.
1.4
1 1 x+2 x −2 x +2−x +2 4
5. (a)
— = 2
x −2 x +2 (x − 2)(x + 2) (x + 2)(x − 2) (x − 2)(x + 2) x −4
= − =
(b) Since 4x + 2 = 2(2x + 1) and 4x2 − 1 = (2x + 1)(2x − 1), the LCD is 2(2x + 1)(2x − 1). Then
6x + 25 6x2 + x − 2 (6x + 25)(2x − 1) − 2(6x2 + x − 2) 21(2x − 1) 21
− = = =
4x + 2 2 4x2 − 1 2(2x + 1)(2x − 1) 2(2x + 1)(2x − 1) 2(2x + 1)
18b a 18b − a(a − 3b) + 2(a2 − 9b2)
2
a(a + 3b) a
(c) − +2 = = =
a2 − 9b2 a + 3b 2 + 3b)(a − 3b)
(a (a + 3b)(a − 3b) a − 3b
1 1 1 a − 4 − a(a − 2) + 8a 2(5a − 2) 5a − 2
(d) − + = = =
! + 2) b(a "− 4) 8ab(a2 − 4) 8ab(a2 − 4) 4ab(a2 − 4)
8ab 8b(a 2
2t − t 2
5t 2t t(2 − t) 3t −t(t − 2) 3t −3t 2
(e) = · =
· t −2 − t −2 t +2 t −2 · =
t +2 # $ t +2 t −2 t +2
# $ a− 1
a 1 − 2a1
a 1 − 2a 1
= 4a − 2, so 2 − = 2 − (4a − 2) = 4 − 4a = 4(1 − a)
(f) = 41 2
2 0.25 0.25
6. 1 2 − 3x2
2(x + 1) + x − 3x(x + 1)
(a) + −3 = =
x t x +1 t x(x + 1) x(x + 1)
(b) − t(2t − 1) − t(2t + 1) −2t
= =
2t + 1 2t − 1 (2t + 1)(2t − 1) 4t 2 − 1
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
, 3x 4x 2x − 1 3x(x − 2) + 4x(x + 2) − (2x − 1) 7x2 + 1
(c) − − = = x2 − 4
x +2 2−x (x − 2)(x + 2) (x − 2)(x + 2)
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
Essential Mathematics for
Economic Analysis
3rd edition
Knut Sydsæter
Arne Strøm
Peter Hammond
, Preface
This student’s solutions manual accompanies Essential Mathematics for Economic Analysis (3rd edition, FT
Prentice Hall, 2008). Its main purpose is to provide more detailed solutions to the problems marked⊂ ⊃SM in the
text. This Manual should be used in conjunction with the answers in the text. In some few cases only a part
of the problem is done in detail, because the rest follows the same pattern. We are grateful to Carren Pindiriri
for help with the proofreading. We would appreciate suggestions for improvements from our readers as well
as help in weeding out inaccuracies and errors.
Oslo and Coventry, July 2008
Knut Sydsæter ()
Arne Strøm ()
Peter Hammond ()
Version: 9 July 2008
Contents
1 Introductory Topics I: Algebra ................................................................................................................ 1
2 Introductory Topics II: Equations............................................................................................................ 4
3 Introductory Topics III: Miscellaneous ................................................................................................... 6
4 Functions of One Variable ....................................................................................................................... 8
5 Properties of Functions .......................................................................................................................... 12
6 Differentiation ....................................................................................................................................... 14
7 Derivatives in Use ................................................................................................................................. 18
8 Single-Variable Optimization................................................................................................................ 22
9 Integration ............................................................................................................................................. 26
10 Interest Rates and Present Values .......................................................................................................... 33
11 Functions of Many Variables................................................................................................................. 35
12 Tools for Comparative Statics ............................................................................................................... 37
13 Multivariable Optimization ................................................................................................................... 42
14 Constrained Optimization...................................................................................................................... 48
15 Matrix and Vector Algebra .................................................................................................................... 57
16 Determinants and Inverse Matrices ....................................................................................................... 61
17 Linear Programming.............................................................................................................................. 67
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
, CHAPTER 1 I N T R O D U C T O R Y T O P I C S I: A L G E B R A 1
Chapter 1 Introductory Topics I: Algebra
1.1
1. (a) True (b) False. −5 is smaller than −3, so on the number line it is to the left of −3. (See Fig. 1.1.1
in the book.) (c) False. −13 is an integer, but not a natural number. (d) True. Any natural number is
rational. For example 5 = 5√
/1. (e) √ False, since 3.1415 = 31415/10000, the quotient of two integers.
