Mat 170 Homework problems Introduction Exercise Set 0.1 In problems 1-10, write down each of the following without absolute value sign (do not simplify), x is a real number: 1. 4 3 2. 13 3 3. 4 4. 3 5. 2 10 6. 3 8 1 7. 3 2 8. 2 4 3x 9. 3 15 x , if 5 x 10. 3 4 x x , if 3 4 x In problems 11-13, use order operation to simplify the expressions (always perform division before multiplication). One may remember PEDMAS (P for parentheses, E for exponents, D for division, M for multiplication, A and S for addition and subtraction). 11. 15 5 4 6 8 6 ( 5) 8 2 12. 2 21 37 25 5 2 13. 3 8 ( 4)( 2 ) 16 2 4 3 6 2 In problems 14-17, evaluate the expressions 14. , 0 x x x 15. 3 , 3 3 x x x 16. 2 3 , if 4 9 x x x 17. 2 3 , 3 9 x x x In problems 18-20, determine the value of the following expressions 18. 13.2 19. 13.2 20. 13.2 + 13.2 21. Classify the following numbers as whole number, rational number, and/or irrational number: 3 1 , 3.6666..., 13, 81, 64, 30, 5, 0 3 Simplify the expressions (22-24) 22. 2[5 7( 2) 2] x x 23. 3( 7) 4 x x 24. 1 (12 8) (4) 4 x x Exercise Set 0.2 Simplify with positive exponents
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Mat 170 Homework problems
Introduction
Exercise Set 0.1 In problems 1-10, write down each of the following without absolute value sign (do not simplify), x is a
real number:
1. 4 3 2. 13 3 3. 4 4. 3
5. 2 10 6. 3 8 1 7. 3 2 8. 24 3x
9. 3 15x , if 5x 10. 3 4x x , if 3 4x
In problems 11-13, use order operation to simplify the expressions (always perform division before
multiplication). One may remember PEDMAS (P for parentheses, E for exponents, D for division, M for
multiplication, A and S for addition and subtraction).
11. 15 5 4 6 8
6 ( 5) 8 2 12. 221 3 7 25 5 2 13.
38 ( 4)( 2 ) 16 2
4 3 6 2
In problems 14-17, evaluate the expressions
14. , 0x
xx
15. 3
, 33
xx
x 16.
2
3, if 4
9
xx
x 17.
2
3, 3
9
xx
x
In problems 18-20, determine the value of the following expressions
18. 13.2 19. 13.2 20. 13.2 + 13.2
21. Classify the following numbers as whole number, rational number, and/or irrational number:
31, 3.6666..., 13, 81, 64, 30, 5, 0
3
Simplify the expressions (22-24)
22. 2[5 7( 2) 2]x x 23. 3( 7) 4x x 24. 1
(12 8) ( 4 )4
x x
Exercise Set 0.2
Simplify with positive exponents
1. 4 45 2 2. 4 4( 5) 2 3. 3
3
( 15)
3 4.
3
3
( 2)
3
5.
03
3
15
3 6. 3 2( 3 ) 7. 2 42 2 8.
0
3
( 8)
3 2
9. 143
3
x
x 10. 7 14( )( )x x 11. 2 4 2( 3 )x y 12.
3
6
y
13.
3
2
10
x 14.
25
3
3x
y 15.
3 6
5 10
35
5
x y
x y 16.
25 4
10 2
10
30
x y
x y
17.
33 5
2 3
20
100
x y
x y 18.
30
60 1/ 2( )
x
x 19.
5
1/5 4 5( )
x y
x y 20.
3 2 2
4 1 3
( )
( )
x y
x y
Express the given numbers in scientific notations
21. 2860000000 22. 1220000 23. 0.0000000142 24. 0.00808
Simplify the numbers
25. 51.23 10 26. 51.23 10 27. 5
3
1.23 10
2.54 10 28. 51.23 10 40
Exercise Set 0.3 Simplify the expressions
1. 1/3( 125) 2. 1/416 3. 4 625 4. 1/ 25 5. 1/ 7( 128)
6. 1/7128 7. 2/ 7(128) 8. 3 27 /125 9. 3 27 /125 10. 3 24 / 81
Rationalize the denominator
11. 2 5
7 3 12.
12 3
6 13.
3 11
3 11 14.
2 3
2 3 3 2
15. 4 5
1 2 16.
3
5 17.
10
7 3 18.
13
10 3
19. 5
1 3 20.
1 2
(2 3)(1 2)
Simplify the expressions
21. 5
31
2700
100
x
x 22.
8
2
40
10
x
x 23. 2 36 3 2x x x 24.
5 12
5 2
64
2
x
x
25. 3 3 3( 9 )( 81)x 26. 32 32 8x x
Exercise Set 0.4 1. Determine the polynomial and its degree
a) 22 5 13x x b) 22 5 13x x
c) 3 2 122 9x x
x d) 2 4 99 13 10x x x
e) 5 1/4 99 9 13 10x x x f) 2 4 9
2
9 13 10x x x
x
g) 2 4 9 3
2
9 13 10x x x x
x h)
2 4 9 2
2
9 13 10x x x x
x
i) 2 3x j) 2 2 3(9 4 2)x x
k) 4 513 2 9x x x l) 4 2503 9x x
m) 6 3256 32 1x x n) 2 4 919 13 10x x x
2. Perform the indicated operations, write the result in standard form of a polynomial and indicate its degree.
a) 2 4 9 2 12( 9 13 10) (3 12 9)x x x x x
b) 2 5 2(3 9 1) (15 12 9)x x x x
c) 9 2 4 2 93 2 5 3 (15 3 ) 2( 9)x x x x x x
d) 2 4 9 2 12( 9 13 10)(3 12 )x x x x x
e) 33( 9)( 9)x x
f) ( 1)( 2)( 3)x x x
Exercise Set 0.5
Factor completely
1. 2 13 36x x
2. 3 25 36x x x
3. 2 15 26x x
4. 23 24 36x x
5. 2 312 6 8ax ax ax
6. 2 3 3 212 48x y x y
7. 3 29 9x xa
8. 2 481 36x a
9. 2 ( ) 3 3x a b a b
10. 2 236 18x x
11. 22 72 648x x
12. 2 2( 3) ( 3)x x
13. 3 38 27x a
14. 210 17 7a a
15. 4 3 29 18 27x x x
16. 2 (2 5) 4(2 5)x x x
17. 2 2( 1) ( 1)x x x
Exercise Set 0.6
1. Find domain of the following functions
a) 21
13 x b)
2 3
3
x c)
2
2 16
64
x
x d)
2
3 3
2 3
x
x x
e) 2
5
2 11 5
x
x x f)
2
13
169
x
x g)
3
3
9 4
x
x x h)
2
100
5 14x x
2. Simplify and determine the domain
a) 1 2 2 13
1 2 1
x x x
x x x b)
2 6( 2)1
2 6
x x
x x
c) x b a a b a
a x b a a b d)
2 2
2 2
2(45 2 ) 3( 9)7
9 3
x x
x x
e)
22 21 5 2 1
92
x x x
x x f) ( )
x bc x ac x aba b c
b c c a a b
g) 2 4 3
23
x x
x h)
2
1
2 1
x
x x
3. Simplify and find domain
a) 2
2
2 8 3
9 2
x x x
x x b)
2 3 2
12 6 2
8 32 3
x x
x x x
c) 2 3 9( 9)
2 5
xx
x d)
2 2
2
25 4 5
4 4 10 25
x x x
x x x
e) 2
2 3 4 2 10
5 25 8 12
x x x
x x x f)
2
2 3 4 2 10
5 25 8 12
x x x
x x x
Exercise Set 0.7 - 0.8
Solve the following Inequalities and write
answers in interval(s), use real line test:
1. 13419 x
2. 6 (2 3) 4 5x x x
3. 6 (2 3) 4 5x x x
4. 3 1 5
32 2 6
x x
5. 4 7
2 53
xx
6. 4 7
2 5 03
xx
7. 4
3 45
x
8. 4
3 45
x
9. 5
14x
10. 5
14x
11. 2 1
53 4
x
x
12. 2 1
53 4
x
x
13. 3 4
2 1x x
14. 3 4
2 1x x
15. 4 3
2 1x x
16. 4 3
2 1x x
17. 4 3
2 1x x
18. 7 1
2 2x x
19. 7 1
2 2x x
20. 3
05
x
x
21. 3
05
x
x
22. 1
1 02
x
x
23. 062 xx
24. 22 5 12 0x x
25. 22 5 12 0x x
26. 2 12 0x x
27. 22 9 4 0x x
28. 22 9 4 0x x
29. 2 9 0x x
30. 2 4x x
31. 2 4 3 0x x
32. 2 16x
33. 2 16x
34. 2 16x
35. 2 5 6 0x x
36. 2 5 6 0x x
37. 2 6 0x x
38. 2 6 0x x
39. 2 5 6 0x x
40. 2 5 6
01
x x
x
41. 2 5 6
01
x x
x
42. 2 5 6
01
x x
x
43. 2 5 6
02
x x
x
44. 2
2
3 2 3
4 2
x x x
x x
45. 2
2
3 2 3
4 2
x x x
x x
Hint: Do not cross multiply to solve. Use
3 12 0 0
1 1
x x
x x etc
46. 1 3x
47. 2 5 7x Hint: This inequality
has no solution, as the left side is
always positive.
