Appendix C
Appendix C.1 (page A14)
1.Rational 3.Irrational 5.Rational 7.Rational
9.Rational 11. 13.
15.(a) True (b) False (c) True (d) False
(e) False (f) False
17. is greater than and less than 3.
The interval is bounded.
19. is no more than 5.
The interval is unbounded.
21. 23.
25. 27.
29. 31.
33. 35.
37. 39.
41. 43.
45. 47. (a) (b)
49. 51.
53.(a) (b)
55.1 57.(a) 14 (b) 10
59. units 61. or 63.(a) (b) 65.b
67.False. The reciprocal of 2 is which is not an integer.
69.True 71.True 73–79. Proofs
81. Proof
Appendix C.2 (page A21)
1.(a) 3.(a)
(b) (b)
(c) (c)5.(a) (b)
(c)
7.Quadrant II 9.Quadrants I and III
11.Right triangle:
13.Rhombus: the length of 15.
each side is 17.
Collinear, because 19.
Not collinear, because 21. 23.
25.
27.c 28.b 29.a 30.d 31.
33.
35.
37. 39.
41. 43.
45. 47.
y
1x
−1 −2 −31
−3−2
1
2, −5
4−y
1
2,1
2
x
2 12
−11
−2−1−2/H20849x/H110011
2/H208502/H11001/H20849y/H110015
4/H208502/H110059
4 /H20849x/H110021
2/H208502/H11001/H20849y/H110021
2/H208502/H110052xy
3 12 −1
−2
−4
−5−3 (1, −3)−2xy
3 12 −1
(1, −3)
−5−2/H20849x/H110021/H208502/H11001/H20849y/H110013/H208502/H110050 /H20849x/H110021/H208502/H11001/H20849y/H110013/H208502/H110054x2/H11001y2/H1100526,0002x2/H11001y2/H110026x/H110024y/H110013/H110050x2/H11001y2/H110012x/H110024y/H110050x2/H11001y2/H110024x/H110012y/H1100211/H110050x2/H11001y2/H110029/H110050/H20898x1/H110013x2
4, y1/H110013y2
4/H20899/H20898x1/H11001x2
2, y1/H11001y2
2/H20899, /H208983x1/H11001x2
4, 3y1/H11001y2
4/H20899,y/H11005±/H2090655 x/H11005±3d1/H11001d2 > d3.d1/H11005/H209062, d2/H11005/H2090613, d3/H110055d1/H11001d2/H11005d3.d1/H110052/H209065, d2/H11005/H209065, d3/H110053/H209065y
3
2
1(1, 2)(3, 3)
(0, 0)(2, 1)
x
3 2 1/H209065.
Number of stores
Year (2 ↔ 2002)xy
2468 1 0 1 2100011001200130014001500160017001800/H20849d1/H208502/H11001/H20849d2/H208502/H11005/H20849d3/H208502d3/H11005/H2090650d2/H11005/H209065d1/H11005/H2090645y
2
x2d
(4, 0)(2, 1)
3d1 −2
−2d2
(−1, −5)/H208980, 1/H11001/H209063
2/H20899/H209068/H110022/H209063
x1 + 3
20, ,23
12
−1−1 −2(−1, 1)(1, 3 )y
/H20849/H110021
2, /H110022/H20850 /H208493, 3/H208502/H2090610 2/H209065y
2
4 2 −2x
−63
2, −5 −1
2, −2 −1
2, 1
−4(4, 5)
(3, 3)
(2, 1)
5x
4 3 2 15
4
3
2
1y/H113413/H110021/H11341/H11005/H113413/H11341/H11002/H113411/H11341; /H11341/H110023/H110021/H11341 > /H11341/H110023/H11341/H11002/H113411/H11341; 1
2,22
7 > /H9266355
113 > /H9266x/H1135059 x/H1134941 x/H1135036/H11341x/H1100212/H11341/H1135010 /H11341x/H1100212/H11341/H1134910/H11341x/H110022/H11341 > 2 /H11341x/H11341/H11349251; /H1100251; 51 /H1100251; 51; 51 4; /H110024; 4x
2 0 −2 −4x
2 −2 0 −4/H110023/H11349x/H113492 /H110023/H11349x/H113491x
4 0 2 −2 2 0 −2x
−40 < x < 3 /H110023 < x < 2x
a + b a − ba x
10 013
−10−7a/H11002b < x < a/H11001b x /H1135013, x/H11349/H110027x
2 1 0 −1 −2x
8 6 4/H110021 < x < 1 x > 67
x2
4 21
2
0 −2−1
x
22
1 0 −1 −2/H110021
2 < x < 7
2 x/H113501
20.03 <r/H113490.07; /H208490.03, 0.07 /H20852 y/H113504; /H208514, /H11009/H20850x
−10123456xx
−4−3−2−101234/H110023 x11
374
11Answers to Odd-Numbered Exercises A1439781285774770_AppC_ans.qxd 9/11/13 10:37 AM Page A143 r 8 ft 15 in. 85 cm 24 in.12,963
/H9266 mi
s 12 ft 24 in. 63.75 cm /H9266 96 in. 8642 mi
/H9258 1.5 1.63/H9266
442/H9266
3A144 Answers to Odd-Numbered Exercises
49. 51.
