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A P P E N D I X A 6
Selected Answers
Chapter 2 Answers
2.5 a) all -2, d) all
2.8 = 3, = = 4x 3, d) = 2,
= + = 3.
2.9 additional revenue from the (q + output, that is, marginal revenue.
increase in consumption when national income goes up I unit.
2.1 I + + 3. + --
+ + 30).
2.12 = 9. = (I +
2.16 continuous and differentiable at =
continuous = 0,
continuous, not differentiable =
continuous and differentiable at =
2 . 2 0 6 +
+ 6. 2.22 86. XX.
2.23 increases 30.78 dollars.
2.25 9.9992s.
Chapter 3 Answers
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900 SELECTED ANSWERS
3.9 a) no global or min on at 1 and min at 2 on
b) no min on , min at 0 and at 1 on
c) min at -4, at -2 min at no on
d) min at 0, at 10 on min at 0, no on
3.10 (x =3.11 = 20 = (20 = 12.50.
3.16 No positive output maximizes profit.
3.19 Elasticity equals for all prices.
3.22 = + kb), = + +
Chapter 4 Answers
4.1 a) h)(z) = + 4, g)(x) = + 19.b) h)(z) = + g ) ( s ) = +
h)(z) = =
4.2 a) inside = + 1, outside =
inside = l/x, outside + + 4,
c) inside = 2.x 7, outside =
4.3 a) h)(l) = = = 5 2x =
b) h)(z) = + =
= 2x =
4.4 a) + =
+
4 . 5
d )
4 . 9 = =
b) = l/f(l) = --(I + = -4
Chapter 5 Answers
5 1 a) 8 b) 2 e) 1 h) 125 i )
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SELECTED ANSWERS 90
= =
= = (2 In
5.10 = In
5.11
5.12 a) $1870.62 $4754.17
5.13 48.05 years
5.14 =
5.15 = 3.393
5.16 a) = + +
= +
Chapter 6 Answers
6.1 Net contribution cost =
6.2C=6070. =2875. F=36.422.
6.3 = + =Or , + 3: = =4.
6.4 = = + + 3; no positive solution.
6.5 8.82 percent and 18.28 percent, respectively.
6.6 percent for white females.
Chapter 7 Answers
7.1 and
7.6 = = both decrease as
increases.
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90 2 SELECTED ANSWERS
7.17 a) General solution is = + = To get an integ
solution, take = 4, then = 6 and = 3. Just change the right ha
side in the system in part a and apply the same analysis.
7.18 a = -8.
7.19 a) one, = =
7.20 a) 2, 3, 3.
7.21 a) Only the solution for iii and iv; infinitely many other solution
for the other two. Unique solution for every RHS for and infini
many solutions for every RHS for ii; or infinitely many solutions
pending the RHS for or one solution depending on the RHS
for iii.
7.23 Only
7.25 : a) 2, and 1 of the other 3, = = +
7.26 C = 0.05956 P, = 0.04702. P. F = 0.35737
7.29 a) and any of the other 3 can be if is chosen as
exogenous and set to 0, = 0.6, = 0.6, = if is the o
exogenous variable and set equal to 0, the corresponding system has i
finitely many solutions.
7.30 No, the 3 X 4 coefficient matrix has rank 2; no submatrix can have
rank 3.
Chapter 8 Answers
8 . 1
8 . 5 =
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SELECTED ANSWERS 9
8.18 Carry the multiplication.
8.19 ( (
0 -6 - 1
0 13 -3 2
2 -1
8.20 =
8.21 A and defined implies has rows.A and BA defined implies B has
0 0
a symmetric matrix.
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904 SELECTED ANSWERS
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SELECTED ANSWERS
One row echelon form is
One row echelon form is
9.9 All nonsingular since det 0.
1 3
0 6
2
0
9.14 = = = = -4.
= 24: = det AR = 432; + B) = = = d e t A B =
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906 SELECTED ANSWERS
10.12 a) acute, 45" right, 90 acute, 30 obtuse, 106.8 acute, 63.4"
10.13 a)
10.14
10.16 + + +
10.19 42.03"
10.24 expand determinant along top row
10.25 a) 1)
10.26 3.64
10.28 (I) (3+ (1 (1 + 1
10.29 no
10.30 = + 15 = + 8 = 5
10.32 no
10.33 =
= ( ) + = 2
= 10.34
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SELECTED ANSWERS 9
10.38 yes; no.