48. 2 5 7x
49. 2 5 6x x
50. 2 1 8x
51. 23 6 3
5 5
x x
52. 2 5 6x x
53. Evaluate the following and write as
a ib :
a) (3 5 ) (7 ) 3i i i
b) (3 ) ( 17 ) 3i i i
c) (3 5 ) (7 2 )i i
d) (3 5 )(7 )i i
Exercise Set 1
Plot the points in the same xy – plane
1. (5, 1), (0,0), ( 2, 1), (3, 2), ( 1,0), (4,4)
2. ( 4, 4), (4, 4), ( 4,4), ( 4, 4)
3. (1,5), (0,2), (5,0), ( 3,0), (0, 1), (3,3), ( 3, 3)
4. (100,50), (30, 50), ( 70,20), ( 100, 100)
5. ( 5,0), (0, 2), (3,1), (1,5)
Find the distance and mid point
6. (3,5), (7,3) 7. ( 5, 3), (1,3)
8. ( 10, 10), (10,10) 9. (0,0), ( 4, 6)
10. (10, 2), (4,0) 11. (3,9), (6,9)
12. (18,2), (2,2) 13. ( 3, 1), ( 5, 7)
14. (0,6), (6,0) 15. (7,0), (9,0)
Write in the standard form of y mx c ,
find slope, y – intercept and determine if
the straight lines are parallel, perpendicular
or neither. Plot the straight lines.
16. 3 2 5; 2 3 10 0x y x y
17. 4 2 10; 2 9 0x y x y
18. 3 4 12; 2 5 0x y x y
19. 1; 5 03 4
x yx y
20. 5 0; 10 0x y x y
21. 2 5 8
1; 5 03 4 15
x yx y
22. 11 4 10; 3 5 0x y x y
Plot the following functions:
23. 23 12 3y x
24. 2 10y x
25. 23 15 18 0y x
26. 215 4
2y x
27. 215 4
2y x
In exercises 28 – 70, discuss the
transformation of the function using the
techniques of shifting, stretching, shrinking
and/or reflecting. Compare the given
function with its basic function:
In exercises 28 – 37 use the basic function 2( )f x x
28. 2( ) 1F x x 29. 2( ) 2 3F x x
30. 2( ) 2 4F x x 31. 2( ) 2( 1) 1F x x
32. 2( ) ( 1) 3F x x 33. 2( ) 3( 2) 5F x x
34. 2( ) (2 1) 1F x x 35. ( ) 2 ( 1) 3h x f x
36. ( ) ( 1) 3F x f x 37. 1
( ) ( ) 33
F x f x
In exercises 38 – 49 use the basic function 3( )f x x
38. 3( ) 1F x x 39. 3( ) 2 3F x x
40. 3( ) 2 4F x x 41. 3( ) 2( 1) 1F x x
42. 3( ) ( 1) 3F x x 43. 3( ) 3( 2) 5F x x
44. 3( ) (2 1) 1F x x 45. ( ) 2 ( 1) 3h x f x
46. ( ) ( 1) 3F x f x 47. 1
( ) ( ) 33
F x f x
48. 1
( ) ( 1)2
F x f x 49. 1
( ) 32
F x f x
In exercises 50 – 56 the basic function is
( )f x x
50. ( ) 1F x x 51. ( ) 2 1 1F x x
52. ( ) 2 1 3F x x 53. 1
( ) 1 12
F x x
54. ( ) 2 ( 1) 10F x f x
55. ( ) ( 0.5 ) 7F x f x
56. ( ) 2 (3 1) 2F x f x
In exercises 57 – 64 the basic function is
( )f x x
57. ( ) ( 1) 3F x f x 58. 1
( ) ( ) 33
F x f x
59. 1
( ) ( 1)2
F x f x 60. 1
( ) 32
F x f x
61. ( ) ( 0.5 ) 7F x f x 62. ( ) 2 (3 1)F x f x
63. ( ) 2 3 6 2F x x 64. ( ) 2 (3 ) 2F x f x
In exercises 65 – 70 the basic function is 1
( )f xx
65. 1
( ) ( 1)2
F x f x 66. 1
( ) 32
F x f x
67. ( ) ( 0.5 ) 7F x f x 68. ( ) 2 (3 1)F x f x
69. 1
( ) 23
xF x 70.
5( ) 2 9
2
xF x f
In exercises 71 – 76 perform the operation
using the functions ( ) 2 1, ( )f x x g x x .
Also determine their domain.
71. ( ) ( ) 2 ( )F x f x g x
72. ( ) 3 ( ) 2 ( )F x f x g x
73. ( ) ( ) ( ) 3F x f x g x
74. ( )
( )( )
f xF x
g x
75. ( ) ( ) ( )F x f x g x
76. ( ) ( ) ( )F x g x f x
In exercises 77 – 87 find inverse of the
given function and verify that
f g g f x , where f is the inverse of
g and g is the inverse of f respectively.
77. ( ) 3 4f x x 78. ( ) 3 4f x x
79. ( ) 4f x x 80. 3( ) 3 4f x x
81. 2
( ) , 23 6
xf x x
x
82. 1 4
( ) ,3 4 3
f x xx
83. 2
( ) 3, 0f x xx
84. 3( ) 2 6f x x
85. 4 5 3
( ) ,2 3 2
xf x x
x
86. ( ) 1, 1g x x x
87. 2
( ) , 22
g x xx
88. The function 2( ) ( 3)f x x is not one-
to-one. Choose a largest possible
domain containing the number –30 so
that the function restricted to the
domain is one-to-one. Find the inverse
on the restricted domain.
89. The function 2( ) ( 3)f x x is not one-
to-one. Choose a largest possible
domain containing the number 100 so
that the function restricted to the
domain is one-to-one. Find the inverse
on the restricted domain.
90. Plot the following points for ( )y f x :
(1,1) , ( 1, 1) , (2,8) , ( 2, 8) . Then plot
the points for 1( )y f x
91. Plot the function 2
, 55
y xx
, choose
several points on the function to plot the
inverse function.
Chapter 2
In exercises 1 – 12 identify the polynomial as
monomial, quadratic, cubic, etc., and as even, odd
or neither.
