answerspapalg2b wbk 2012-13 · page 1 section 5.1 1. 25 2. 2187 3. 1024 4. 256 5. 64 27 6. 729 7....
TRANSCRIPT
Page 1
Section 5.1
1. 25 2. 2187 3. 1024 4. 256 5.64
27 6. 729 7.
512
19,683 8.
1
125
9. 52.139 10 10. 9.72 × 106 11. 1.6 × 105 12. 2.0 × 103 13. 1.0 × 106
14. 2.5 × 101 15. x6 16. 7
6
2m
n 17.
13
5
45c
d 18. 1 19.
14 2
9
x y 20.
6
3 3
27r
q s
21. 12
1
z 22. 18b 23.
2 22n nx y 24. 6 2
2 4
n
n
y
x
25. 4 4n nx y
26. 4
32 2 38 2 2 210
24 2 23
1 1 1 11330 2 2 2 2
1 9 3b 1) b) ) 1 d) 5 e) f) g) 3 12 h) i) 3a-4a
24 a
)2 ( 4 ) k) 2 l) 1/2 m) 25/9 n) 1/4 o) -8 27. ) ( ) ) (1 )
na b ba c x x x ab
a b a ba
j a a a a b a b b a b ab
Section 5.2 through 5.9 1. 2. 3. 2 2 3 1 0 6 20
4 2 6 12 12
2 1 3 6 6 32
(2) 32f
3 8 5 0 14
24 87 261
8 29 87 275
( 3) 275f
As , ( )
As , ( )
x f x
x f x
4. 5. 6. As , ( )
As , ( )
x f x
x f x
As , ( )
As , ( )
x f x
x f x
As , ( )
As , ( )
x f x
x f x
7. 211 2 2x x 8a. 3 3 34 ; 16 ; 36x x x 8b. 356 448x 8c. 2x 8d. 4, 2; 8, 2; 12, 2r h r h r h
9. 3 2
( 2)( 1)( 2) 0
4 4 0
x x x
x x x
10.
4 3 2
( 4)( 3)( 3) 0
4 9 36 0
x x x
x x x x
11. 2
( 6 5)( 6 5) 0
12 31 0
x x
x x
12.
3 2
( 5 2 3)( 5 2 3) 0
10 13 0
x x x
x x x
13. 2 282 4 9
2 3x x
x
14. 2
2
8 715 20
4
xx
x
15. 2
3 2
10 10 92
5
x xx
x x
16. 23 6 1x x 17. 2 107
2 8 284
x xx
18. 2 346 12 17
2x x
x
19. 22 5 1x x 20. 4 3 2 1
3 3 5 51
x x x xx
21. ( ) ( 2)(2 5)(2 5)f x x x x 22. 2( ) ( 4)( 2)( 2 4)f x x x x x
Page 2
23. 2 2( ) ( 1) ( 4)( 1)f x x x x x 24. no other zeros 25. 2 2
other zeros and3 3
26. 1 3
other zeros and3 2
27. remaining factors are (2 3) and ( 1)x x
28. remaining factors are (4 3) and (2 1)x x 29. remaining factors are (2 1) and ( 5)x x 30. remaining factors are (4 1) and (3 2)x x 31. 210k 32. 42k
33. ( ) 2( 2)( 1)( 2)f x x x x 34. 2( ) ( 1)( 2)f x x x Section 6.1
1. 633/5 2. (25)4/3 3. 1247/6 4. 43 57 5. 313 6. 58 204 7. 9 8. 64 9. 16
10. 216 11. 1
125 12. 2 13.
1
8 14.
1
59,049 15.
1
3,125 16. 186.01 17. 22.89
18. 47.29 19. 132.26 20. 0.14 21. 32 22. 32.15 23. 0.09 24. 64311.53 25. 4.21 26. 2.34 27. 0.18, 5.82 28. 3.21 29. 2.06 30. 1.97, 0.63
31. 1.32 32. 1.78 33. 3.58 34. 6.19 cm 35. 1.81 in. 36. n na a when a < 0
and n is even 37.
