Precalculus Final Exam Review: NON-CALCULATOR section
1
1

csc  sin 
1.) Simplify:
2.) Simplify:
cos 2 x
1  csc 2 x
3.) Find all solutions in the interval [0,2 ) : csc x + 2 = 0
4.) Find all solutions in the interval [0,2 ) : 2 cos 3 x  cos 2 x  0
6.) Simplify: cos 6x cos 3x – sin 6x sin 3x
5.) Evaluate: cos 165° (Hint: 165° = 210° - 45° )
7.) Given sin u 
 12
3
,  u
13
2
8.) Given cos  
4
and sin   0 , find tan 2
5
and csc v 
4
7
,

2
 v   , find cos (u + v)
9.) Find an equation of a line that passes through (5, 1) and is perpendicular to the line 4x – 2y = 5
 x  3, x  3
find f(-5)
2
x

8
,
x


3

10.) Given f ( x)  
11.) Find the vertex of the parabola: y = x2 – 2x + 8
12.) Find the x- and y-intercepts of: y = 2x2 – 5x – 3
13.) Find the vertical asymptote(s): f ( x) 
1
( x  2)( 2 x  3)
14.) Find the horizontal asymptote(s): f ( x) 
2x 2  9
3x 2  1
15.) The domain of f ( x)  5  e x
16.) Convert from rectangular to polar coordinates: x2 + y2 + 3x – 2y = 0
17.) Evaluate: 3 log b
1
b
18.) Solve for x: 27x = 243
19.) Find a formula for the nth term of the sequence. (Assume n begins with 1)
2 3 4 5
, , , ,...
1 4 9 16
20.) Find an for the arithmetic sequence with a1 = 3, d = -7, and n = 54
21.) Find the sum of the infinite geometric sequence: 2, 1, 0.5, 0.25, …
22.) Eliminate the parameter and find the corresponding rectangular coordinates.
x  4 cos , y  3 sin 
 5 
 in polar coordinates using three different representations.
 3 
23.) Write the point  3,


24.) Convert from polar to rectangular coordinates:   2,
7 

6 
25.) Convert from polar to rectangular coordinates: r cos 2   2 sin 
3, x  2
26.) If f ( x)  
find lim x2 f ( x)
5, x  2
  3x 2  5 

27.) Find lim x  2
 2 x  3x  1 
 2 x 2  3x  2 

x2


28.) Find lim x2 
30.) Find an angle coterminal to  
32.) Convert to degrees:
29.) Find
 4
3
lim x 4
x2  2
31.) Find the angle supplementary to  
6
5
3
7
33.) Convert to radians: 40°
  7 

 6 
35.) Find  if sec  
34.) Give the exact value: csc
2 3
3
2
3
36.) A right triangle has an acute angle  , such that tan   . Find sin 


37.) Given u  2i  3 j and v  4i  2 j , find u  2v and calculate its magnitude and direction
38.) Find the quadrant in which  lies if tan   0 and cos  0


39.) Determine the period of f(x) if f ( x)  2 cos 3 x 


2
40.) Determine the amplitude of f(x) if f ( x)  2 sin 4 x   


41.) Describe the horizontal shift to the graph of g(x), given g ( x)  3 sin  2 x 
42.) Determine the period of the function: f(x) = 4 tan(5x)


4

  3 
43.) Evaluate: sin  arctan   
 5 

44.) Find lim h0
( x  h)
2

 3( x  h)  ( x 2  3x)
h
45.) Simplify:
1
1

1  sin x 1  sin x
46.) Solve for x: log(5 – x) – log(2x – 6) = 1
47.) Find the vertices of the hyperbola:
( x  3) 2 ( y  1) 2

4
16
48.) Find the center of the ellipse: 4x2 + 5y2 + 16x – 10y + 1 = 0
49.) Given u  2i  3 j and w  i  j and v  3u  5w , find the component form of v .
50.) A vector has a magnitude of 3 and a direction of   240 °. Find the vector.
51.) A vector w has initial point (4, 6) and terminal point (2, -5). Find the component form of the
vector.
52.) Determine the magnitude of v : v   3,6
53.) Solve x  5  10


54.) Plot the point whose polar coordinates are   4,
3 

4 
x2
55.) Graph the rational function f ( x) 
x3
56.) Graph f(x) = 4 + log x
58.) Sketch the graph: f(x) = 2 + sec 4x
57.) Graph f(x) = log (x + 4)
60.) Sketch the graph: f(x) = -3 sin (2x)


61.) Graph and write the equation for the vertical asymptotes of y  tan  2 x 
62.) Sketch a graph of f(x) = 3 – ex
63.) Sketch a graph of


4
( x  1) 2 ( y  3)

1
9
4
Precalculus Review: Calculator Active
1.) Given a triangle with a = 42, b = 10, and A = 94°, find C.
2.) Find the number of years required for a $3500 investment to triple at a 7% interest rate
compounded continuously.
3.) The sun is 23° above the horizon. Find the length of a shadow cast by a flagpole 17 feet
tall.
4.) Find the direction of v if v   3,6 .
5.) A triangle has b = 20, c = 28 and C = 50°. Find the area of the triangle.
6.) A triangle has a = 50.2 cm, b = 29.7 cm, and c = 63 cm. Find the area.
7.) Ship A is 60 miles from a lighthouse on shore. Its bearing from the lighthouse is S 17° W.
Ship B is 74 miles from the same lighthouse with a bearing of S 48° W. Find the number of
miles between the ships.
8.) Solve 2x2 – 3x – 7 < 0
9.) Find all exact, real solutions of 4x3 – 38x – 6 = 0.