Honors Math Analysis
Fall Final Exam 2010 – 11
ALL numbers with decimals
MUST be stated to three digits
beyond the decimal point!!!
1. State the domain and range of: k  x   4  x 2  2 .
Domain : 2  x  2
Range : 2  y  0
2. Find a formula for f
y
6
x
x4
–1
(x), if f  x  
6

y4
6
.
x4
6
y4 
x
2
36
6
y  4     y  f 1  2  4
x
x
3.
g
If f  x   x and g  x   x 2  4, find an expression
for  g f  x  and give its domain.
f  x  =
 x
2
4  x4
g
f  x   x  4
Domain : x : x  0
Honors Math Analysis
Fall Final Exam 2010 – 11
4. A rectangle is inscribed between the x-axis and the
parabola y = 36 – x2 with one side on the x-axis as
shown in the figure.
a. Let x denote the x-coordinate of the point
highlighted in the figure. Write the area A of the
rectangle as a function of x.
y
A  2 xy  2 x  36  x 2   A  72 x  2 x 3
x
b. What values of x are in the domain of A?
Domain : x : 0  x  6
c. Find the maximum area such a rectangle can
have.
Maximum area = 166.277
5. Describe how to transform the graph of f (x) = x2
into the graph of g(x) = – (x + 3)2 + 1.
Translate the graph of f three units left and one unit up
and reflect about x-axis to obtain the graph of g.
x
Honors Math Analysis
Fall Final Exam 2010 – 11
6. Write an equation for the quadratic function given
that the vertex is at (– 4, 5) and a second point on the
graph is (0, –3).
2
2
y  5  a  x   4    y  5  a  x  4 
 3  5  a   0   4 
2
1
 16a  8  a  
2
1
2
y  5    x  4
2
7. State the zeroes and relative minimum and
maximum of f (x) = x3 – x2 – 20x – 2.
Relative maximum: y = 26.550
Relative minimum: y = 44.032
8. Larry launches a rock straight up with a slingshot.
The rock is initially six feet above the ground and has
an initial velocity of 170 feet/second.
Honors Math Analysis
Fall Final Exam 2010 – 11
a. Write an equation that models the height of the
rock t seconds after it is launched.
h(t) = 6 + 170t – 16t2
b. What is the maximum height the rock reaches?
When does it reach that height?
The rock reaches a height of 457.563 ft at t = 5.313 s.
c. When will the rock hit the ground?
The rock hits the ground at t = 10.660
s.
9. Determine the angle measure in both degrees and
radians if the terminal side is reached by two and onethird counterclockwise rotations.
Honors Math Analysis
Fall Final Exam 2010 – 11
7
2 1 rev  rev
3
3
7rev 360

14
 840

 14.661rad
3 rev
180
3
10. The point (2, – 4) is on the terminal side of an angle
in standard position. Give the smallest positive angle in
both degrees and radians.
y




2





x






-4


4
  tan
 63.435
2
1
63.435  360  296.565

180
 5.176rad
11. The angle x is in the standard position and sec x < 0
and csc x > 0 with 0 ≤ x ≤ 2π. Determine the quadrant
of x.
1
1
sec x 
, csc x 
 sin x  0, cos x  0  QII
cos x
sin x
Honors Math Analysis
Fall Final Exam 2010 – 11
Honors Math Analysis
Fall Final Exam 2010 – 11
12. State the amplitude, period, phase shift, domain,
and range of g(x) = – 2cos (3x + 1).
 
1 
g  x   2 cos  3 x  1  2 cos  3  x   
3 
 
2
1
Amplitude  2, period 
, phase shift 
3
3
Domain : x : x , Range : y : 2  y  2
13. The windshield wiper on a certain minivan is 20
inches long and the blade is 16 inches long. If the
wiper sweeps through an angle of 110°, how large an
area does the wiper clean?
110
2
2
2
Area 

20


4

368.614
in


360
Honors Math Analysis
Fall Final Exam 2010 – 11
14. Prove the identity:
2sin  cos3   2sin 3  cos   sin 2 .
2sin  cos3   2sin 3  cos   sin 2
 2sin  cos  1
 2sin  cos   cos 2   sin 2  
 2sin  cos3   2sin 3  cos 
16. Solve sin 2 x  x3  5 x 2  5 x  1 graphically, stating
all solutions in the interval 0 ≤ x ≤ 2π.
x  1.849,3.591
Honors Math Analysis
Fall Final Exam 2010 – 11
17. Solve Δ ABC if a = 14.7, A = 29.3°, C = 33°.
B  180   29.3  33   117.7
b
a
a sin B 14.7 sin117.7

b

 26.595
sin B sin A
sin A
sin 29.3
a sin C 14.7 sin 33
c

 16.360
sin A
sin 29.3
18. A parallelogram has sides 15 and 24 and an angle of
40°. Find the lengths of the diagonals.
B  180  40  140
d1  152  242  2 15 24 cos 40  15.794
d 2  152  242  2 15 24 cos140  36.777
Honors Math Analysis
Fall Final Exam 2010 – 11
19. Find the angle between vectors u  2, 4 and
v  6, 4 .
u v  2 6  4 4  4, u 
v 
 6   4
2
  cos 1
2
 2    4 
2
 52    cos
1
2
 20
uv
u v
4
 82.875  1.446rad
20 52
20. Find the two sets of polar coordinates
corresponding to rectangular coordinates (2, –3) for
0 ≤ θ ≤ 2π.
r
 2    3  13
2
2
3
  tan
 0.983    2.159    5.300
2
1


13, 2.159 ,
13,5.300

Honors Math Analysis
Fall Final Exam 2010 – 11
21. Write the complex number 8(cos 210° + i sin 210°)
in standard form (a + bi).

3  1 
8  cos 210  i sin 210   8  
 i      4 3  4i
 2 
 2
 6.928  4i
22. Use De Moivre’s Theorem to find the indicated
8
 

 
power of  2  cos  i sin   .
12
12  
 
8
 

 
  
  
8
 2  cos 12  i sin 12    2  cos  8 12   i sin  8 12  





 

2
2 

 256  cos
 i sin

3
3 

 128  221.703i
23. Convert the polar equation r  3cos   2sin  to
rectangular form.
r  3cos   2sin   r 2  3r cos   2r sin 
x 2  y 2  3x  2 y
Honors Math Analysis
Fall Final Exam 2010 – 11
24. A company manufactures boxes with no tops by
cutting squares of x inches from each corner of an 18 by
32 inch piece of cardboard and then folding up the
resultant tabs.
a. Write an equation for the volume of the box.
V  x  32  2 x 18  2 x 
b. What is the domain of the equation you wrote in
part a?
Domain: x: 0≤ x ≤ 9
c. If the box is to have a volume of 875 in3, what
value of x will minimize waste?
x = 2.483 in will minimize waste.