3 33 3 3 5 5 3
. 27 . 1252 2
i ia b
c. cube roots of 1
are 1, 1 3
2
i ; cube roots of 8 are 2,
2 2 31 3
2
ii
; cube roots of 64 are 4,
4 4 32 2 3
2
ii
Section 6.2
1. 4400 5 2. 33/2 3. 76/5 4. 51/2 5. 4 6. 24 6 7. 6
3 8. 59 81 9.
70
35
10. 3 44 1331 11 11. 33 2
4 12. 2 13.
5 45
3 14. 3 3x 15.
4 4
2
x x
y 16. 1 4x
17. 1 3 1 6x y 18. 4 1 5
3
8x y
z 19.
1 4
1 2 1 3
3z
x y 20.
9 2
3 2 3 28
y
x z 21. 4 349x x 22. 23 120x
23. 42 3 24. 2 233x z xyz 25. 8 x 26. 3 50 25
5
x x 27. 32 x 28. 24x z xyz
29. 42 2 30. 5 31. 8 8y x 32. 6 33 2ab ab
Section 6.3 1. 2x2 + x x1/2 4 2. 6x2 + 2x1/2 5 3. 4x2 + x 5x1/2 + 1 4. 2x2 x + 7x1/2 6
5. 2x2 4x1/2 + 5 6. 4x2 + x 3x1/2 + 1 7. 5 31 3
33x
x 8. x5/2 + x1/2 9. 3x1/6
Page 3
10. 7 3 1 3
3
x x 11. 3 2
1 2
1x
x 12.
5 6
3
x 13.
1 22 x
x
; positive real numbers
14. 1 2
3
1x
; all real numbers greater than 1 15. 22 7 5
9
x x ; all real numbers
16. 1 22
1
2x x ; all real numbers < 0 and <>
1
2 17.
22 1
3
x x ; all real numbers
18. 1 4x ; positive real numbers 25. Sample answer:f(x) = , g(x) = x + 1 26. Sample answer: f(x) = x3 + 2, g(x) = 27. Sample f(x) = g(x) = x 1
Section 6.4
1.
1 1 1
-1
1 1 1( ( )) ( ) 1 1 1 , 1; ( ( )) ( 1)
111
1 1 1 1, 0 Therefore, the inverse of f(x)= is f ( ) 1.
1 1 x-11 1
f f x f x x x f f x fx x
x
x x xx
x x
2.
1 1
1
2 34 32 3 4(2 3) 3( 4) 8 12 3 12 54( ( )) ( ) , 4;
2 34 2( 4) (2 3) 2 8 2 3 524
4 32 34 3 2(4 3) 3(2 ) 8 6 6 3 52g( ( )) ( ) ,4 32 (4 3) 4(2 ) 4 3 8 4 542
xx x x x x xxg g x g x x
xx x x x xx
xx x x x x xxg x g x x
xx x x x xx
-1
2
2 3 4 3Therefore, the inverse of g(x)= is g ( ) .
4 2
x xx
x x
3. 1 1) ( ) 3 , 0; 3 ) ( ) 3 , 0; 3 a f x x x y b f x x x y
O
f^-1
f
O
f^-1
f
3 x
x2
1
x
x
Page 4
4. 1 1) ( ) 2 2, 2; 2 ) ( ) 2 2, 2; 2 a f x x x y b f x x x y
O
f
f -̂1
O
f
f^-1
5. 1 2 1 2) ( ) 4 ,0 2; 2 0 ) ( ) 4 , 2 0;0 2 a f x x x y b f x x x y
O
f^-1
f
O
f^-1
f
6.2 2
1 15 5 5 5) ( ) , 0; ) ( ) , 0;
2 2 2 2
x xa f x x y b f x x y
O
f
f^-1
O
f
f^-1
7. 1
1
) ( ) 2, 2, 0; ( ) 2, 0; 2
) ( ) 2, 2, 0; ( ) 2, 0; 2
a f x x x y f x x x y
b f x x x y f x x x y
8. 12 2( 1)( ) , 2, 1; ( ) , 1, 2
2 1
x xf x x y f x x y
x x
9. 13 4 3 3 3 4 3 3( ) , , ; ( ) , ,
2 3 2 2 2 3 2 2
x xf x x y f x x y
x x
10. k = 3
11. 1 1 1(0) 3, ( 1) 0, (1) 4, (0) 1, (3) 0, and (4) 1f f f f f f
Inverse Functions 1-6 Show that f (g(x)) = x and g( f (x)) = x
7. f 1
(x) 8. f 1
(x) = 5x15 9. f 1
(x) = x2 + 3, x ≥ 0
10. x ≥ 0 11. 12. , x ≥ 1
x 2 1
2 3
, 2 5
2 1 )( 2 1 x x f 71
4)( xxf
41)(1 xxf
Page 5
13. 14. 15. f 1
(x) = x 1, x ≥ 1
1. 3. 2. 3. Sample answer: 4. Sample answer:
; f 1 = + 3;
domain x ≥ 1
f 1 = x 2; domain x ≥ 0
Section 6.5
1. 2.