1. ( ) 2P x x 2. ( ) 5 1P x x
3. 2( ) 2 1P x x x 4. 3( ) 4P x x
5. ( ) 12 4
x yP x 6. 7( ) 2 3 1P x x x
7. 3( ) 3 8 1P x x x 8. 5 3( ) 9 2 5P x x x
9. 3( ) 8P x x x 10. 6( ) 9P x x
11. 2( )P x x x 12. 3( )P x x x
In exercises 13 – 20 consider the standard form for
the quadratic polynomial 2( ) ( )P x a x h k to
find maximum or minimum value of the given
polynomials:
13. 21 1( ) 10
3 5P x x x
14. 2( ) 3P x x x
15. 2( ) 3P x x x
16. 2( ) 8 5P x x x
17. 2( ) 3 23 14P x x x
18. 2( ) 5 20 1P x x x
19. 2( ) 2 84P x x x
20. 2( ) 3 26 16P x x x
In exercises 21 – 34 discuss the end behavior
21. 19 15 2( ) 17 3 2 13P x x x x
22. 2( ) 5 13P x x x
23. 3 2( ) 3 2P x x x x
24. 7 3( ) 5 2 9P x x x
25. 2( ) ( 7)( 3)P x x x x
26. 2( ) 3( 3) ( 8)P x x x
27. 4 2( ) 3 23 14 1P x x x x
28. 4 2( ) 23 14 1P x x x x
29. 7 2( ) 3 1P x x x x
30. 7 2( ) 3 14 1P x x x x
31. 3( ) ( ) 27P x x
32. 2( ) 9 78 160P x x x
33. 3( ) 27P x x
34. 3 2( ) 2 2P x x x x
In exercises 35 – 41 solve for x using
complete factors.
35. 2( ) 2( ) 4 , ,P x x a b x ab a b are
constants.
36. 4 3 2( ) 2P x x x x
37. 3 2( ) 16 16P x x x x
38. 3 2( ) 2 2P x x x x
39. 4( ) 16P x x
40. 2( ) 2 5 3P x x x
41. 2( ) 2 6 1P x x x
In exercises 42 – 44 divide the first polynomial by
the second polynomial.
42. 2( ) 3 7P x x x , ( ) 3P x x .
43. 2( ) 5 3 7P x x x , ( ) 2 3P x x
44. 2( ) 5 7P x x x , ( ) 3P x x
In exercises 45 – 54 find domain, vertical,
horizontal, and/or slant Asymptote, hole (if any)
45. 2
2
2 5 1( )
36
x xf x
x
46. 3
4 3 2
3 5( )
12
x xf x
x x x
47. 3
2
5 5( )
12
x xf x
x x
48. 2
3( )
12
xf x
x x
49. 2
3 2
4( )
12
x xf x
x x x
50. 1
( )12
f x xx
51. 236 6 2
( )5 9
x xf x
x
52. 2
2
7 62 80( )
4
x xf x
x
53. 2
2
16( )
3 27
xf x
x
54. (3 8)( 8)
( )( 8 6)(8 8)
x xf x
x x
In exercises 55 – 64 use rational zero test to
list a) all possible rational zeros, and
b) find real zeros by factoring
c) use synthetic division to find zeros
55. 2( ) 289 289 42P x x x
56. 3 2( ) 3 13 15P x x x x
57. 3 2( ) 2 9 4P x x x x
58. 2( ) 4P x x x
59. 3 2( ) 5 4 20P x x x x
60. 4 2( ) 4 15P x x x x
61. 3 2( ) 4 8 15P x x x x
62. 3 2( ) 6 14 15P x x x x
63. 4 3 2( ) 12 54 108 81P x x x x x
64. (3 8)( 8)
( )( 8 6)(8 8)
x xf x
x x
65. Find a third degree polynomial with
zeros 1, 2, 3 and leading coefficient –2.
66. Find all zeros of 4( ) 16P x x
67. Find a polynomial with zeros 1, 2i and
leading coefficient 1.
68. Find a rational function having vertical
asymptotes at 7 5
,5 2
x x and
horizontal asymptote at 7
10y .
69. Find a polynomial of degree 3 with
leading coefficient –1 and zeros at
4, and –5i. Simplify the polynomial
with real coefficients.
70. Find a polynomial of degree 4 with
leading coefficient –1 and zeros of
multiplicity 2 at 4, and –5i.
Simplify the polynomial with real
coefficients.
71. Find a rational function having
vertical asymptotes at 3, 3x x
horizontal asymptote at 1y
and x intercepts at (4, 0), and ( 4,0)
In exercises 72 – 76 use Descartes rule of sign to
determine the nature of roots
72. 2( ) 289 289 42f x x x
73. 3 2( ) 5 4 20f x x x x
74. 3 2( ) 6 14 15g x x x x
75. 4 3 2( ) 12 54 108 81h x x x x x
76. 2( ) 4g x x x
77. The concentration of a drug t
seconds after injection is given by
2
4 3( )
2 1
tC x
t
Estimate the time when will the
concentration be maximum.
Determine the horizontal asymptote
(if any) and explain in this context.
78. The population of a certain species
in millions is given by the rational
function 20 2
( )3 2
tP t
t , where t is
in months after January 1st, 2000.
a) Graph the polynomial b) Estimate the initial population c) Estimate population for January 1st,
2010 d) Estimate population in the long run.
Exercise Set 3.1
In Exercises 1 – 9, classify the angles as
acute, right, obtuse or straight and reflex
and draw each angle. If degree sign is not
given the figure is in radian measure.
1. 177 2. 137.5 3. 96.33
4. 1.2 5. 17
3
6. 17
8
7. 25.7 8. 9. 278
In Exercises 10 – 21, convert each angle in
degrees to radians, express your answer as
multiple of
10. 177 11. 137.5 12. 96.33
13. 70 14. 90 15. 300
16. 270 17. 27 18. 25.5
19. 18 20. 720 21. 5
In Exercises 22 – 30, convert each angle in
radians to degrees
22. 23. 3
2 24. 5
25. 2.3 26. 2
13 27.
5
6
28. 2
3 29.
4 30. 7
In Exercises 31 – 39, find an angle between 0
and 2 that is coterminal with the given
angle ( 3.142 radian). Find also the
reference angle if any.
31. 13
2 32. 7 33.
30
7
34. 16
3 35.
21
11 36. 7
37. 15
38. 17
7 39.
3
2
In Exercises 40 – 48, find an angle between 0
and 360 degrees that is coterminal with the
given angle ( 180 )
40. 380 41. 7 42. 380
43. 16
3 44.
21
11 45. 7 radian
46. 1044 47. 1634 48. 3
2
In Exercises 49 – 56, find the arc length, and
area of the sector, where r is the radius in
inches and is the central angle
49. 2, 380r 50. 5, 322r
51. 4, 30r 52. 3, 90r
53. 2, 1.3r 54. 7, 2r
55. 3,r 56. 3
5,5
r
57. A wheel has a radius of 12 feet, and is
rotating at 6 revolutions per minute. Find the
angular speed and linear speed in feet per
minute.
58. The blades of a wind machine are 12 feet
long and rotating at 5 revolutions per second.
Find the angular and linear speed.
59. A mountain bike with 26 inches wheels (13
inch radius) is rotating at 10 revolutions per
second. Find the angular and linear speed.
60. The diameter of car wheel is 185 mm, it
rotates at 30 revolutions per second, find its
angular and linear speed.
61. Find the angle in radians formed by the
hands of a clock at 1:30.
62. Find the radian measure of a central
angle that cuts off an arc of length 8
inches with a radius of 4 inches.
Exercise Set 3.2
In Exercises 1 – 12, show the approximate
location of ( ) (cos ,sin )P t t t on the unit
circle for the given value of t.
1. 3
t 2. 3
t 3. 45t
4. 2
3t 5.
7t 6.
19
3t
7. 5
6t 8. t 9.
37
3t
10. 412
3t 11.
512
19t 12.
33
2t
13. If 3 4
( ) ,5 5
P t , find the coordinates of
given point.
a) ( )P t b) ( )P t c) ( )P t
d) ( )P t e) ( )P t f) ( )P t
g) (5 )P t h) (240 )P t
14. If 2 21
( ) ,5 5
P t , find the
coordinates of given point.
a) ( )P t b) ( )P t c) ( )P t
d) ( )P t e) ( )P t f) ( )P t
g) (5 )P t h) (240 )P t
15. If ( ) 0.0981,0.196P t , find the
coordinates of given point.
a) ( )P t b) ( )P t c) ( )P t
d) ( )P t e) ( )P t f) ( )P t
g) (5 )P t h) (240 )P t
16. If the point ( ) , 0.8P t x lies on the
unit circle find x.
