Download 2nd Semester Final Review

2nd Semester Final Review
1. Given the graph of f as shown, determine the following:
a. f 0  
b. The roots (zeros) of the function f
c. lim f ( x) 
x 
lim f ( x) 
x  
d. The interval(s) of x on which the function f is increasing.
2. Use the graph below to determine the following limits. If the limit does not exist, write DNE.
a. lim k ( x) 
x 
b.
c.
d.
e.
f. lim k ( x ) 
x 5
x  
lim k ( x) 
g. lim k ( x ) 
lim k ( x) 
h. lim k ( x ) 
x  6 
x 5
x 5
lim k ( x) 
x 6 
lim k ( x) 
x 6
3. A ball is thrown upward from a bridge and falls to the ground (past the bridge). The height of
the ball above the ground d (measured in feet) t seconds after it was thrown is modeled by the
formula, d  f (t)  16t 2  22t  54 .
a. Using the formula, approximate the height of the bridge.
b. Using the formula, determine the number of seconds that have elapsed (since the ball is
thrown) when the ball hits the ground.
c. As the time since the ball was thrown increases from 2 seconds to 2.5 seconds, how
does the height of the ball change? Be specific about the amounts!
4. Consider an object moving on a circular path with a radius of 4.2 meters.
a. How many degrees are swept out when the object travels 19 meters on the circular
path?
5
b. How many meters does the object travel when it sweeps out
radians on the circular
9
path?
c. What does it mean for the object to travel 2.5 radians?
d. Suppose the distance the object traveled on the circular path varied from 3.2 meters to
8.3 meters. How many radians did the object sweep out over this distance?
5. Find the exact values of the six trig functions of . Sketch the angle in standard position.
sin  = 
7
;  in Q3
13
cos  =
sec  =
sin  =
csc  =
tan  =
cot  =
6. Find the degree measure of the central angle associated with:
a. An arc that is 14.5 cm in a circle with a diameter of 8 cm.
b. An arc that is 20.2 cm in a circle with a radius of 4 cm.
7. Find the area of each sector given its central angle  and the radius of the circle. Leave answers
in terms of 
3
, r  7.5
a.  =
b.  = 120, r = 6.3
4
8. Evaluate:


a. tan  Sin 1  1  
4

c.



3
  Sin 1  1
csc  Arc tan  



 3 

3
3

 Tan 1
b. sin  Sin 1

2
3



 1  7
d. cot  Cos 1    
 2 3




9. The radius of the earth is approximately 3960 miles. How many miles does a 48 longitudinal
angle intercept at the equator?
10. A pendulum of length 1.8 m swings through an angle of  degrees. If its tip travels 2.6 m,
what is the value of  .
11. Calculate the area of a sector of a circle of radius 2.2 cm that is intercepted by a central angle
of 14 .
12. The area of a circle is 36 m 2 . Determine the area of a sector intercepted by a central angle of
50 .
13. Determine the phase shift and vertical shift of each function. Then write an equation of each
graph.
a)
b)
14. Jose is the last person to board a Ferris wheel that has a radius of 52 feet. The
counterclockwise arc swept out by Jose is measured from the 3 – o’clock position. Since Jose
is the last one to board, the Ferris wheel starts up and doesn’t stop again until the ride is over.
a) Draw a diagram that show’s Jose’s location and the angle swept out by Jose at an
arbitrary moment in time.
b) Define a function f that relates Jose’s vertical distance above the center of the Ferris
wheel (measured in feet) as a function of the measure of the angle (in radians) swept
out by Jose. Over what interval of input will Jose complete on revolution?
c) Define a function g that relates Jose’s horizontal distance to the right of the center of
the Ferris wheel (measured in feet) as a function of the measure of the angle (in
radians) swept out by Jose. Over what interval of input will Jose complete on
revolution?
d) Suppose Jose’s cart is traveling 0.3 radii per second on a circular path as the wheel
rotates. Define a function h that relates Jose’s horizontal distance to the right of the
center of the Ferris wheel (measured in feet) as a function of the time since beginning
the ride (measured in seconds). How long will it take for Jose to complete one
revolution?
e) Sketch a graph of the function in part (d). Label your axes. Identify the interval of
input over which Jose completes one revolution.
f) Define a formula that relates the distance Jose travels d (measured in feet) in terms of
the number of seconds n, since the he began to move.
15. The lark is a unit of angle measure, where any circle’s circumference is 150 larks.
a) If a protractor for measuring angles in larks has a radius of 8.2 inches, explain how a
circle and be used to create this protractor. What is the arc length on this protractor, in
inches, of 1 lark, 25 larks, 50 larks?
b) How many radians are equivalent to 100 larks?
c) Define a function that converts a number of radians to a number of larks.
16. The diagram below shows an object that begins at (4,0) and travels 19 meters counterclockwise around a circle with a radius of 4 meters. Determine the vertical component y of its
position and the horizontal component x of its position in meters from the center of the circle.
Provide the position coordinates in both exact form (e.g., in terms of sine and cosine) and
decimal approximations (use your calculator).
x = _______________________________
y = _______________________________
Simplify:
sin 
17.
tan 
18. sec 2   tan 2 
19. cos  tan 2   cos 
20.
1  sin 2 
1  cos 2 
Find a numerical value of one trigonometric function of x if:
22. tan x cos x 
1
, find the value sin x
2
21.
sin 2 x sec x cot x  3 find the value of csc x.
23.
Find the value of sin(    ) if tan  
4
5
, cot  , 0     90  , and 0     90 
3
12
24.
Find the value of tan(    ) if cos  
5
3
, sin   , 0     90  , and 0     90 
13
5
25.
If cos   0.6, and 0     90  , find the exact value of sin 2 .
26. If  is an angle in the first quadrant and cos  .5 , find the exact value of cos 2
4
27. If cos    and  has its terminal side in Quadrant II, find the exact value of tan 2 .
5
Solve:
28. 2 sin x  1  0 for 0  x  2 .
29. tan x  3  0 for principal values of x. Express the solution(s) in radians
30.
sec x
 1  0 for all real values of x.
csc x
31. tan x  3  0 for 0  x  2 .
32. 4 sin 2 x  1  0 for principal values of x. Express the solution(s) in radians.