domain: x ≥ 4, range: y ≥ 2 domain: x ≥ 2, range: y ≥ 3 3. 4.
domain: x ≥ 0, range: y ≤ 3 domain: x ≥ 2, range: y ≤ 2 5. 6.
domain and range: all real numbers domain and range: all real numbers 7. 8.
domain and range: all real numbers domain and range: all real numbers
1 3 4)(1
xxf
54
51)( 51 xxf
2 1
1x
Page 6
9. ; ; domain and range: all real numbers 10. (0, 0) and (1, 1)
11. On the interval 0 to 1, the larger the root, the steeper the graph. On the interval 1 to , the larger the root, the less steep the graph. 12. 13. (1, 1), (0, 0) and (1, 1) 14. On the interval 1 to 1, the larger the root, the steeper the graph. On the intervals to 1 and 1 to , the larger the root, the less steep the graph. 15.
16.
a)
O
b)
O
c)
O
17.
a)
O
b)
O
c)
O
Page 7
18.
a)
O
b)
O
c)
O
d)
O
Section 6.6
1. 15
2 2. 21 3. 4 4. 5 5. 39 6. 3 7. 6 8. 27 9. 120 10. 169
11. 397
81 12. 32 3 13. 1,000 14. 6 15. 9 16. 33 6 6 17. no solution
18. 23 19. 3 20. 7
3 21. −5 22. 3 23. no solution 24.
81
16 25. 0
26. 3
4 27. no solution 28.
25
12 29. 12 in. 30. 24 in 1.
3 2
2 2.
1 5 5
2
3. 1 4. 3
0, ,15
5. a) k =2 b) k<2 c) k>2 6. 9 4 66
1.56615
Section 7.1
1. 2. 3.
: , ; : 3,D R : , ; : 1,D R : , ; : ,3D R
4. 5. 6.
Page 8
: , ; : 5,D R : , ; : , 2D R : , ; : 1.5,D R
7. sample answer: 13 3xy 8. y-intercept of 1. As a get larger the graph gets steeper. 9. $8,396.71 10.$8,719.45 11.$8,121.19 12. 2.13% 13. y = 12,941,197 (1.0213)t
14. 19,726,093 1. 64(8)xy 2. 3(4)xy 3. 1
(5)2
xy 4. 3(3)xy
Section 7.2 1. decay 2. growth 3. growth 4. decay 5. decay 6. growth 7. 8. 9.
: , ; : 2,D R : , ; : 3,D R : , ; : 1,D R
10. 11. 12.
: , ; : , 2.5D R : , ; : 3,D R : , ; : 5,D R
13. V = 175,000(0.82)t 14. $24,053.41 15. 16. after 5 yr 17. V= 1600(0.8)t
18. a. V 6000t 30,000 b. V 30,000(0.775)t c.
During the first 2 years, the exponential model represents a faster depreciation. d. Book value after 1 year: Linear model: $24,000Exponential model: $23,250Book value after 3 years: Linear model: $12,000Exponential model: $13,965e. According to the linear model, the car will have no value after 5 years. According to the exponential model, there will never be a time when the car will have no value because the x-axis is a horizontal asymptote of the graph of the model. f. Linear model: An advantage for the buyer is the older the car, the less the buyer would have to pay compared to the exponential model. A disadvantage is if the buyer would resell the car, they would lose more money compared to the exponential model. An advantage for the seller is during the first two years, the seller would be able to sell the car for more money compared to the exponential model. A disadvantage is after 5 years the car is not worth anything and would then be difficult to sell.