17. Find exact value of sin , cost t and
tan t for the given value of t.
a) 3
4t b)
19
3t c)
3t
d) 3
4t e)
19
4t f)
4t
g) 3
2t h)
8
3t i)
19
6t
j) 3
6t k)
32
4t l) 7t
m) 135t n) 240t o) 210t
p) 150t q) 720t r) 540t
18. Find all vales of t in the interval [0, 2 ]
satisfying the equation 2cos cos 2t t .
19. Find all vales of t in the interval [0, 2 ]
satisfying sin cos cos sin 1 0t t t t .
20. Find all vales of t in the interval [0, ]
satisfying sin cos cos sin 1 0t t t t .
21. Find the angle in radians formed by the
hands of a clock at 1:30.
22. Find the radian measure of a central
angle that cuts off an arc of length 8
inches with a radius of 4 inches.
23. Find the radian measure of a central
angle that cuts off an arc of length 10
inches with a radius of 6 inches. Also
find the area of the sector.
Exercise Set 3.3
In exercises 1-10, find all trigonometric
functions, from the given information.
1. 3
sin ,5
t t is in quadrant II
2. 3
sin ,5
t t is in quadrant I
3. 3
cot ,4
t t is in quadrant IV
4. tan 5,t t is in quadrant III
5. 1
csc ,5
t t is in quadrant IV
6. 2 3
sin ,5 2
t t
7. 2sec 5 0,t t is in quadrant II
8. 7
tan ,5
t t is in quadrant III
9. 3
cot , 05 2
t t
10. 4
2cos ,5 2
t t
In exercises 11- 20, determine all values of
t in the interval 0 2t
11. cot 1 0t 12. 2sin 2 0t
13. 2(2sin ) 2t 14. cot tan 0t t
15. 23tan 1 0t
16. 8 sin 2 2 tan 2 2t t
17. 22cos cos 1t t
18. 2cos sin cos 0t t t
19. 2cos sin cos 2sin 1 0t t t t
20. 2sin 2 2 tan 2 0t t
In exercises 21-30, determine exact values
21. 3
2sin 2 csc6 2
22. 2 tan cot6 3
23. 2sec 3 tan2 6
24. csc405 sin 45 sec360
25. 19 31
sin csc6 3
26. 19 31
sin csc tan356 3
27. 2 219 38sin cos
6 12
28. 31
tan cot6 3
29. 5 3
sin csc tan36 4
30. sin( ) cos
csc csc
x x
x x, 0
2x
31. Simplify 1
sec(2 ) 5cos
xx
32. Show that sin(3 ) sin cos 1x x
33. Find exact value: sin72 cos18 csc90
34. Find exact value: cos72 sin18 sec120
Exercise Set 3.4
In exercises 1-10 find amplitude, period,
horizontal and vertical shift
1. 2sin(3 )y x
2. 2cos(3 1)y x
3. 2sin( 3 2) 3y x
4. 2tan(3 4)y x
5. 3sin 3 103
y x
6. 2 3
sin 43 2 3
xy
7. 2cos 2 3y x
8. 4sin 33
xy
9. 1 sin 2y x
10. 3
sin 22 2
xy
In exercises 11-18 use the graphs of the
sine and cosine to sketch one period of the
graph of the function.
11. 1 2cosy x
12. 2 3sin 23
y x
13. 2 sin 2 1y x
14. 1
tan 13 3
y x
15. 2sin 3 13
y x
16. 3csc 3 13
y x
17. 1
cos 2 33 3
y x
18. 1
cot2 3
y x
19. Find exact value of x so that
sin cos .x x 20. Use calculator to find x so that
cos sin 0x x x .
21. Use calculator to find x so that
sin cosx x x
22. Use calculator to find x so that tan cotx x
In exercises 23-30 find all values of in the
interval [0,2 ] satisfying the given
equations.
23. 2tan 1x
24. 1 tan 0x
25. 2sin 2 2 tan 2 0x x
26. 2tan 2tan cotx x x
27. cos cotx x
28. 22sin sin 1x x
29. 2sin cos cos 0x x x
30. 2cos cot csc 2 0x x x
31. Given that 5cos 4,2
x x .
Find tan , csc .x x
32. For 0 2x , solve for x when
tan 1 02
x
33. An object is thrown from the point A
on the inclined plane (see the figure).
The object hits the inclined plane at B.
Find the distance between A and B, if
AC = 20 cm.
B
A 30 C
Exercise Set 4.1
In exercises 1 – 8 find the exact vale(s) of the
expression:
1. 1 1tan
3 2. 1 1
sin2
3. 1 1cos
2 4. 1 1
sin2
5. 1tan 0 6. 1 3sin
2
7. 1 3cos
2 3 8. 1tan 3
In exercises 9 – 15 find exact value(s )
9. 1 3tan tan
2 3
10. 1 2sin sin
2
11. 1 3tan sin
2 3
12. 1 3sin tan
2 3
13. 1 1cos tan
3
14. 1 12tan sin
13
15. 1 5tan cos
13
In exercises 16 – 20 find exact value(s) on
[ / 2, ]
16. 1sin sin3
17. 1 3sin sin
4
18. 1 3cos cos
4
19. 1cos cos4
20. 5
arctan tan4
In exercises 21 – 25 find the value(s) in terms
of x
21. tan arcsin3
x
22. 2
cos arcsin3
x
23. 1
sin 2arcsin2 3
x
24. sin arccos3
x
25. cos arctan3
x
26. Rewrite the expression sin(arctan( ))x
as a expression of x.
In exercises 27 – 45 find all solutions of the
equation on [0,2 ]
27. 22 cos sin 2t t
28. 2cos cos 2t t
29. sin cos cos sin 1 0t t t t
30. 3sin 2 5sin 1t t
31. sin tan 0t t
32. 22cos cos 1t t
33. 2tan sin 3tan 0t t t
34. cos sin 1/ 2t t
35. sin cos 1x x
36. 2sin 3 0x
37. 22cos 4cos 6t t
38 2cos 1 0t
39. cot (1 tan ) 0t t
40. cos2 sinx x
41. cos sin 3 / 4t t
42. sin( /3) sin( /3) 1t t
43. cos sin 2cos 0x x x
44. sin2 cos 0x x
45. 2 2cos2 sin 5cos cos 3x x x x
Exercise Set 4.2
In exercises 1 – 20 prove the identities
1. 1 cos sin 1 cos
sin cos sin cos
x x x
x x x x
2. sin cos sin
sin cossec csc sec
x x xx x
x x x
3. 2sin( / 2 )cos( ) cos 0x x x
4. 1 cos 2
tansin 2
5. 2sin cos sec2 tan2x x x x
6. sec cos tan sinx x x x
7. sin 1 cos
1 cos sin
x x
x x
8. tan sin( )
tan1 cos
x xx
x
9. 21 12 2cot
1 cos 1 cosx
x x
10. 1 sin
sec tancos
xx x
x
11. 21 cos
sin tancos
xx x
x
12. 21 sin
cos cotsin
xx x
x
13. 4 4 2sin cos 1 2cosx x x
14. sin cos cos sin
sec cscsin cos
x x x xx x
x x
15. 2 2
2
2
sin coscos
tan 1
x xx
x
16. 2 2 2 2cot 2 sin 2 cos 2 csc 2x x x x
17. 2 1 sin(sec tan )
1 sin
xx x
x
18. ln sec ln cosx x
19. 2 3sin sin cos cosx x x x
20. 1 2sin cos sin cosx x x x
21. A pole of 100 feet is supported by a
cable of length 230 feet. Find the angle
of elevation from the top of the pole to
the point on the ground.
22. It is given that 2
2
2
2 tan1 ( ( ))
sec
xf x
x.