(f) False. Counterexample: 2 + (− 2) = 0. (g) True. (h) True.
1.3
9. (a) (2t −1)(t2 −2t +1) = 2t (t 2 −2t +1)−(t 2 −2t +1) = 2t 3 −4t 2 +2t −t 2 +2t −1 = 2t 3 −5t 2 +4t −1
(b) (a + 1)2 + (a − 1)2 − 2(a + 1)(a − 1) = a2 + 2a + 1 + a2 − 2a + 1 − 2a2 + 2 = 4
(c) (x + y + z)2 = (x + y + z)(x + y + z) = x(x + y + z) + y(x + y + z) + z(x + y + z) =
x2 + xy + xz + yx + y2 + yz + zx + zy + z2 = x2 + y2 + z2 + 2xy + 2xz + 2yz (d) (x − y − z)2 =
(x−y−z)(x−y−z) = x 2 −xy−xz−xy+y 2 −yz−xz−yz+z 2 , so (x+y+z)2−(x−y−z)2 = 4xy+4xz
13. (a) a2 +4ab+4b2 = (a +2b)2 using the first quadratic identity. (d) 9z2 −16w2 = (3z−4w)(3z+4w),
according to the difference-of-squares formula. (e) − 1 x2 + 2xy − 5y2 = − 1 (x2 − 10xy + 25y2) =
5 5
− 15 (x − 5y) 2 (f) a — b = (a — b )(a + b ), using the difference-of-squares formula. Since
4 4 2 2 2 2
a2 − b2 = (a − b)(a + b), the answer in the book follows.
1.4
1 1 x+2 x −2 x +2−x +2 4
5. (a)
— = 2
x −2 x +2 (x − 2)(x + 2) (x + 2)(x − 2) (x − 2)(x + 2) x −4
= − =
(b) Since 4x + 2 = 2(2x + 1) and 4x2 − 1 = (2x + 1)(2x − 1), the LCD is 2(2x + 1)(2x − 1). Then
6x + 25 6x2 + x − 2 (6x + 25)(2x − 1) − 2(6x2 + x − 2) 21(2x − 1) 21
− = = =
4x + 2 2 4x2 − 1 2(2x + 1)(2x − 1) 2(2x + 1)(2x − 1) 2(2x + 1)
18b a 18b − a(a − 3b) + 2(a2 − 9b2)
2
a(a + 3b) a
(c) − +2 = = =
a2 − 9b2 a + 3b 2 + 3b)(a − 3b)
(a (a + 3b)(a − 3b) a − 3b
1 1 1 a − 4 − a(a − 2) + 8a 2(5a − 2) 5a − 2
(d) − + = = =
! + 2) b(a "− 4) 8ab(a2 − 4) 8ab(a2 − 4) 4ab(a2 − 4)
8ab 8b(a 2
2t − t 2
5t 2t t(2 − t) 3t −t(t − 2) 3t −3t 2
(e) = · =
· t −2 − t −2 t +2 t −2 · =
t +2 # $ t +2 t −2 t +2
# $ a− 1
a 1 − 2a1
a 1 − 2a 1
= 4a − 2, so 2 − = 2 − (4a − 2) = 4 − 4a = 4(1 − a)
(f) = 41 2
2 0.25 0.25
6. 1 2 − 3x2
2(x + 1) + x − 3x(x + 1)
(a) + −3 = =
x t x +1 t x(x + 1) x(x + 1)
(b) − t(2t − 1) − t(2t + 1) −2t
= =
2t + 1 2t − 1 (2t + 1)(2t − 1) 4t 2 − 1
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
, 3x 4x 2x − 1 3x(x − 2) + 4x(x + 2) − (2x − 1) 7x2 + 1
(c) − − = = x2 − 4
x +2 2−x (x − 2)(x + 2) (x − 2)(x + 2)
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