Page 9
Exponential model: An advantage for the buyer is during the first two years, the buyer would have to pay less for the car compared to the linear model. A disadvantage is if the buyer buys the car after two years, they would pay more compared to the linear model. An advantage for the buyer is that after 5 years the car is still valuable compared to the linear model. A disadvantage is that if the car were sold during the first two years, the seller would get less money for it compared to the linear model. Section 7.3
1. 104e 2. 4
9
e 3. 1227e 4. 632 xe 5.
9 3
8
xe 6. 332 2x xe e
11. 12. 13.
: , ; : 3,D R : , ; : 1,D R : , ; : 2,D R
14. 15. 16.
: , ; : 1,D R : , ; : 4,D R : , ; : 3,D R
17. exponential growth 18. 19. 3 20. 23 21. 21
22.
2 2 22 2
[ ( )]2 4
x x x xe e e ef x
; 2 2 2
2 2[ ( )]
2 4
x x x xe e e eg x
2 2 2 22 2 2 2 4
[ ( )] [ ( )] 14 4 4
x x x xe e e ef x g x
Section 7.4
1. 41/2 2 2. 34 81 3. 3
164
4
4. 3 5. 3
2 6.
5
2 7.
2
3 8.
1
4
9. 3
2 10. 3 2 2125
) ) ) ) ) )10004
eta b c d e f
x k
13. 1( ) 7xf x
Page 10
14. 1 3( )
4
x
f x 15. 2
1 1( )
2
x
f x
16. 1 2( ) xf x e 17. 1 3( ) 1xf x e
18. 1 1 1( )
3 3f x x
11. 12. 13.
: 2, ; : ,D R : 2, ; : ,D R : 1, ; : ,D R
1. log
5 125 = 3 2. 3. ln 7.3890 = 2 4. ln 0.6065 = 0.5
) 3 ) 3 ) 1 ) 0 and 1 or(0,1) (1, )a x b x c x d x x
Section 7.5
1. 3.332 2.0.560 3. 2.772 4. 3.738 5. 1.659 6. 0.980 7. 43
log2
x 8. log x
log y log z 9. log7 y log
7 z log
7 3 10. 3 log
2 x log
2 y 2 log
2 z
11. 3
ln ln2
x y 12. 1
log 2 log 2log2
x y 13. ln 7 ln y 2 ln x
14. 5[log9 2 2 log
9 x log
9 y log
9 z] 15. 2 log
6(x 3) log
6 3 3 log
6 y
16. 35
log3
17. 2
25log
y
x 18. 2 3
9log
x y 19.
2
3
( 3)ln
6( 2)
x
x
20. 3
3
( 4)log
6
x
x
21. 4
2
3ln
( 1)
x
x 22. 1.869 23. 1.032 24. 2.665 25.
23
2
( 1) ( 8)ln
( 1)
x x
x x
26. 5 34log ( 2)( 2)x x x 27.
36 2 6 12
2
10 ( 4) ( 1)log
x x
x
28.
3 3 34
9 6ln
125
x y z
w v
1. –0.8 2. –0.5 3. 2 4. 16 5. 4 6. –2.5 7. ln (x2 1) 4 ln x
8. 26 6
1[3log log ( 9)]
4x x 9.