Determine the function ( ).f x
23. It is given that 1tan cot ( ( ))x x f x .
Determine the function ( ).f x
25. It is given that sec 1 cos
sec 1 cos
x p x
x p x.
Determine the constant p .
26. It is given that
2 sin(tan sec )
sin
p tt t
q t.
Determine the constants ,p q .
Exercise Set 4.3
In exercises 1 – 10 find exact value(s)
1. 10
cos12
2. 5
cos6
3. sin12
4. 7
cos12
5. 7
sin12
6. 5
tan6
7. tan12
8. 13
cos12
9. 13
sin12
10. 13
tan12
11. Given that 3
cos7
x , x is in quadrant
IV and 4
sin5
y , y is in quadrant II.
Find
a) cos x y b) sin x y
c) cos x y d) sin x y
e) The quadrant where x y lies
f) The quadrant where x y lies
12. Find the exact value of
2 2
cos cos sin sin12 3 12 3
13. Find the exact value of
3 2 3 2
cos cos sin sin7 21 7 21
14. Find the exact value of
2 2
sin cos cos sin12 3 12 3
15. Find the exact value of
2 2
sin cos cos sin12 3 12 3
In exercises 16 – 20 simplify
16. cos( )sin( )x x
17. cos( / 6) sin( /3)x x
18. sin( / 2) cosx x
19. Show thatsin2 2sin cosx x x
20. Use cos x y formula to prove
2 2
2
2
cos 2 cos sin
2cos 1
1 2sin
x x x
x
x
Exercise Set 4.4
1. Given that 3
cos ,csc 05
x x , find
cos2 , sin2 , tan2x x x
2. Given that 3 3
cos ,5 2
t t
cos2 , sin2 , tan2t t t
3. Given that 3
sin ,5 2
, find
cos2 , sin 2 , tan 2
4. Given that 3 3
cos ,5 2
t t
cos , sin , tan2 2 2
t t t
5. Use half angle formula to write Given
that 1
sin12 2
x y , then find x
and y.
6. If 2 2 2sin 7 cos 7 ( ( ))x x f x , find
The function ( )f x
7. Given that 19
cos ,20
x x is in
quadrant IV. Find cos2 sin 2x x
8. If 4 4cos 9 sin 9 cos( ( ))x x f x , find
the function ( )f x .
9. If 2
2
2
2 tan1 ( )
secg , find the
function ( )g
In exercises 10 – 15 find A and B
10. 2 sin(sec tan ) , sin
sin
A tt t t B
B t
11. sec 1 cos
, cossec 1 cos
x A tt A
x A t
12. 19 ( 1)
sin12 4
A B
13. cos
tan , sin 02 sin
x A xx
x
14. 7
tan , 08
A AA
A A
15. 2 2 2(1 cos )secx x B
In exercises 16 – 20 find all values of x in [0,
2π] that satisfy the given equation
16. sin 2 sinx x
17. sin2 cosx x
18. sin2 cos2x x
19. 2cos 3sin 3x x
20. cos2 5cos 2x x 21. Show that
4 3 1 1sin cos 2 cos 4
8 2 8x x x
22. Show that
21 cos
sin2 2
x x
23. Show that
21 cos
cos2 2
x x
24. Use double angle formula to find exact
value of sin cos8 12
25. Show that 4 4cos sin cos2t t t
Exercise Set 4.5
In exercises 1 – 10, rewrite each product as a
sum or difference
1. sin3 cos2x x
2. cos2 cos5x x
3. tan3 tanx x
4. sin 2 cos2x x
5. cos5
csc10
x
x
6. 7 5
sin cos8 4
7. 7 5
sin sin8 4
8. 7 5
cos cos8 4
9. 7 5
cos sin8 4
10. 3
sin cos4 4
11. Solve for x if sin 2 sin3 0x x
12. Solve for x if sin 2 sin3 0x x
13. Solve for x if cos2 cos3 0x x
14. Solve for x if cos2 cos3 0x x
15. Show that
2sin( )
tan tancos( ) cos( )
x yx y
x y x y
Exercise Set 4.6
In exercises 1 – 10, solve for x on [0, 2π]
1. cos2 0x
2. tan 2 sin 2 0x x
3. 2sin cos2 0x x
4. sin 2
cot 2cos 2
xx
x
5. 1
1csc2x
6. tan5 cos5x x
7. 2sec csc tan 0x x x
8. 2csc sec cotx x x
9. 2sin3 2x
10. 21 sin 2cosx x
11. Find all solutions for x when
xx 2sin2cos1
12. Find all solutions for x when
2cos 1 0x
13. Find all solutions for x when
2sin cos cos 0x x x
14. Find all solutions for x when
2sin cos cos 2sin 1 0x x x x
15. Find all solutions for x when
sin cos 2cos 0x x x
16. Find all solutions for x when
2cos 2.5cos 1.5 0x x
17. Find all solutions for x when
3tan 4tan 0x x
Exercise Set 4.7
In exercises 1 – 10 draw the triangle and find
all missing information
1. Angle 52CAB , angle 70CBA ,
27.6AB cm
2. Angle 38CAB ,
22AC cm , 43AB cm
3. Angle 38CBA ,
112BC cm , 133AB cm
4. Angle 38CAB ,
22AC cm , 43AB cm
5. Angle 115C , 10a ft , 8b ft
6. Sides 42, 38, 67a b c
7. Side 7b , angles 110C ,
40A
8. Side 6a , angles 80C , 40A
9. Side 10c , angles 110C ,
30B
10. Sides 9, 4, 11a b c
11. Find the distance AB across a river, a distance BC = 250 ft is laid off on one side of the river. Also given that angle
110B , angle 35C . See the adjacent graph below B C A
12. Two ships leave the St. Martin Island at the same time, traveling on courses that have an angle of 100 degrees between them. The first ship travels at 25 miles per hour and the second ship travels at 40 miles per hour. Find how far apart the ships are after 3 hour 30 minutes.
13. Two ships leave the St. Martin Island at the same time, traveling on courses that have an angle of x degrees between them. The first ship travels at 30 miles per hour and the second ship travels at 40 miles per hour. The distance between the ships after 2 hours is 100 miles, find x.
14. The path of a satellite orbiting the earth causes it to pass directly over two stations situated at A and B, which is 50 miles apart. The angle of elevation of the satellite at A is 85 degrees and the angle of elevation at B is 78 degrees. See the graph below
C
85 78
A B
Find how far is the satellite from
station A
a. Find how far is the satellite from station B
b. Find the height of the satellite above the ground
Round your answer to three
decimal places.
Exercise Set 5.1
In exercises 1 – 10, sketch the graphs, showing
any horizontal asymptote.
1. ( ) 3 1xf x 2. ( ) 10 2xf x
3. ( ) 3 1xf x 4. 2( ) 4(3 ) 1xf x
5. ( ) 10xf x e 6. 2( ) 5 7xf x e
7. 1( ) 10 5xf x 8. ( ) 5 1x
f x
9. 2
( ) 1x
f x e 10. ( ) 5 1x
f x
In exercises 11 – 15, use a graphing device to
plot and compare
11. The graphs of ( ) 2xf x , ( ) 2 xh x ,
( ) 2xg x , and ( ) 3xl x , ( ) 3 xk x ,
( ) 3xp x
12. The graphs of
( ) 2xf x and 1( ) 2 2xf x
13. The graphs of 1( ) 3xf x ce
for 2, 1,0,1,2c
14. The graphs of ( ) 2xf x , ( ) 2x
g x
and ( ) 2x
h x
15. The graphs of ( ) 5xf x , ( ) 4(5 )xg x
and ( ) 4(5 )xf x
16. Given that ( ) xf x e , determine and
simplify ( ) ( )f x h f x
h
17. Given that ( ) xf x e , determine and
simplify ( ) ( )f x f a
x a
18. Given that ( ) xf x e , determine (2)f
19. Use a graphing device to compare to compare the rates of growth of the
functions ( ) 5xf x and 5( )h x x by
graphing the two functions in the following viewing windows
a) [ 5,5] [ 5,5]
b) [ 10,10] [ 10,10]
c) [0,5] [0,5]
Approximate the solutions to 55x x correct
to three decimal places.