53.Proof 55.True 57.True 59–61. Proofs
Appendix C.3 (page A31)
1.(a) (b)
3.(a) (b)
5.(a) (b)
(c) (d)
7.(a) (b) (c) (d)
9.
11.(a) (b)
13.(a) Quadrant III (b) Quadrant IV
15. 17.
19.(a) (b)
(c) (d)
21.(a) (b)
(c) (d)
23.(a) 0.1736 (b) 5.7588 25.(a) 0.3640 (b) 0.364027.(a) (b)
29.(a) (b)
31. 33.
35. 37. 39. 5099 ft
41.(a) Period: 43.Period:
Amplitude: 2 Amplitude: 3
(b) Period: 2
Amplitude:
45.Period: 47.Period:
49.(a) (b)
Change in amplitude Change in period
(c)
Horizontal translation
51. 53.
55. 57.
59. 61.
63.a/H110053, b/H110051
2, c/H11005/H9266
23
2 22
π
2ππ π−y
xy
x
21
−12ππy
x
23
21
ππ πy
3
x
312
−1π πy
x
3 3
21
−1y
x
31
−1π π−1.5−c = −2
c = −1 c= 1c= 2
2 21.5−1−c = ±2 c = ±11
−2.5−
c= 2c= 1c = −2
c = −12.52/H9266
5/H9266
21
21
2 /H9266/H9258/H110050, /H9266
2, /H9266 /H9258/H11005/H9266
3, 5/H9266
3/H9258/H110050, /H9266
4, /H9266, 5/H9266
4/H9258/H11005/H9266
4, 3/H9266
4, 5/H9266
4, 7/H9266
4/H9258/H110055/H9266
6, 11/H9266
6/H9258/H11005/H9266
4, 5/H9266
4/H9258/H110053/H9266
4, 5/H9266
4/H9258/H11005/H9266
4, 7/H9266
4
tan 11/H9266
6/H11005/H11002/H209063
3tan 5/H9266
3/H11005/H11002/H209063cos 11/H9266
6/H11005/H209063
2cos 5/H9266
3/H110051
2sin 11/H9266
6/H11005/H110021
2sin 5/H9266
3/H11005/H11002/H209063
2tan/H20849/H11002225/H11034/H20850/H11005/H11002 1 tan 225 /H11034/H110051cos/H20849/H11002225/H11034/H20850/H11005/H11002/H209062
2cos 225 /H11034/H11005/H11002/H209062
2sin/H20849/H11002225/H11034/H20850/H11005/H209062
2sin 225 /H11034/H11005/H11002/H209062
2tan 5/H9266
4/H110051 tan /H9266
4/H110051cos 5/H9266
4/H11005/H11002/H209062
2cos /H9266
4/H11005/H209062
2sin 5/H9266
4/H11005/H11002/H209062
2sin /H9266
4/H11005/H209062
2tan 120 /H11034/H11005/H11002 /H209063 tan 60 /H11034/H11005/H209063cos 120 /H11034/H11005/H110021
2cos 60 /H11034/H110051
2sin 120 /H11034/H11005/H209063
2sin 60 /H11034/H11005/H209063
24
3/H209063
2cot /H9258/H1100512
5 tan /H9258/H110055
12 cot /H9258/H110053
4 tan /H9258/H110054
3sec /H9258/H11005/H1100213
12 cos /H9258/H11005/H1100212
13 sec /H9258/H110055
3 cos /H9258/H110053
5csc /H9258/H11005/H1100213
5 sin /H9258/H11005/H110025
13 csc /H9258/H110055
4 sin /H9258/H110054
5/H11002151.1 /H11034 /H11002 105/H11034 210/H11034 270/H110342/H9266
3; 2.0947/H9266
4; 5.4985/H9266
6; 2.618/H9266
6; 0.52410/H9266
3, /H110022/H9266
319/H9266
9, /H1100217/H9266
9240/H11034, /H11002480/H11034 396/H11034, /H11002324/H11034y
x
5 3 1
(2, −1)
−4−3−1
−2123
−9−999781285774770_AppC_ans.qxd 9/11/13 10:37 AM Page A144 65.