1 1log 2 4log log 6 log log3 5log log
2 2x y w z
10. 2 21ln 4 2ln ln(2 )
2y x y 15. ln 2 0.6931, ln 3 1.0986, ln 4 1.3862, ln 5
1.6094, ln 6 1.7917, ln 8 2.0793, ln 9 2.1972, ln 10 2.3025, ln 12 2.4848, ln 15 2.7080, ln 16 2.7724, ln 18 2.8903, ln 20 2.9956
364 1
log 4
Page 11
Section 7.6
1) 2 2) 4 3) 41 1 e 4) 1
4,4
5) 15
2 6)
11
2 7) 21, e 8) 1, 9, 5 34
9) 7
1,3
10) 2,e e 11) 7
2 12) 0 13) ln(2 5) 14) 6 log38 15) 2 ln 75
16) 2 ln 5 2ln 3
ln 5 3ln 3
17) 1 18) 16 19) 81 20) a) 0.567 b) 2.787
Section 7.7 – Practice B
1. 1
(3)2
xy 2. 1
(2)5
xy 3. 3(4)xy 4. 1
82
x
y
5. 2
53
x
y
6. 3 1
4 3
x
y
7. y = 37.86(6.85)x 8. y = 0.0019(1.33)x 9. y = 62.12(0.03)x
11. y = 3x3 12. 21
2y x 13. y = 2x1.5 14. y = 4x2 15. 0.51
4y x
16. 2.51
2y x 17. y = 12.94x3.3 18. y = 0.014x1.05 19. y = 93.34x2
20. 0.81x0.79;$6.14 Section 7.7 – Practice A 7. yes 8. yes 9. no 19. yes 20. no Section 7.7 – Practice C
4. ; y = 3(1.8)x 5. ; y = 0.9(4x) Exponential and Logarithmic Application Problems 1. $789.24 2. $156.83 3. $5913.70 4. 7.123%(semiannually); 7.251%(cont.) 5. 6.882%( semiannually); 6.7661%(cont.) 6. 7.177% 7. 6.960
8. a) 1
( ln 2)3
0( )t
N t N e b) 4.755 hr c) 6 hr 9. 33062.445 years 10a). 18.616 min later
10b) The temperature of the object will approach the room’s temperature of 30°C. 11. b) ( ) 0.033860(1.94737)xN t c) 0.66648( ) 0.033860 xN t e e) 1.847 f) 6.193 hr later Exponent and Logarithm Practice 1. $6,414.27 2. $4,429.41 3. 7.727 years 4. $4,931.94 5. $4,204.80 6. 19.56% 7. 0.0488 ml 8. 10,400 years 9. 209 days 10. 75.044 pascals 11. no, it is 3,561 years
Page 12
12. 1971 earthquake was 100.7 times more intense than 1987 earthquake 0.7(Note: 10 5.012)
13. 1,396.7 volts 14. pH = 7 15. 8.71 minutes 16a. ln 0.8
60k
16b. 1:42 p.m.
17. 322 P 18. 6,031,455,620 years Section 8.1
1. direct 2. neither 3. inverse 4. inverse 5. 8.64
; 34.556yx
6. 15
; 60yx
7. 5
; 2.58
yx
8. direct 9. neither 10. inverse 11. 18 ; 540z xy
12. 15
; 2252
z xy 13. 25 125
;84 14
z xy 14. 10 yr 15. 43,725 16. 43725
dp
17. 7950 units 18. 0.22; H = 0.22mT 19. 4.541 kilocalories Variation Word Problems
1. a. Inversely; 1890
fl
b. inch-pounds c. 630 pounds, 126 pounds, 63 pounds,
31.5 pounds d. 6.3 inches, 37.8 inches 2. a. 16560
vp
c. 60 pounds
1e. 2b. d. no, 16560/p can never equal 0
e. Yes. The equation can be rewritten in the form 16560vp so any product of volume and pressure pairs will be equal to the same number and therefore to each other.
Page 13
3. a. 2
4000s
d b. 40 units c. 400,000 units 4. a. The intensity is inversely
proportional to the square of the distance; 2
320I
d b. 1.25 mr/hr; 3.2 mr/hr; 32,000
mr/hr c. 25.2982 meters 5 a. 21
150b s ;
7
30r s b. b(60) = 24 m; r(60) = 14
m; b(120) = 96 m; r(120) = 28 m; b(180) = 216 m; r(180) = 42 m
c. 21 7
150 30t s s ; t(60) = 38 m; t(120) = 124 m; t(180) = 258 m
4d. 5d.
e. 940 1
2.8563 109.7
football fieldm fields
m 6 a. 76.492s p b. 33.123 knots
c. No, because the power is inside the 7th root d. It would require too much power 7. a. 3 21.297d h b. 458.56 cm; more than 10 times as big c. 83.24 m 8a. because the time divided by the number of slices is not a constant b. 0.51461.225t n c. 3.572 minutes; 1.225 minutes
d. about 38 slices 9. a. 2125bd
sl
b. 250 pounds
c. 12,000 pounds d. i. 2 ii. 4 iii. ½ 10. a. 42500f pr b. 163.84 cubic millimeters per second c. 244.14 units
11. a. 2 21 ;
kw k s g
w b.