20. Determine the value of an investment
of $5000 in 10 years at the interest rate of
5% compounded as indicated.
a) Annually b) Monthly
c) Semiannually d) Quarterly
e) Weekly f) Biweekly
g) Daily h) Continuously
21. What initial investment at 5% compounded semiannually for 10 years will accumulate to $10000?
22. A population of bacteria doubles every hour. At noon the number of bacteria was 1,000. Set up an exponential function to model the growth of the bacteria, and forecast the population at 5 p.m. Also, estimate the population at 10 a.m., two hours before the number of bacteria were counted.
23. The number of bacteria in a culture is
given by 0.05( ) 150 tn t e , where ( )n t
is measured in grams.
a) Find the mass at time t = 0 b) How much of the mass
remains after 18 years?
Exercise Set 5.2
In exercises 1 – 12, evaluate the expression.
1. 2
7log 7 2. 2
2log 2 3. 2
7log 49
4. 10log 10000 5. 10log 0.0001 6. 10
1log
100
7. ln xe 8. 3 1log x
e e 9. 2ln xe
10. 4log 64 11. log1010 12. 21/ 2ln xe
In exercises 13 – 20, determine domain, x
intercept and vertical asymptote if any.
13. 10log (2 6)x 14. log5 log x
15. 5
2log
2
x
x 16.
10
2log
2x
17. 2
10log (16 )x 18. 2ln( 2) lnx x
19. 5log ( 2)x x 20. 2
3log ( 1)x
In exercises 21 – 30, express the equation in
exponential form
21. 5log 125 3 22. ln x y
23. 10log 0.01 2 24. log 125 3x
25. 3log (8 8 )x y 26. ln ln 2 3x x
27. log 125 log 5 2x x 28. 5log 12 x
29. 2ln ln 2 3x x 30. 5log 3x
In exercises 31 – 35, rewrite the expression as
a single logarithm
31. 5 5 5log 2 log (2 2) logx x x
32. 23ln 5ln( 2) ln( 3)x x x
33. 5 22log 5log 1x x
34. ln5 2ln ln1x x
35. 35ln( 1) lnx x
In exercises 36 – 40, solve for x
36. 2
5log ( 20 ) 3x x
37. ln(3 5) 1 0x
38. ln( 1) ln ln8x x
39. 2log(3 5 3) 0x x
40. 1
log 23
x
41. Plot the graph of
2
2log (5 2 5) 3x x and
solve algebraically for x.
42. Use logarithm to solve the exponential
equation 4 15x
43. Given ( ) lnf x x , simplify the
expression ( ) ( )f a h f a
h
44. Evaluate the expression
8 8log 1536 log 3
In exercises 45 – 50, determine the constants
45. 3 1/ 2
3 3 3log ( ) log logx y A x B y
46. 3 1/ 2
log log log log3
x yA x B y C
47. 10
5
5ln ln ln( 3)
3
yA y B z
z
48. 2 10
5ln ln ln ln
x yA x B y C z
xyz
49. ln ln( ) ln( )x y
A x y B x yx y
50. 10
ln ln ln3
xA x B C
51. An investment of $5000 will grow to
$12500 at 5% interest compounded quarterly
in t years. Find t.
52. The number of bacteria in a culture is
modeled by the exponential function 0.57( ) 2000 tn t e , where t is in hours. Find
a) The initial count of this bacterium b) The relative rate of growth c) After how many hours will the bacteria
count reach 10000? 53. The count in a bacteria culture was 8000
after 5 hours and 16500 after 7 hours. Assume
the growth model by the function
0( ) rtn t Pe , where t is in hours. Find the time
when the count will be double to its initial size.
54. The half-life of strontium-90 is 29 years.
How long will it take a 56 mg sample to decay
to a mass of 10 mg?
55. A culture has initial bacteria count
9000. After one hour the count is 4500.
Find the relative growth rate and the
number of bacteria after 2 hours.
56. The radioactive isotope strontium 90
has a half-life of 28.5 years. Find
a) How much strontium 90 will remain after 15 years from an initial amount of 450 kilograms.
b) How long will it take for 75% of the original amount to decay?
c) The time when the amount is 100 kilograms.
Exercise Set 6
In exercises 1 – 14 Find Cartesian form of
the polar equation
1. 4r 2. 2sinr
3. 16r 4. 2cosr
5. 1 sinr 6. 1 sinr
4. 2 cosr 8. 2 cosr
9. sin 2r 10. lnr
11. r 12. 4 2sinr
13. 2 36cos2r 14. 2 8sin 2r
In exercises 15 – 20, convert the
rectangular coordinates to polar
coordinates.
15. (1,2) 16. (1, 3)
17. (1, 2) 18. ( 1, 2)
19. ( 3, 1) 20. (1, 3)
21. ( 3, 3) 22. (0, 1)
In exercises 23 – 28, convert the
rectangular equation to a polar equation.
23. y x 24. 2 2 25x y
25. 2 2 6x y x 26. 3y x
27. 2 2 6x y y 28. 2 3y x
29. Find the polar equation of the given
Cartesian equation
2 2 2 2
2 2 4
h k h kx y
where h, k are the constants representing
center of the circle.
30. Identify the conic and write its polar
form: 2 2 2 2( 1) 1 ( 1)x y x y
31. Find the point(s) of intersection of the
curves sin 2 , and 1r r
Answers to odd number problems
Section 0.1
1. 3 4 3. 4 5. 10 2
7. 2 3 9. 3 15x 11. 6/5
13. 1/ 3 15. 1, 2; 1, 2x x
17. 1 1
,3 3or 3; , 33 3
x x xx x
19. 13
21. Rational numbers are: 1/3,
3.6666…, 3 64, 81,30, 5,0
Irrational number is 13
23. 21 x
Section 0.2
1. 410 or -10000 3. 35 5. 36
7. 22 9. 140x 11. 4 89x y 13. 6
310
x
15. 2 4
7
x y 17.
3 15 6
1
5 x y 19.
4
19
x
y
21. 92.86 10 23. 81.42 10
25. 123000 27. 0.004842520
Section 0.3
1. 5 3. 5 5. 2
7. 22 9. 3 / 5
11. 14 2 3 7 5 15
46 13. 3 11 10
15. 4 2 5 4 10 17. 5( 7 3)
2
19. 5( 3 1)
2 21. 23x
23. 36x 25. 9x
Section 0.4
1. a) Second order b) Not a polynomial
` c) Not a polynomial d) Ninth degree
e) Not a polynomial f) Not a polynomial
g) Not a polynomial h) Not a polynomial
i) First degree j) Twelfth degree
k) Fifth degree l) Fourth degree
m) Third degree n) Ninth degree
Section 0.5
1. ( 4)( 9)x x
3. ( 2)( 13)x x 5. 22 (3 4 6)ax x x
7. 9 ( )( )x x a x a 9. ( )(2 3)a b x
11. 22( 18)x
13. 2 2(2 3 )(4 6 9 )x a x xa a
15. 29 ( 3)( 1)x x x 17. 3( 1)x
Section 0.6
1. a) 13x b) all real values of x
c) 8x d) 3,1x
e) 1/ 2, 5x f) all real values of x
g) 0, 2 / 3x h) 2, 7x
3. a) 4
2, 3;3
xx
x
b) 9
2,0,1/ 4;4 16
xx
c) ( 3)(2 5)
5 / 2,3;3
x xx
d) 5
5,1;4
xx
e) 2 25
4, 3/ 2, 5;2 8
xx
x
f) 2
2
2 33/ 2, 5; (2 8)
25
xx x
x
Section 0.7-0.8
1. 2 3, (2,3]x 3. 2 / 3, ( , 2 / 3)x
5. 2.2, [ 2.2, )x 7. 16 19, ( 16,19]x
9. 4 1, ( 4,1]x
11. 21/13 4 / 3, [ 21/13, 4 / 3)x
13. 0 4 /11 or1/ 2x x , (0,4 /11] (1/ 2, )
15. 2 or1 2x x , ( , 2] (1,2)
17. 2 or1 2x x , ( , 2) (1,2)
19. 2 , ( 2, )x 21. 5 3, ( 5,3]x
23. 3, or 2 , ( , 3] [2, )x x
25. 4 3/ 2, [ 4,3/ 2]x
27. 1/2, or 4 , ( ,1/2] [4, )x x
29. 0, or 9 , ( ,0] [9, )x x
31. 3 1, [ 3, 1]x 33. ( , )
35. 1, or 6 , ( , 1] [6, )x x
37. 3 2, [ 3,2]x 39. 2 3, [2,3]x
41. 1, or 2 3, ( ,1) [2,3]x x
43. 3, [3, )x 45. 2 2, ( 2, 2)x
47. No solution
49. 2 1,or 3, [ 1,2] [3, )x x
51. 1 2 1 2 , [1 2,1 2]x
53. a) 10 7i b) 20 i c) 10 7i
d) 26 32i
Chapter 1
For 1 – 5 follow example 1 on section 1.1.
