Download PRECALCULUS GT/H

PRECALCULUS GT/H
REVIEW FOR FIRST SEMESTER FINAL
Work the following on notebook paper.
1. Find all of the zeros of f  x   6 x 4  x3  32 x 2  5x  10.
2. Find all of the zeros of f  x   21x 4  95x3  109 x 2  105x  200.
3. Find two positive numbers such that their product is 192 and the sum of the first plus
three times the second is a minimum.
4. A manufacturer wants to design an open box having a square base and a surface area of
108 sq. in. What dimensions produce a box with maximum volume?
5. Four feet of wire is to be used to form a square and a circle. Find the length of the side
of the square and the radius of the circle so that the enclosed total area is:
(a) a maximum (b) a minimum
6. Find all of the asymptotes of:
x2  9
2 x 2  3x  2
(a) y  2
(b) y 
x 1
x  16
________________________________________________________________________________
Solve.
7. x  6  x  14
8. x  1  1  x  6
9. x3  4x2  x  4
x2  x  6
10.
0
x 1
11. x  2  3x  1
12.Find two angles, one positive and one negative, that are coterminal with the given angle.
5
a. 35
b.
6
13. a. Convert 165 to radians. Give answers in terms of  .
b. Convert 208 to radians. Give the answer to the nearest hundredth of a radian.
14. a.
Convert

7
6
to degrees.
b. Convert 1.8 radians to nearest tenth of a degree.
15. Write an equation for a sine wave that has amplitude 3 and period 8.
2
16. If  is a third quadrant angle and cos   , find sin  .
3
17. Express each of the following in terms of a reference angle.
a. cos236 b. sin485 c. sin( 62 )
18. Give the exact value of the six trigonometric functions for each angle.


a.
b. 225 c. 
6
2
5
3
Let cot x  , where   x  . Find the values of the other five trigonometric functions.
12
2
19.
20. Without using your calculator or a table, find the value of each expression. Leave your
answers in terms of  whenever appropriate.
3
2
a. Tan 11
b. Sin 1
e. Sin 1 0
f. sin(Cos 1 )
21.
3
)
2
5
g. sec(Tan 1 )
12
c. Cos 1 ( 
3
5
Evaluate without using a table or calculator.
5
a. sin120
b. sec
4
22. Evaluate without using tables or calculators.
F 3 I b. cos( Arc tan
G
H2 J
K
a. Cos 1 
3)
3 I
F
G
H2 J
K
c. cos 
d. Tan 1 3
h. cos( Sin 11)
1I
3
F
) d. Arcsin(sin )
G
J
H2 K
4
c. sin( Sin 1 
7 I
F
G
H6 J
K
d. tan 
1I
F
G
H2 J
K)
e. csc(Tan 1 
23. If sec x  5 and   x  2 , find exact value for the other five trigonometric functions of x.
24. Determine the sinusoidal function of sine with amplitude
1
3
, period  , and translation 2 units
up.
25. Determine the angle coterminal with 
26. A
4
9
7
2
that has a radian measure between 0 and 2 .
clockwise rotation would terminate in which quadrant and yield what angle measurement?
27. Express the function sin

8
in terms of its cofunction.
28. Find the value of cos if sin   
2
3
and angle  is in standard position with terminal side in
the fourth quadrant.
29. Sec and cot  have opposite signs in which quadrant?
30. Determine the following: sin( Cos 1
5
).
6
31. Match the function to the graph.
A. y  2 sin 2 x
B. y  3cos 3x
1
2
5
2
E. y  sin x
F. y  cos
3
2
C. y  cos
x
x
3
2x
H. y  cos
3
x
2
D. y  2 sin
G. y  2sin x
2
3
3
3
3
2
2
2
2
1
1
1
1
0
1
2
3
4
5
6
1
1
1
1
2
2
2
2
3
3
3
3
1. __________
2. _________
3. _________
4. _________
3
3
3
3
2
2
2
2
1
1
1
1
1
1
1
1
2
2
2
2
3
3
3
3
0
5. __________
32. Find
sin 
, given that
6. _________
     2 .