2
kg
s c.
2
80,000g
s
d. 5.5556 km/l; 3.125 km/l; 2 km/l e. No because the km/l would be too large Section 8.2
1. x = 1; y = 5 2. x = 2; y = 3 3. 1 3
;2 2
x y 4. 1
; 24
x y 5. x = 2; y = 1
6. 2
; 53
x y
7. 8. 9.
Page 14
: 4; : 3D x R y : 0; : 2D x R y 2
: ; : 45
D x R y
10. 11. 12.
: 1; : 1.5D x R y : 0.25; : 1.25D x R y 2 4
: ; :7 7
D x R y
13. 2 12
4
xy
x
14.
6 14
3
xy
x
15.
0.75 12.5) ( ) ) 0 950
50
xa C x b x
x
c) Section 8.3 1.
3 3 5) : 4, 4; int : 2, ; int : ; . : ; . : 4;
5 16 2f(x) does not have a removable discontinuity point;
74The graph of f(x) crosses its horizontal asymptote at x = - .
7
a D x x x y H A y V A x
3) : 3, 7; int : 3; int : ; . : 1; . : 7;
73
f(x) has a removable discontinuity point (3, ); 5
The graph of f(x) never crosses its horizontal asymptote.
b D x x x y H A y V A x
d) As the tank is filled, the rate at which the concentration of brine is increasing slows. The concentration of brine appears to approach 75%.
Page 15
3) : 2; int : 3; int : ; . : 0; . : ;
45
f(x) has a removable discontinuity point (2, ); 12
The graph of f(x) crosses its horizontal asymptote at (-3, 0).
c D x x y H A y V A None
1) : 0; int : ; int : ; . : ; . : 0; Oblique Asy: 1
4d D x x None y None H A None V A x y x
3 1 2 3 1) : ; int : , 2; int : ; . : ; . : ; Oblique Asy: 2
2 4 3 2 2e D x x y H A None V A x y x
) : 0, 2, 1; int : 2; int : ; . : ; . : 0, 1; Oblique Asy: 1;f D x x x x y None H A None V A x x y x
) : 1; int : 1.109( ); int : 2; . : ; . : 1; Oblique Asy:g D x x calculator needed y H A None V A x None
a)
O
b)
O
c)
O
d)
O
e)
O
f)
O
Page 16
g)
O
2. 2 20) ( ) 0.1 )3.169 )9.466a C r r c cm d cents
r
3. 2( 3)(2 54)) ( ) ) (0, ) ) 9 . 6 . ) 96
x xa A x b c in by in d in
x
4. a. The graph of f(x) crosses its horizontal asymptote at 5
( ,0)2
.
b. The graph of g(x) crosses its horizontal asymptote at 41
( , 2)6
.
c. The graph of h(x) never crosses its horizontal asymptote. Section 8.4
1. 7
2
x
x
2. not possible 3.
2 3 9
3
x x
x
4. 15
4
y
x 5.
7 4
3
55
32
x y
z 6. 2x
7. ( 5)( 4)
3 ( 7)
x x
x x
8. 2
2(2 3)
( 2)
x
x x
9. 1 10. (2 5)( 4)
3( 2)
x x
x
11. 1
x
x 12.
2 1
4 ( 2)
x
x x
13. ( 6)
(2 1)( 2)
x x
x x
14. ( 2)( 6)
2
x x
x
15.
2
1
x
x 1.
(4 1)(5 6)( 2)
(4 5)( 1)( 6)
x x x
x x x
2. 3
5( 5)
4
t
t
3.
4
2( 1)n
x
x 4.
( 8)( 5)
( 1)
n n
n n
x x
x x
5. horizontal asymptote: y = 1; vertical asymptote: x = 1
6. horizontal asymptote: y = 0; vertical asymptote: x = 3
Page 17
7. horizontal asymptote: y = 0; vertical asymptotes:
8. horizontal asymptote: y = 0;
vertical asymptotes:
9. 10. 11. 12.