7. 6 2, ( 2,0) 9. 2 13, ( 2, 3)
11. 3, (9 / 2,9) 13. 2 10, ( 4, 4)
15. 2, (8,0)
17. 2 5, 2,5; 2 9, 2,9y x y x , parallel
19. 4 / 3 4, 4 / 3,4; 5, 1,5y x y x , neither
21. 8/15 4 / 5, 8 /15,4 / 5;y x
15/8 75/8,15/8, 75/8y x , perpendicular
23 – 26 follow example 9-12 of section 1.5.
29. Vertically stretched by a factor 2 and shift 3 units upward.
31. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward
33. Vertically stretched by a factor 3, horizontal shift by 2 units to the right, 5 units upward and reflection about x
axis.
35. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 3 units upward
37. Vertically compressed by a factor 3, and shift 3 units downward
39. Vertically stretched by a factor 2, and shift 3 units upward
41. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward
43. Vertically stretched by a factor 3, horizontal shift by 2 units to the right and 5 unit upward and reflection about
x axis
45. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 3 units upward
47. Vertically compressed by a factor 3, and 3 units upward
49. Horizontally stretched by a factor 2, shift by 3 units upward then reflection about x axis
51. Vertically stretched by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward then reflection about
x axis
53. Vertically compressed by a factor 2, horizontal shift by 1 unit to the right and 1 unit upward then reflection
about x axis
55. Horizontally stretched by a factor 2, shift by 7 units downward then reflection about y axis
57. Horizontal shift by 1 unit, shift 3 units upward then reflection about x axis
59 .Vertically compressed by a factor 2, shift by 1 unit to the right
61. Horizontally stretched by a factor 2, shift by 7 units downward
63. Horizontally compressed by a factor 3, vertically stretched by a factor 2, shift by 2 units to the left, and shift 2
units upward
65. Vertically compressed by a factor 2, shift by 1 unit to the right, and reflection about x axis
67. Horizontally stretched by a factor 2, shift 7 units downward and reflection about x axis
69. Vertically compressed by a factor 3, shift by 1 units to the right, and shift 2 units upward
71. 2 1 , [0, )x x
73. (2 1) 3, [0, )x x
75. ( ) 2 1, [0, )F x x
77. 1 1( ) ( 4)
3f x x
79. 1 2( ) 4f x x
81. 1 6( )
3 2
xf x
x
83. 1 2( )
3f x
x
85. 1 5 3( )
2 4
xf x
x
87. 1
2
4( ) 2f x
x
89. 1( ) 3f x x
Chapter 2
1. Monomial, odd, linear
3. Quadratic, neither
5. Neither, linear
7. Cubic, neither
9. Cubic, odd
11. Cubic, neither
13. 21( ) ( 0.3) 10.03
3P x x , max
15. 2( ) 3( 1/ 6) 1/12P x x , max
17. 2( ) 3( 23/ 6) 361/12P x x , max
19. 2( ) 2( 21) 882P x x , min
21. , ; ,x P x P
23. , ; ,x P x P
25. , ; ,x P x P
27. , ; ,x P x P
29. , ; ,x P x P
31. , ; ,x P x P
33. , ; ,x P x P
35. 2 , 2x a b 37. 1,4, 4x
39. 2, 2x 41. 0.158, 3.158x
43. 5 9 55
,2 4 4
Q x R
45. : 6, : 6, 2D x VA x Slant y x
47. : ( , ), 5 5D x Slant y x
49. : 0, 0, 0D x Hole x HA y
51. 36 354
9 / 5, 5 9,5 25
x VA x SA y x
53. : 3, 3, 3 1D x VA x HA y
55. a) 1, 2, 3, 6, 7, 14, 21, 42
( 1, 2, 3, 6, 7, 14, 21, 42) /17
( 1, 2, 3, 6, 7, 14, 21, 42) / 289
b) Irrational zeros are 0.176, 0.824 (correct to three decimal places)
57. a) 1, 2, 4 b) One rational zero 4, two irrational zeros are 2.414,0.414
59. a) 1, 2, 3, 4, 5, 10, 20
b) One irrational zero -1.48 (correct to 2
decimal places)
61. a) 1, 3, 5, 15 b) Rational zero -3
63. a) 1, 3, 9, 27, 81
b) Rational zero -3 with multiplicity 3.
65. 3 2( ) 2 12 22 12P x x x x
67. 3 2( ) 4 4P x x x x
69. 3 2( ) 4 25 100P x x x x
71. 2
2
16( )
9
xP x
x
73. Two positive, one negative
75. Four negative zeros
77. 3.56t , horizontal asymptote is ( ) 0y C t , the meaning of zero is that the is no concentration of the drug
in the long run.
Section 3.1
1. Obtuse 3. Obtuse 5. Acute 7. Acute
9. Reflexive 11. 55
72 13.
7
18 15.
5
3
17. 3
20 19.
10 21.
