Final Exam Review – S18
Name________________________________
Pre-AP Precalculus
Inverse Trig
*All completed without a calculator.
Evaluate.
1.

  
arccos  tan   
 4 

4.
sin arccos 3x 
7.
sin arctan  0 




3.

 5
sin  1  sin 
 6


 12  
sin  tan 1   
 5 

6.

  
cos 1  cos    
 3 

8.

 2 
sec  sin  1 
 

 5 

9.


3 

cot  sin  1  
 2 




11.
cos2 x
1  sin x
12.

2.
tan1 cos  
5.


 

Identities
*All completed without a calculator.
Simplify.
10.
cot2 x  csc2 x
sec2 x  1
sin2 x
Verify.
13.
1  tan x
 csc x  sec x
sin x
16.
Find exact values for sin 2A, cos 2A, and tan 2A if A terminates in Quad III and tan A 
17.
Given: sin A 
14.
sec x  cos x
tan x
2
 cos x
  
tan   
6 4
1
 sec2 x  sec x tan x
1  sin x
3
12
and tan B   . A & B are both in quadrant II. Find cos A  B  .
4
5
Give the exact value for each.
18.
15.
19.
 5  
sin 
 
 4 6
4
.
3
Solve each equation. Give the general solution and the particular solution for x  0,2  .
20.
4sin x cos x   2
21.
sin2x  2 cos x
22.
23.
2cos2 x  7 cos x  3  0
24.
2sin 2x   1  0
25.
26.
tan x sin x   sin x
27.
4 sin2 2x   3  0
28.
2sin2 x  cos x  1
2 tan x cos x  tan x  0
2sin2 x  3sin x  1  0
Law of Sines/Cosines and Vectors
*Calculators allowed. Be sure to put calculator in degree mode.
29.
Given PET , m T  5719', m E  7245', and p  10.3. Find side e.
30.
Given MIK , m I  3817', i  13.3, and k  17.7. Find the m K .
31.
Given NAP , n  10.3, a  9.8, and p  6.5. Find m N .
32.
Given
DOG , m D  48 12', d  12.5, g  15.5 . Find side o .
33.
Given
ABC , m A  43 45', b  57, c  85 . Find the area.
34.
Find the area of
35.
A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 12° with the
ABC if a  12.32, b  8.46, and c  15.05 .
horizontal. The flagpole’s shadow is 16 meters long and points directly up the slope. The angle of elevation
from the tip of the shadow to the sun is 20°. Find the height of the flagpole.
36.
An airplane flies 370 miles from point A to point B with a bearing of N 24° E. It then flies 240 miles from
point B to point C with a bearing of N 37° E. Find the distance from point A to point C.
37.
Find the component form and linear combination form of the vector, and then determine its magnitude and
direction.
initial point: (6, 3)
38.
Given u  3i  2 j and v  1, 2 . Find the following:
a.
39.
terminal point: (-8, 7)
2u  3v
b.
geometric sketch of v  2u
A force F1 of 36 N pulls at an angle of 20° above due east. Pulling in another direction is a force F2 of 48 N
acting at an angle of 42° below due west. Find the magnitude and direction of the resultant force.
40.
An airplane flies east for 210 km before turning 70° south and flying 100 km. Find the distance and direction
of the plane from its starting point.
Chapter 3 – Exponential/Logarithmic Functions
*No calculator unless otherwise stated.
Sketch the graph of each. Be sure to give domain, range, and the equation of the asymptote.
41.
1
f  x   2  
2
x 3
4
42.
f  x   2log3  x  3  1
Y
43.
f  x   ln  x  5   3
Y
Y
X
X
X
Evaluate.
44.
log3 7 81
47.
Expand.
49.
Factor and simplify completely.
ln
x
y z3
2
4log3 2
45.
3
48.
Condense.
46.
2ln e 7  ln e 6
ln x  4 ln  x  2  ln x  2
2
5
2
1
x  4  x 3   4  x  3
3
6
Solve without a calculator.
50.
1
 
5
x 1
 1252x 3
51.
a 3x 2  b 4x 1
52.
log2 x  log2  x  2  log2  x  6
For #57-61, solve with a calculator. Be sure to show all work. Round to the nearest thousandth.
54.
8 4
6 2x
53.
e 2x  10e x  24
56.
The half-life of a radioactive isotope is 1200 years. How long will it take a 10 gram sample to decay to 2.4
 13  41
55.
6  5ln2x  2
grams?
57.
How long will it take an initial investment of $1000 to grow to $5,000 if the investment is earning 4.5%
annual interest, compounded quarterly?
Sequences & Series
*Calculators may be used, but no decimal approximations unless in the context of a word problem. Show all
formulas and work.
58.
Find a formula for an if in the arithmetic sequence a3  54 and a20  3 .
59.
Find the 30th partial sum of 5, 8, 11, 14,...
60.
If ak 1  2ak  3, find the first 5 terms if a4  45 .
61.
Find a formula for an if in the geometric sequence a3 
16
64
and a5 
. (Assume r is positive.)
3
27
Find the sum of the series if it exists.
62.
n 1
1
 16  
n 1
2
12
63.

n
 4  .2 
n 0
64.
1 1
  1  3  ...
9 3
Write the series in sigma notation.
66.
1 2 7 14



 ...
7 10 13 16
68.
Find the fourth term of the expansion 2x 2  3y
65.
3 9 27
729
 
 ... 
2 6 24
5, 040
67.
Simplify.
n  3 !
n  1 !
69.
Expand x  2y 2


70.
An object with negligible air resistance is dropped from a plane. During the 1st second of fall, the object
5


8
.
.
falls 4.9 meters; during the 2nd second, it falls 14.7 meters; during the 3rd second, it falls 24.5 meters,
etc. If this pattern continues, how many meters will the object fall in 10 seconds?
71.
An investment firm has a job opening with a salary of $30,000 for the first year. Suppose that during the
next 39 years, there is a 5% raise each year. Find the total compensation over the 40-year period.
Polars and Parametric Equations
*All problems except projectile motion should be completed with no more than a 4-function calculator.
Write each of the following in standard form, parametric form (76 & 77 only), and then sketch.
72.
x 2  4y 2  10x  16y  37  0
73.
9y 2  x 2  54y  8x  56  0
Y
Y
X
X
74.
y 2  8x  4y  28  0
75.
Convert the rectangular coordinate

3,  1

to polar form.
Y
X
76.
 3 
Convert the polar coordinate  2,
 to rectangular
 4 
form.
Convert each rectangular equation to polar form.
77.
3x  y  2  0
78.
3x 2  3y 2  27
79.
y 4
82.
r 
Convert each polar equation to rectangular form.
81.
2
1  sin 
80.
r  3 sec 
83.
An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an
r  2 cos 
angle of 10° with the horizontal and at an initial speed of 240 feet per second.
a.
Assuming the ground is level and ignoring air resistance, find the distance the arrow travels before it
hits the ground.
b.
Find the maximum height of the arrow.
Limits
*All problems in this section should be completed without a calculator.
Find the limit if it exists.
84.
 5x  3 
lim 

x  2  2x  9 
87.
 3 
lim 

x   x 2 
85.
 x  7 2

lim 

x 3 
 x 3 
86.
 1
1
 

x

4
4

lim 
x
x 0 





88.
 t2 

lim 
x   t  3 


89.
 3x 2  4 

lim 
x    1  x 2 