The graph of will reflect any parts of the graph of below the -axis about the -axis. The graph of will
reflect the part of the graph of right of the -axis about the
-axis.
67.
January, November, December
69.
2
−2−1 3f/H20849x/H20850/H110054
/H9266 /H20898sin /H9266x/H110011
3 sin 3 /H9266x/H110011
5 sin 5 /H9266x/H11001. . ./H20899100
001 2yy f /H20849x/H20850f/H20849/H11341x/H11341/H20850 x xf/H20849x/H20850 /H11341f/H20849x/H20850/H113412
−12
−2y
x
π πh(x) = sin (⏐x⏐)2
−12
−2−2 −y
x
π π π πg(x) = ⏐sin x⏐
2 −12
−2y
xf(x) = sin x
π π πAnswers to Odd-Numbered Exercises A1459781285774770_AppC_ans.qxd 9/11/13 10:37 AM Page A145 © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copi ed or duplicated, or posted to a publicly accessible webs ite, in whole or in part. CHAPTER 1 Preparation for Calculus Section 1.1 Graphs and Models................................................................................. 2 Section 1.2 Linear Models and Rates of Change.................................................... 11 Section 1.3 Functions and Their Graphs................................................................. 22 Section 1.4 Fitting Mode ls to Data.......................................................................... 34 Section 1.5 Inverse Functions.................................................................................. 37 Section 1.6 Exponential and Loga rithmic Functions ............................................. 54 Review Exercises .......................................................................................................... 63 Problem Solving ........................................................................................................... 73
Appendix C.1 (page A14)
1.Rational 3.Irrational 5.Rational 7.Rational
9.Rational 11. 13.
15.(a) True (b) False (c) True (d) False
(e) False (f) False
17. is greater than and less than 3.
The interval is bounded.
19. is no more than 5.
The interval is unbounded.
21. 23.
25. 27.
29. 31.
33. 35.
37. 39.
41. 43.
45. 47. (a) (b)
49. 51.
53.(a) (b)
55.1 57.(a) 14 (b) 10
59. units 61. or 63.(a) (b) 65.b
67.False. The reciprocal of 2 is which is not an integer.
69.True 71.True 73–79. Proofs
81. Proof
Appendix C.2 (page A21)
1.(a) 3.(a)
(b) (b)
(c) (c)5.(a) (b)
(c)
7.Quadrant II 9.Quadrants I and III
11.Right triangle:
13.Rhombus: the length of 15.
each side is 17.
Collinear, because 19.
Not collinear, because 21. 23.
25.
27.c 28.b 29.a 30.d 31.
33.
35.
37. 39.
41. 43.
45. 47.
y
1x
−1 −2 −31
−3−2
1
2, −5
4−y
1
2,1
2
x
2 12
−11
−2−1−2/H20849x/H110011
2/H208502/H11001/H20849y/H110015
4/H208502/H110059
4 /H20849x/H110021
2/H208502/H11001/H20849y/H110021
2/H208502/H110052xy
3 12 −1
−2
−4
−5−3 (1, −3)−2xy
3 12 −1
(1, −3)
−5−2/H20849x/H110021/H208502/H11001/H20849y/H110013/H208502/H110050 /H20849x/H110021/H208502/H11001/H20849y/H110013/H208502/H110054x2/H11001y2/H1100526,0002x2/H11001y2/H110026x/H110024y/H110013/H110050x2/H11001y2/H110012x/H110024y/H110050x2/H11001y2/H110024x/H110012y/H1100211/H110050x2/H11001y2/H110029/H110050/H20898x1/H110013x2
4, y1/H110013y2
4/H20899/H20898x1/H11001x2
2, y1/H11001y2
2/H20899, /H208983x1/H11001x2
4, 3y1/H11001y2
4/H20899,y/H11005±/H2090655 x/H11005±3d1/H11001d2 > d3.d1/H11005/H209062, d2/H11005/H2090613, d3/H110055d1/H11001d2/H11005d3.d1/H110052/H209065, d2/H11005/H209065, d3/H110053/H209065y
3
2
1(1, 2)(3, 3)
(0, 0)(2, 1)
x
3 2 1/H209065.