36 23. 270
25. 414 27. 150 29. 45
31. ,2 2
33. 2 2
,7 7
35. 21
,11 11
37. 29
,15 15
39. ,2 2
41. 20 , 20
43. 120 , 60 45. 0.717 , 0.717
47. 166 , 14 49. 12.22 inches, 12.22 sq. inch
51. 2.09 inches, 4.19 sq. inch
53. 8.17 inches, 8.17 sq. inch
55. 9.42 inches, 14.14 sq. inch
57. 12 , 144 59. 20 , 260 61. 3
4
Section 3.2
1. Quadrant I 3. Quadrant I 5. Quadrant I
7. Quadrant II 9. Quadrant I 11. Quadrant IV
13. a) ( 3/ 5,4 / 5) b) ( 3/ 5, 4 / 5)
c) ( 3/ 5,4 / 5) d) (3/ 5,4 / 5)
e) ( 3/ 5, 4 / 5) f) ( 3/ 5,4 / 5)
g) ( 3/ 5,4 / 5) h) (3/ 5, 4 / 5)
15. a) ( .0981, 0.196) b) ( .0981, 0.196)
c) ( .0981, 0.196) d) (0.0981, 0.196)
e) ( .0981, 0.196) f) ( 0.0981, 0.196)
g) ( .0981, 0.196) h) (0.0981, 0.196)
17. a) sin(3 / 4) 2 / 2 , cos(3 / 4) 2 / 2
tan(3 / 4) 1
b) sin(19 / 3) 3 / 2 , cos(19 / 3) 1/ 2
tan(19 / 3) 3
c) sin(19 / 3) 3 / 2 , cos(19 / 3) 1/ 2
tan(19 / 3) 3
d) sin( 3 / 4) 2 / 2 ,
cos( 3 / 4) 2 / 2
tan( 3 / 4) 1
e) sin( 19 / 4) 2 / 2 ,
cos( 19 / 4) 2 / 2
tan( 19 / 4) 1
f) sin( / 4) 2 / 2 , cos( / 4) 2 / 2
tan( / 4) 1
g) sin(3 / 2) 1 , cos(3 / 2) 0
tan(3 / 2) does not exist
h) sin(8 / 3) 3 / 2 , cos(8 / 3) 1/ 2
tan(8 / 3) 3
i) sin( 19 / 6) 1/ 2 ,
cos( 19 / 6) 3 / 2
tan( 19 / 6) 1/ 3
j) sin(3 / 6) 1 , cos(3 / 6) 0
tan(3 / 6) does not exist
k) sin( 11 / 4) 2 / 2 ,
cos( 11 / 4) 2 / 2
tan( 11 / 4) 1
l) sin( 7 ) 0 , cos( 7 ) 1
tan( 7 ) 0
m) sin( 135 ) 2 / 2 ,
cos( 135 ) 2 / 2 , tan( 135 ) 1
n) sin(240 ) 3 / 2 ,
cos(240 ) 1/ 2 , tan(240 ) 3
o) sin(210 ) 1/ 2 ,
cos(210 ) 3 / 2 , tan(210 ) 1/ 3
p) sin(150 ) 1/ 2 ,
cos(150 ) 3 / 2 , tan(150 ) 1/ 3
q) sin( 720 ) 0 ,
cos( 720 ) 1 , tan( 720 ) 0
19. 0, 2 , / 2 21. /12
23. 5 / 3 rad, 30 sq. inch
Section 3.3
1. sin 3/ 5t csc 5 / 3t
cos 4 / 5t sec 5 / 4t
tan 3 / 4t cot 4 / 3t
3. cot 3/ 4t tan 4 / 3t
cos 3/ 5t sec 5 / 3t
sin 4 / 5t csc 5 / 4t
5. csc 7 / 5t sin 5 / 7t
cos 2 6 / 7t sec 7 6 /12t
tan 5 6 /12t cot 2 6 / 5t
7. sec 5 / 2t cos 2 / 5t
sin 21 / 5t csc 5 21 / 21t
tan 21 / 2t cot 2 21 / 21t
9. cot 3 / 5t tan 5 / 3t
cos 3 34 / 34t sec 34 / 3t
sin 5 34 / 34t csc 34 / 5t
11. cot 1; / 4, 5 / 4t t
13. sin 1; / 2, 3 / 2t t
15. tan 3 / 3; / 6, 5 / 6,7 / 6,11 / 6t t
17. 0,2 , 2 / 3, 4 / 3t
19. , / 6, 5 / 6t 21. 1 2
23. Does not exist 25. 1.6547
27. 1 29. 2 / 2
31. 5 33. 1
Section 3.4
1. 2, 2 / 3, 0, 0 3. 2, 2 / 3, 2 / 3, 3
5. 3, 2 / 3, / 9, 10 7. 2, 1, 3/(2 ), 0
9. 1, 1, 0, 1
11.
3
0 2
-1
13.
3
0 1
15.
3
0 2
17.
3
0
19. / 4, 5 / 4x
21. 0, 4.49, 7.73, 10.9
23. / 4,3 / 4,5 / 4,7 / 4x
25. 0, , /8,7 /8x 27. 0, / 2,3 / 2x
29. / 2,3 / 2,4 / 3,5 / 3x
31. tan 3/ 4, csc 5/ 3x x 33. 40 3 / 3
Section 4.1
1. / 6 3. 3 / 4 5. 0 7. / 6
9. 3 / 2 11. 3 13. 3 / 2 15. 12/5
17. / 4 19. 3 / 4 21. 29
x
x
23. 29
9
x x 25.
2
3
9 x
27. 0, / 4, 3 / 4, 2t
29. 0, / 2, 2t 31. 0, / 2, 2t
33. 0, / 3, 2 / 3,4 / 3,5 / 3, , 2t
35. / 2t 37. t 39. 3 / 4,7 / 4t
41. / 3, / 6t 43. / 2, 3 / 2x
45. 2 / 3, 4 / 3x
Section 4.2
21. 25.77 23. ( ) sin cosf x x x 25. 1p
Section 4.3
1. 3 / 2 3. 6 2
4 5.
6 2
4
7. 2 3 9. 6 2
4
11. a) 8 10 9
35 b)
6 10 12
35 c)
8 10 9
35
d) 6 10 12
35
e) x y is in quadrant I and x y is in quadrant II.
13. 1
2 15.
6 2
4 17. 0
Section 4.4
7 241. cos 2 , sin 2
25 25
24tan 2
7
x x
x
7 243. cos 2 , sin 2
25 25
24tan 2
7
5. 2, 3x y 7. 0.2117 9. ( ) cosg
11. A = 1 13. A = 1 15. tanB
17. / 2,3 / 2, / 6,5 / 6x 19. 3 / 2x
Section 4.5
1. 1
(sin5 sin )2
x x 3. cos 2 cos 4
cos 2 cos 4
x x
x x
5. 1
(sin15 sin 5 )2
x x
7. 1
(cos3 /8 cos17 /8)2
9. 1
(sin 3 /8 sin17 /8)2
11. 2 2
5
nx or (2 1)x n
13. 2 1
5
nx or (2 1)x n
Section 4.6
Assume 1,2,3,4....n
1. 3
,4 4
x 3. 3
,2 2
x
5. 5 7
,8 8
x 7. 0,2x
9. ,12 4
x 11. (2 1)
3
nx
13. (2 1) (4 3)
,2 6
n nx
15. (2 1)
2
nx 17.
(4 3),
8
nx n
Section 4.7
1. Angle 58 , 30.58, 25.64ACB AC BC
3. 57.02 , 84.98 , 82.2A C AC
5. 36.55 , 28.45 , 15.22A B AC
7. 30 , 9, 13.16B a b
9. 40 , 6.84, 5.32A a b
11. 250 ft
13. 73.2x
Section 5.1
1.
Horizontal asymptote 1y
3.
Horizontal asymptote 1y
5.
Horizontal asymptote 10y
7.
Horizontal asymptote 15y
9. No horizontal asymptote
11.
3 x 3x
2 x 2x
2x
3x
13. 2c
1c
0c
1c
2c
4(5 )x
15.
5x
4(5 )x
17.
2
2
1
1
x
x
e
e
19. a) 8144.47 b) 8235.05 c) 8193.08
d) 8218.10 e) 8241.63 f) 8239.65
g) 8243.32 h) 8243.61
21. 6102.71 23. a) 150 b) 61 years
Section 5.2
1. 2 3. 4 5. 4 7. x 9. 2x 11. 10
13. 3x ; 3.5, 3x x
15. 2x , 0x ; 2x , 0x
17. 4 4x ; 15x , 4x
19. 2x , 0x ; 0.414,2.414x ,
2, 0x x
21. 35 125 23. 210 0.01 25. 3 8 8y x
27. 2 25x 29. 32x e 31. 2
5log1
x
x
33. 3log( )x x 35. 5
3
(1 )ln
x
x
37. 5
3
ex 39. 2 / 3, 1x
41. 0.6, 1 43. 1
ln 1h
h a
45. 3, 1/ 2A B 47. 5, 1/ 2A B
49. 1, 1A B 51. 18.44 years
53. 1309.23,36.2%,1.915
55. 1.2 years
57. 69.31%, 2250.21r
Chapter 6
1. 2 2 9x y , a circle of radius of 3
3. 2 2 256x y
5. 2 2 2 2x y y x y
7. 2 2 2 22x y x x y
9. 2 2 3 2 2( ) 4 0x y x y
11. 2 2tan( )y x x y
13. 2 2 2 2 2 2( ) 2 36( )x y y x y
15. 15, tan (2)r
17. 13, tan ( 2)r
19. 12, tan ( 1/ 3)r
21. 16, tan (1)r
23. 1 tan 0
25. 6cosr
27. 6sinr
29. 2 2 24 ,r h k
where cos / 2, sin / 2x r h y r k
31. 2 2 arctan( / )y xx y e
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