Precalculus Spring Final Review
Name_________________________
Due the day of the final exam. Show work where applicable.
Section I (trig only):
Write an equation for each of the following graphs. There are multiple correct answers.
1.
2.
Solve each equation.
3. 3  2 sin x  0
4. 2 sin2 x  x 1  0
5. 2 cos 2 x  sin x  2
Find the exact values of each.

2

6.   sin 1  

2



3

7.   arccos 
 2 


8. arcsin 1
3
1
, cos    and  is in Quadrant II and  is in Quadrant III, find the exact values of
5
3
each expression below:
sin   
cosa   
cos   
9.
10.
11.
If sin  
12.
cos 2
13.
sin 2
Section II (MC trig and seq/series):
1. Solve 4sin2x – 3 = 0.
3
a) 
b) 60 , 120  c) 60 , 120  , 240  , 300 
2
2. Given sin x = 5/13 find sec x.
a) 12/13 b) 13/12
c) 13/5
d) 30 , 150  , 210  , 330 
d) 5/13
3. Find the nth term of the arithmetic sequence with a1 = 10 and a8 = 38.
a) an = 10 + 38n
b) an = 10 + 38(n-1)
c) an = 10 + 4(n-1)
4. Write the first three terms of the sequence a1 = 5, ak+1= 4ak + 3
a) 5, 8, 11
b) 5, 23, 95
c) 5, 20, 80
d) 5, 7, 12
20
5. Find the sum
 4n  5
n 1
a) 740
b) 74
c) 1480
d) 10
6. Find the infinite sum of the geometric sequence 20, 10, 5, ...
a) 35
b) 40
c) not possible
d) 30
7. Find the exact value of cos 2x if sin x = 3/5 and angle x is in the 2nd quadrant.
a) -4/5
b) 7/25
c) -7/25
8. Simplify cot x + tan x
a) sec x csc x
b) 1
d) 8/10
c) -1
d) sin2x + cos2x
9. Find the exact value of sin 2x if tan x = 5/12 and x lies in the first quadrant.
a) 120/169
10. Simplify
b) 24/26
c) 10/26
d) 60/169
cos x  1
sin x

sin x
cos x  1
a) -sec x/(cosx -1)
b) sec x / (cos x -1)
11. Simplify tanx cos x
a) cot x
b) sin x
c) -1
c) cos2x / sin x
d) 0
d) cos x
n!
assuming that n begins with 1.
n2
c) 1, 2/4, 3/9
12. Find the first 3 terms of the sequence an =
a) 1, ½, 2/3
b) 1, ¼, 1/9
13. Given cos x = 5/13 and sin x < 0 find tan x.
a) 5/12
b) 12/5
c) -5/12
d) - 12/5
14. Use the properties of inverse functions to evaluate cos(arctan ¾).
a) 3/5
b) 4/5
c) 4/3
d) 3/4
15. Find the solutions in radians for cos x = -1/2.
2
4
a)
b)
c) a & b
d) no solution
3
3
16. Find the solutions in radians sin x = ½
 5
 2


,
,
a)
b)
c)
d)
6 6
3 3
6
3

 4( 3) n 1
17. Find the sum
1
n 1
a) 4
b) 6
c) 4/3
d) no solution
c) 1100
d) 2200
24
18. Find the sum
 4n  3
n 5
a) 17
b) 93
19. Find the nth term of the sequence 5, 8, 11, 14, ….
a) an = 5 + 3(n-1)
b) an = 5 + 3n
c) an = 14 + 5(n-1)
20. Find the nth term of the sequence 3, 12, 48, 196, ….
a) an = 4(3)n-1
b) an = 3(4)n-1
c) an = 12n-1
21. Find the exact value of cos 75
a)
3 1
2
b)
6 2
4
c)
6 2
4
Section III (limits):
Compute the following limits.
1. lim
x
8x 5  6x 2  1050
16x 5  7 x
3. lim
x  
5. lim
y  
7. lim
10 x 3  9 x
5x 3  4 x
2. lim
5x 2
x  0
4. lim
n  3
n 2  7n  12
n2  9
30 y 3
2y4
6. lim
n
n  16
12 cos x
8. lim
2
x  6x  9
n 4
x 3
2
2
x
9. lim
x2
7x  14
x2
Using the function below, determine these limits:
10. lim
n
n2
n
11. lim f (x) =
x2
16. lim f (x) =
x0
12. lim f (x) =
17. f (0) =
13. lim f (x) =
18. lim f (x) =
x2
x2
14. lim f (x) =
x4
x
19. lim f (x) =
x
15. lim f (x) =
x2
IN ADDITION TO THIS HANDOUT, YOU SHOULD CAREFULLY REVIEW THE
LIMITS HANDOUT WE HAVE BEEN WORKING ON THESE LAST TWO WEEKS.