Number of stores
Year (2 ↔ 2002)xy
2468 1 0 1 2100011001200130014001500160017001800/H20849d1/H208502/H11001/H20849d2/H208502/H11005/H20849d3/H208502d3/H11005/H2090650d2/H11005/H209065d1/H11005/H2090645y
2
x2d
(4, 0)(2, 1)
3d1 −2
−2d2
(−1, −5)/H208980, 1/H11001/H209063
2/H20899/H209068/H110022/H209063
x1 + 3
20, ,23
12
−1−1 −2(−1, 1)(1, 3 )y
/H20849/H110021
2, /H110022/H20850 /H208493, 3/H208502/H2090610 2/H209065y
2
4 2 −2x
−63
2, −5 −1
2, −2 −1
2, 1
−4(4, 5)
(3, 3)
(2, 1)
5x
4 3 2 15
4
3
2
1y/H113413/H110021/H11341/H11005/H113413/H11341/H11002/H113411/H11341; /H11341/H110023/H110021/H11341 > /H11341/H110023/H11341/H11002/H113411/H11341; 1
2,22
7 > /H9266355
113 > /H9266x/H1135059 x/H1134941 x/H1135036/H11341x/H1100212/H11341/H1135010 /H11341x/H1100212/H11341/H1134910/H11341x/H110022/H11341 > 2 /H11341x/H11341/H11349251; /H1100251; 51 /H1100251; 51; 51 4; /H110024; 4x
2 0 −2 −4x
2 −2 0 −4/H110023/H11349x/H113492 /H110023/H11349x/H113491x
4 0 2 −2 2 0 −2x
−40 < x < 3 /H110023 < x < 2x
a + b a − ba x
10 013
−10−7a/H11002b < x < a/H11001b x /H1135013, x/H11349/H110027x
2 1 0 −1 −2x
8 6 4/H110021 < x < 1 x > 67
x2
4 21
2
0 −2−1
x
22
1 0 −1 −2/H110021
2 < x < 7
2 x/H113501
20.03 <r/H113490.07; /H208490.03, 0.07 /H20852 y/H113504; /H208514, /H11009/H20850x
−10123456xx
−4−3−2−101234/H110023 x11
374
11Answers to Odd-Numbered Exercises A1439781285774770_AppC_ans.qxd 9/11/13 10:37 AM Page A143 r 8 ft 15 in. 85 cm 24 in.12,963
/H9266 mi
s 12 ft 24 in. 63.75 cm /H9266 96 in. 8642 mi
/H9258 1.5 1.63/H9266
442/H9266
3A144 Answers to Odd-Numbered Exercises
49. 51.
53.Proof 55.True 57.True 59–61. Proofs
Appendix C.3 (page A31)
1.(a) (b)
3.(a) (b)
5.(a) (b)
(c) (d)
7.(a) (b) (c) (d)
9.
11.(a) (b)
13.(a) Quadrant III (b) Quadrant IV
15. 17.
19.(a) (b)
(c) (d)
21.(a) (b)
(c) (d)
23.(a) 0.1736 (b) 5.7588 25.(a) 0.3640 (b) 0.364027.(a) (b)
29.(a) (b)
31. 33.
35. 37. 39. 5099 ft
41.(a) Period: 43.Period:
Amplitude: 2 Amplitude: 3
(b) Period: 2
Amplitude:
45.Period: 47.Period:
49.(a) (b)
Change in amplitude Change in period
(c)
Horizontal translation
51. 53.
55. 57.
59. 61.