13. a) b) about 6,594 gal c) 30 ft by 15 ft by 5 ft
Section 8.5 1. 2x(x + 3)(x 3) 2. 2(x 3)(x+ 4) 3. 3(x 1)(x 2)(x 3) 4. (x + 1)(x 3)(x + 5)(x + 6) 5. 6. 7.
8. 9. 10. 11. 12.
13. 14. 15. 16.
1. 2. 3. 4.
5. a.domain: all real numbers except x = 1, range: all real numbers except y = 0
b. domain: all real numbers except x = 0 and x = 1 ;1
( ( ))x
f f xx
c. ( ( ( )))f f f x x ; the graph is not a line because the graph has holes at x= 0 and x = 1.
6. A = 4, B = 2, C = 2 4 2 2 4( 1)( 1) 2 ( 1) 2 ( 1)
1 1 ( 1)( 1)
x x x x x x
x x x x x x
8
5 x
1 x
x h x 1
11
1
xhx 22
1
xh x xhx
1
25
4
xx
x 2 33
13 2
xx
xx2
2 323 x
xx
1 6
x x
211
6
xx 3 123
9 227 2
xx
xx
112 1137
2
2
xxx
xx
6 6 x
2 1 1 5
x x
2 2
34223
2
xxxx
xx
2 3 33 11 2
x x x x
2610
532
xx
xx
x
x
2 1 2
1 1
2 2
tt x
x
6
3
3
422 xx
Page 18
2 2 2 2 2
2 3 3
4( 1) 2 2 2 2 4 4 4 4
( 1)
x x x x x x x
x x x x x x
Section 8.6
1. no solution 2. 23 3. 1
2 4.
9
5 5. 8 6. 1, 7 7. no solution 8. 8
9. no solution 10. 9, 3 11. 3, 1 12. 1 5
,2 2
13. 3 14. 6, 12
1. 1 2. 2, 5 3. 6 4. 8 5. 6 6. −2 8. −2; 0 and 3 are extraneous.
9. 4( 1)( 2)
( 3)( 4)
x x
x x
Applications of Rational Functions
1a. 550 92 15
128.6715
1b.
550 92( )
nC n
n
1c. 1d. The asymptotes of the rational function are 0n and 92C 1e. The yearly expense of electricity continues no matter how many years the refrigerator works. The cost will never go below the $92, but the cost approaches $92.
1f. Graphing the two functions 550 92
( )n
C nn
and 2
1200 92( )
nC n
n
together or
reviewing a table of values will show the more expensive refrigerator remains more expensive annually although both approach $92 as n approaches infinity.
2a.
2b. 13.76 micrograms which occurs at 18.2 minutes 2c. The graph shows that the concentration is 0 at time 0 . Within the first 15 minutes, the concentration rises sharply to the maximum at 18.2 minutes. After reaching the maximum, the kidneys begin cleansing the blood and rapidly remove the drug. 2d. There is only one asymptote in this rational function which is at 0C . Oddly, this value actually exists in the range of the function at 0C ; however, the graph tends toward 0C as t values increase to infinity. This characteristic gives the asymptotic behavior.
Page 19
2e.
3a. The domains for both are 0x . 3b. asymptotes for both are 0x and 5y 3c. Both Tiffany and Adam are graphing the same function 3d. Answers will vary
4c.
4a. 30 mph 3.33 hours; 55 mph 1.82 hours; 65 mph 1.53 hours. 4b. 100
yx
4d. Faster you travel, fewer hours on the road 5. If x is the number, then 1
xis the
reciprocal. Therefore the sum of a number and its reciprocal is represented by 1
xx
.
Graph the function 1
y xx
for positive x-values. The minimum occurs at the point
(1,2). Section 12.1
1 2 3 4 5 2 1 2 1 1 11. ) 0, , , , , ) 2, 1, , , , ) 1, 2, , 4, ,6
2 3 4 5 6 3 2 5 3 3 5a b c
2 11
1 12. ) ) ) 2 1 ) ( 1) ) ( 1)
3
nn n
n n n n nn
ea a b a c a n d a n e a
n n
1
92
-11 1 1
1 1 1 1 13. ) 1,2,6,24,120,720 ) 1,1,2,3,5,8 4. ) 1 ...
