.Precalculus- Spring Final Review 2015
Directions: Do all work on a separate paper. Please organize your work. Leave all answers in simplified exact form unless otherwise stated. Only use a calculator if the problem states so and round your answers to the hundredth.
(7.1 & 7.2) – Inverse Trig.
Evaluate:
1. cos

1

 
2
3

 2. sec(tan

1
3
3
) 3. sin (cos
2

3
) 4. sin(tan

1
3
4
) 5. cos(sin

1

1
)
3
6. cos



4
 csc

1
5
3


7.
 sin sin

1
 
 cos

1



4
5




(7.3-7.5) – Trig Identities Establish each identity in #8-11.
8. sin


 sin



2 csc

9.
2

 
 cot
  tan

10. tan
  cot
 tan
  cot

1 2 sin 2

11.


 tan

12. Find the exact value of cos 75 .
13. sin
  
3
,
5
3

2
; cos
 
12 3
,
13

2

2

find each of the following: a) sin(
 
) b) cos 2
 c) sin 2
 d) tan(
 
) e) sin

2
(7.7 &7.8) - Trig. Equations
Solve each equation on the interval 0

2

. (Calculator ok for #21)
14. cos
  

1
3 4 2
18. cos 2
  sin

15.
  
1
19. sin
  cos
 
1
16. cos 2
  
2
2
17. 2 sin 2
 
3sin
  
0
20. cos 2
 
5cos
  
0 21. 6sin 2
 
2sin

5 0
(Chapter 8) – Triangle Trig.
Solve for all missing sides and angles in each triangle. (Calculator ok for # 22-26) .
22. a

6, b

8, A

35 23. a

2, B

70 , A

32 24. a

2, b

3, C

60
25. Find the area of the triangle: 26. Find the area of a triangle with side lengths 3, 4, and 6.
4
42º
8
(9.1&9.2) - Polar
27. Change from polar to rectangular coordinates:
a)
 b) 2,
5

2 4
29. Given the point 5,
2

3
28. Change from rectangular to polar coordinates: a) (-5, 5)
find at least 3 other coordinates that represent the same point. b) (1, 3)
30. Graph each polar equation on a polar graph.
a) r

3 b) r

4 sin

31. Graph each of the following by plotting points and using your knowledge of symmetry. a) r
  
b) r
  
c) r
=
q
(9.4 & 9.5) – Vectors
For the following problems use the vectors: v =2 i + 3 j and w = 3 i – 4 j
32. w 33. v
· w 34. angle between the vectors (calculator) 35. 3 v w
36. The vector w has initial point (-3,2) and terminal point (6,5). Find the position vector in the form a i +b j.
37. Draw the vector u =5i+3j+7k and find u
38. A ship sails 50 mph at a bearing of N20°W and then turns and sails 30 mph at a bearing of N80°W. Find the speed and bearing of the resultant vector. (calculator)
39. A plane needs to fly 200 mph N40ºW. (so this is p + w). The air is moving with a wind speed of 60 mph S10ºW. Find the actual speed and direction the pilot needs to direct the plane so that it can go its desired direction . (calculator)

40. The depth of water off the end of a dock varies from 6 feet at low tide to 14 feet at high tide. The first low tide in the morning is at 7am and the next is 12 hours later.
a) Graph the situation. Starting time is midnight this morning (so 7am is 7 on the x-axis.)
b) Write an equation to model the depth of water in relation to time.
c) List the first 4 times that the water is 10 feet deep.
d) Between which times is the tide less than 9 feet deep? (in one tide cycle)
e) List the first 4 times that the water is 12 feet deep.
(10.1-10.4) - Conics
41. Find the vertex, focus, directrix, and two points on the latus rectum of the parabola, then graph. y
2 +
y
=
x
-
42. Find the center, vertices, foci and asymptotes of the hyperbola, then graph. 2 x 2
 y 2

4 x

4 y 4 0
Find the equation of each conic.
43. Hyperbola: vertices at (1,2) and (1,4); focus at (1,6).
45. Parabola: vertex at (2, -3); focus at (2, -4)
44. Ellipse: foci at (-4, 2) and (-4, 8); vertex at (-4, 10)
46. The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 400 feet apart and 100 feet high. If the cables are at a height of 10 feet midway between the towers, what is the height of the cable at a point 50 feet from a tower? (calculator)
47. The arch of a bridge is a semi-ellipse. The arch is 50 ft above the water at the center and 200 feet wide at the base. What is the height of the arch 30 feet from the center?
(14.1-14.4) - Limits
48. Find each of the following limits: a. lim x
®
3
æ
æ x
3 -
27 x
-
3
æ
ææ
b.
lim x
®
5
æ
æ
3 x
3 -
15 x x 2
2 +
9 x
-
45
-
25
æ
æ
c. lim x
®
3
+
2 x
+
4 x
-
3 d. x

0 x
 e x
e) Determine whether f is continuous at 5. 49. x lim

2

 x lim

2

but
Explain how this can be true.
 
is not continuous. f ( x )
=
ì
ï
ï
ï
í
ï x 2
+ x
x
-
5
30
if x
<
11 if x
=
5
5
2
7 x
+
67
7
if x
>
5
50. lim x

1 x

3


 lim x

3

x

3


Optional - Misc. Topics .
Accurately graph. Find all intercepts and asymptotes. You may use a non-graphing calculator to help you find points.
51.
 x
2 x
3

4
Solve each equation:
53. Solve:

3 x 1
6
2 x
56. 3 x

4 x
(calc ok for #56, 57)
Solve the inequality.
58.
2 x

3 x

1

3
62. Given that f ( x ) 
1 x  3 a)
  b)
52. f x
 log ( x

4)
54. 2 x 
8
 x 
4 x
57. 400 e
2 x 
600
55. log
4 x
 log ( x
 
Evaluate: (Leave answers exact, rationalized and simplified)
 and g ( x )
59. log
3

1

 27 

60. ln e
3 x x  1
find the value and domain of each:
   c) f
 g d) f
    e) g f

61. log
16
4