2  7
cos 
33. Graph with the correct trig function.
A. y  2  3cos x
C.
y  3sin x  2
B. y  2 cos x  3
D. y  2  3sin x
1
2
3
4
7. _________
5
6
8. _________
34. A right triangle has an acute angle  such that cot   15. Find cos .
35. Evaluate: sec( Arc tan 3) .
36. Evaluate:
FI.
HK
Arccos
1
2
37. Find the reference angle for   305 .
38. What is the reference angle for 188  ?
39. Graph y  2 sec(2 x)  1.
40. Express 315 degrees in radians.
41. Express 8420'40" in decimal degrees.
42. Express in degrees, minutes, and seconds : 38.405
43. Find the value of cos of the angle in standard position that passes through point (3, 4).
44. An angle with rotation of – 220 degrees terminates in which quadrant?
45. Find the domain of the function : g( x)  x  4 .
46. Graph y = 2tan x +1.
47. Given:
48. Graph
f ( x)  x 2 , g ( x)  3x  4 , evaluate
y  csc
1
x .
bf  ggbg
x 1
2
49. Find the period, amplitude, vertical displacement, and phase displacement of the graph of
each equation.
A. y  3cos(2 x   )  1
B. y  0.5sin(.5x .5 )  3
C. y  3sin 2 x
50. State the type of symmetry:
A. y  x2  1
B. y  x3  x
51. State the domain and range for:
A. y  sin x
C. y  x
B. y  cos x
D. y  sec x
52. State the domain and range for:
A. y  x


C. y  3sec 2  x    1
6



B. y  3sin 2  x    1
4

1
3 
D. y  cot  x 

3
4 
53. Find: csc 90  cot 0  sin180  cos 270 .
54. Graph
y  3 cos
1

 x   1.
2
3
55. Refer to the triangle to find exact value for the following:
A. sin T =
B. cos T =
C. tan R =
D. sec T =
E. csc R =
F. csc T =
G. cot R
56. Describe the transformations of each equation.
A.
B.
C.
D.
f ( x)  2 cos x
f ( x )  cos 21 x
f ( x)  cos( x  2 )
f ( x)  cos x  2
57. Write an equation for the graph of the cosine function with the following characteristics:
period of  , phase shift left 2 , vertical stretch of 3
58. If f ( x)  2 x  4 and g( x)  3x 2  1 , find:
bg
g f bg
x
A. f g x
B.
C. f 2a
D. g 5a
b g
59. If g( x)  x 2  1 and f ( x)  x  2 , find f ( g( x)).
60. Which graph is one-to-one?
A.
B.
C.
D.
61. Describe the graph as odd, even, both, or neither.
A) ____________
B) ___________
C) ____________
62. Name the asymptotes for y  2 sec x  3 .
63. Evaluate: csc14  to 4 decimal places.
.
. (in radians)
64. Find an acute angle such that tan   12617
Solve.
65.
27 x 
3
9x
68. log 32 x  
71. 64 x1  51
3
5
66. 125x1  25x2
67. 92 x  2  9x  3  0
69. x  ln16
70. e4 x  7
72. 42 x1  5x2
73. log3  x  2  log3 x  1
74. 2log4  x  5  3log4  x  5  1
Graph and name the domain and range.
75. y  2 x 1  3
76. f  x   2  e x
77. y = ln x
________________________________________________________________________________
78. The number of bacteria P in a culture after t minutes is modeled by the exponential
growth curve P  Aekt where k = 0.11. If there wer originally 500 bacteria in the
culture, determine in how many minutes there would be 6200 bacteria.
________________________________________________________________________________
79. A paddlewheel on a steamboat makes one complete revolution every 18 seconds. The diameter
of the wheel
is 10 feet. A point is at the bottom of the wheel, 2 feet below the surface of the water, at t = 0.
Write an
equation to represent the point's distance from the surface of the water.
80. A gear makes 6.2 rotations about its axis. what is the angular displacement in radians of a point
on the gear?
81. What is the angular velocity in radians per minute of a notch on a wheel that makes 24 rotations
per second
about its axis?
82. The minute hand of a watch is 1.3 cm long. What is the linear velocity of the tip of the hand?