2 3 -1
1) ! 1 2! 3! ... ( -1)! ! 5. ) ) 6. ) 45 ) 39 ) 32
2
n
k
n n
nk k k
a b ak n n
b k n n a k b a b c
Section 12.2
1 11. arithmetic 2. 3( 1) 5 (3 5) 3, 5, 3n ns s n n s d .
13 13. 50 4. ) 96, 3 ) 96 ( 1)( 3) 3 99 na a a d b a n n
21( ) 3 195) 5. 1305 6. 2360 7. 1
2 2 2n
n
n a ac S n n x
Page 20
Section 12.3 2 3 15 15
2 3 15 15 15
2 12 1
1515 15
3 3 3 3 1 1 3(1 3 ) 1 11. ... (3 3 3 ... 3 ) (1 3 ) (3 1)
9 9 9 9 9 9 1 3 6 6
1 5 5 5 1 1 (1 5 ) 1 12. ... (1 5 5 ... 5 ) (1 5 ) (5 1)
9 9 9 9 9 9 1 5 36 364
(1 34(1 3 ) 33. 2(1 3 ) 2(3 1) 4. 1 3
n nn n n
or
or
or
16
16 16
44 1 5
) 2 2(1 3 ) (3 1)
1 3 3 33 9 27 1 1 4 5 20
5. 1 ... 6. 7. 1, -43 3 14 16 64 7 31 ( ) 1 14 4 4
5 (1 5 ) 1 18. a) r = 5, 5 5 5 b) (1 5 ) (5 1)
1 5 2500 2500
nn n n n
n n
or
a S or
Section 12.4
1. 2 2. 75
56 3.
55
2 4. 10 5. doesn’t exist 6.
3
2 7. 3 8. doesn’t exist
9. –72 10. 162 11. doesn’t exist 12. 2 4 and 3x x 13. 3 1 and 2x x
14. 1 5
2
15. 2 16. a) 11000(1.03)n
na b) 150(1.12)nnp
c) 1
1
1000(1.03)
50(1.12)
n
n
; Since the price of the shares are increasing faster than the investment
amount, the number of shares you are able to buy will decrease from the initial 20 shares each year. Section 12.5 1. 9, 37, 149, 597, 2389, 9557 2. –4, 0, 9, 25, 50, 86 3. 3, 3, 3, 2, 0, –4 4. 1 15, 7n na a a 5. 1 17, 3n na a a 6.
1 2 2 12, 3, n n na a a a a
7. 1 15, n na a na 8. 0 1 0 2 1 3 22, ( ) 1, ( ) 8, ( ) 19,x x f x x f x x f x
4 3( ) 62x f x ; the four iterates are 1,8, 19,62 9. 0 1 05, ( ) 2,x x f x
2 1 3 2 4 3( ) 12, ( ) 82, ( ) 6312x f x x f x x f x ; the four iterates are
2,12,82,6312
10. 0 1 0 2 1 3 2 4 35 27 47
2, ( ) 3, ( ) , ( ) , ( )2 10 18
x x f x x f x x f x x f x ; the four
iterates are 5 27 47
3, , ,2 10 18
11.
0 1 0 2 1 3 21 11
3, ( ) 2, ( ) , ( ) ,2 2
x x f x x f x x f x
Page 21
4 3109
( )22
x f x ; the four iterates are 1 11 109
2, , ,2 2 22
12a. 1 1 1 12000 4
210; 0.0035 0.99653500 7n n n nw w w w w
12b. 19 18 17 16 1520
4 4 4 4 4(210)(.9965) (.9965) (.9965) (.9965) (.9965) ...
7 7 7 7 7w
1819
0
4210(.9965) (.9965 ) 206.988
7n
n
13. 31 4 3 42 144a a a 14. 6
1 6 5 5
45 9
3125 6255
aa a
15.
1
2
3
4
11
5 11
1 1 31 1.5
2 1 21 3 2 17
1.4162 2 3 121 17 12 577
1.4142152 12 17 4081 577 408 665857
1.4142142 408 577 470832
1 665857 470832 8.86731 101.414214
2 470832 665857 6.27014 10
As approaches , approacn
a
a
a
a
a
n a
hes 2.