10.39 a) = -1
10.40
10.41
10.42 If I increases, so does and If increases, increases and
decreases. If increases, so does and
Chapter 11 Answers
11.2 a) independent b) dependent c) independent independent
11.3 independent dependent
11.12 no yes no yesII.14 no yes no
Chapter 12 Answers
1 2 . 1 d)
+ decimal places
the digit after the decimal point in12.2 out of order.
set
second element not in original sequence
12.21 neither. closed. closed. open, closed.
12.31 None bounded.
Chapter 13 Answers
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908 SELECTED ANSWERS
13.13 I for example.
13.23 Domain I - I
3x 7 R R yes + 7) yes
1 R
R R R
R
R R
I
1
R l / e ]
13.24 a) f(x) = = + 1 f(x) = =
f(x) = = = + x; =
Chapter 14 Answers
14.1a) = +6 =
= Y = = =
= =
= =
14.2 =
= ,
= +
14.4 a) 5400 5392.8, 5412.5
5392.798, 5412.471 = 60
14.5 a) own price elasticity =
cross price elasticity =
income elasticity =
1 4 . 6 a ) 4 . 5
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SELECTED ANSWERS
14.15
141, .
14.18
14.19
14.20
14.21
427.5 19.5 14.22 I)) =
Chapter 15 Answers
15.3 2.9X
1.8607: 1.9913
15.5 = I in both cures.
15.6 a) = yes
15.8 no no -0.9449. 0.4961.
15.9
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914 SELECTED ANSWERS
23 .15 a ) l , - 0 . 2 ;
23.17 a) + +
+ +
23.26 a)
where =
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SELECTED ANSWERS
23.40 = = = 1Chapter 24 Answers
24.4 Initial position and initial velocity.
24.5 a) = + k = b) = + k
= + + 2, k = d) = k
24.9 =
24.10 a) = = +
24.13 a) = = +
24.14 = + =
24.16 = = +
24.17 = + +24.25 Equilibria at = 0 (asympt. stable) and at = 2 (unstable);
equilibria at = (unstable) and at I (asympt. stable).
24.26 a. Equilibria at = 0 (unstable) and at = k (asympt. stable);
equilibria at = 0, k (asympt. stable) and at = (unstable).
24.27 Equilibria at 0 (asympt. stable) and = I (unstable);
equilibria at (2n (asympt. stable) and at =
(unstable) all integers
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25.2 = =
25.3 = =
25.6 a) = + = + = = 0.8.
=
= sin = 0, =
= =
coexistence steady state.25.8 (0,
25.9
25.12 all unstable.
25.13 Trace 0 and det 0 implies that is locally asym
totically stable.
25.16 The of Jacobian at are the pure imaginarynumbers
25.17 asymptotically stable, b. d, unstable.
25.20 a) All orbits in interior converge 3); All orbits in interior
converge to (0.6): All orbits in interior converge to (3.0).
25.24 = + = = 0.
25.28 Since Hessian of in (28).
0 is positive definite
the interior of the orthant, Fis convex and its critical point
is a global min.
Chapter 26 Answers
26.1
26.7 n!; II!
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918 SELECTED ANSWERS
27.14 a) has a solution for
has solutions for everyhas unique solutions for every has a solution for every
27.19 a) yes, yes. no, yes, no.
Chapter 28 Answers
28.3 a) 45 X IO!, x II!.
28.4 (123) represents (132) represents
(213) represents (312) represents
(231) represents (321) represents
I represents
I (321). 1 I represents
Chapter 29 Answers
29.1 a) yes, no, yes.
yes. yes. yes; yes.
29.11 a) not open_ closed, not connected.
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SELECTED ANSWERS
1.1; 1.095; 1.0955; 1.095445
1.5; 1.375; 1.4375; 1.414214.
= +
In 1.2 0.18232..
I n 2 I n 2 0 . 5 ;
30.11 = 2, = 2,
= 1.86121, = 1.86816.
30.12 a) +
= 1 + = + += +
= + +
= +
= +
+
= k = P , + abk +
= + + +
+
Appendix Al Answers
Al.1
Al.2 no lub; =
Al.6 = = a,, = 1).
Al.7 2 and + 2 implies + 1)
Appendix A2 Answers
AZ.4 a) sin 120 = = 1 2 0 = =ran 120 = 3 b = =
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SELECTED ANSWERS
AZ.11 I + 0.707429, which compares we
0.707107.
Appendix A3 Answers
A3.3
A3.7 a) + i&)/2, i&)/2.
A3.8 a) 1 + sin i.
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