63.a/H110053, b/H110051
2, c/H11005/H9266
23
2 22
π
2ππ π−y
xy
x
21
−12ππy
x
23
21
ππ πy
3
x
312
−1π πy
x
3 3
21
−1y
x
31
−1π π−1.5−c = −2
c = −1 c= 1c= 2
2 21.5−1−c = ±2 c = ±11
−2.5−
c= 2c= 1c = −2
c = −12.52/H9266
5/H9266
21
21
2 /H9266/H9258/H110050, /H9266
2, /H9266 /H9258/H11005/H9266
3, 5/H9266
3/H9258/H110050, /H9266
4, /H9266, 5/H9266
4/H9258/H11005/H9266
4, 3/H9266
4, 5/H9266
4, 7/H9266
4/H9258/H110055/H9266
6, 11/H9266
6/H9258/H11005/H9266
4, 5/H9266
4/H9258/H110053/H9266
4, 5/H9266
4/H9258/H11005/H9266
4, 7/H9266
4
tan 11/H9266
6/H11005/H11002/H209063
3tan 5/H9266
3/H11005/H11002/H209063cos 11/H9266
6/H11005/H209063
2cos 5/H9266
3/H110051
2sin 11/H9266
6/H11005/H110021
2sin 5/H9266
3/H11005/H11002/H209063
2tan/H20849/H11002225/H11034/H20850/H11005/H11002 1 tan 225 /H11034/H110051cos/H20849/H11002225/H11034/H20850/H11005/H11002/H209062
2cos 225 /H11034/H11005/H11002/H209062
2sin/H20849/H11002225/H11034/H20850/H11005/H209062
2sin 225 /H11034/H11005/H11002/H209062
2tan 5/H9266
4/H110051 tan /H9266
4/H110051cos 5/H9266
4/H11005/H11002/H209062
2cos /H9266
4/H11005/H209062
2sin 5/H9266
4/H11005/H11002/H209062
2sin /H9266
4/H11005/H209062
2tan 120 /H11034/H11005/H11002 /H209063 tan 60 /H11034/H11005/H209063cos 120 /H11034/H11005/H110021
2cos 60 /H11034/H110051
2sin 120 /H11034/H11005/H209063
2sin 60 /H11034/H11005/H209063
24
3/H209063
2cot /H9258/H1100512
5 tan /H9258/H110055
12 cot /H9258/H110053
4 tan /H9258/H110054
3sec /H9258/H11005/H1100213
12 cos /H9258/H11005/H1100212
13 sec /H9258/H110055
3 cos /H9258/H110053
5csc /H9258/H11005/H1100213
5 sin /H9258/H11005/H110025
13 csc /H9258/H110055
4 sin /H9258/H110054
5/H11002151.1 /H11034 /H11002 105/H11034 210/H11034 270/H110342/H9266
3; 2.0947/H9266
4; 5.4985/H9266
6; 2.618/H9266
6; 0.52410/H9266
3, /H110022/H9266
319/H9266
9, /H1100217/H9266
9240/H11034, /H11002480/H11034 396/H11034, /H11002324/H11034y
x
5 3 1
(2, −1)
−4−3−1
−2123
−9−999781285774770_AppC_ans.qxd 9/11/13 10:37 AM Page A144 65.
The graph of will reflect any parts of the graph of below the -axis about the -axis. The graph of will
reflect the part of the graph of right of the -axis about the
-axis.
67.
January, November, December
69.
2
−2−1 3f/H20849x/H20850/H110054
/H9266 /H20898sin /H9266x/H110011
3 sin 3 /H9266x/H110011
5 sin 5 /H9266x/H11001. . ./H20899100
001 2yy f /H20849x/H20850f/H20849/H11341x/H11341/H20850 x xf/H20849x/H20850 /H11341f/H20849x/H20850/H113412
−12
−2y
x
π πh(x) = sin (⏐x⏐)2
−12
−2−2 −y
x
π π π πg(x) = ⏐sin x⏐
2 −12
−2y
xf(x) = sin x
π π πAnswers to Odd-Numbered Exercises A1459781285774770_AppC_ans.qxd 9/11/13 10:37 AM Page A145 © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copi ed or duplicated, or posted to a publicly accessible webs ite, in whole or in part. CHAPTER 1 Preparation for Calculus Section 1.1 Graphs and Models................................................................................. 2 Section 1.2 Linear Models and Rates of Change.................................................... 11 Section 1.3 Functions and Their Graphs................................................................. 22 Section 1.4 Fitting Mode ls to Data.......................................................................... 34 Section 1.5 Inverse Functions.................................................................................. 37 Section 1.6 Exponential and Loga rithmic Functions ............................................. 54 Review Exercises .......................................................................................................... 63 Problem Solving ........................................................................................................... 73
