Last Name:
First Name:
1.)
I
2.)
I
3.)
cD
17.)
4.)
0
18.)
5.)
z
19.)
6.)
0
20.)
x 272=1
7.)
0
21.)
ZZ+pz
8.)
a
0
12.)
0
13.)
cx
15.)
24.)
25.)
11.)
14.)
22.)
3
r
(5
(x-
00
I
‘
2
=1
23.)
9.)
10.)
16.)
Lf
26.)
-
27.)
)
30.)
31.)
(X(-5z
V
II,pse.
32.)
)‘yp€rIoIcS
ad
33.)
28.)
34.)
5
35.) sin(O)
=
12
29.)
36.)
37.)
38.)
39.)
cos(&)
=
tan(9)
=
A
c
A
n
I
&
40.)
D
41.)
I
47.)
48.)
42.)
49.)
43.)
50.)
44.)
F
51.)
45.)
13
52.)
46.)
53.)
54.) sin(O)
A
*
2
Cos(e’)
55.) cos(O)
2
j(@): I
56.) csc(O)
2
2
2.
—2
-2
58.) cot(9)
57.) sec(9)
—I
59.) tan(9)
(
4
—2
60.) arccos(O)
c
61.) arcsin()
62.) arctan(6)
ir
N
-
-
2
—I
7.
—v
-17•
)547
“
63.)
5+18
67.)
64.)
2+11/
68.)
—i
69.)
—1
65.)
66.)
Nf(hi-)
70.) and 71.)
4:
‘1
************************************************
Equations in one variable
For #72-77, give the real number solutions of the equations. If there is no
solution, explain why there is no solution. For at least one of the problems,
the domain of the equation will play an important role. Worth 2 points each.
72.)
i/3x
—
4
=
—9
ro
sZLt€
73.) loge(x) + loge(x
—
1)
=
(x-izO
.x
2’Q
2
SiCe
z_z 1O
17)2
=
7L—j7
L—7- 3
9
—::
20
2I
2
I,
2
S0
‘S
I-c?
2.
—
av6(
-
1
J
I ,i-t, c—j
74.) (x
b€
0
x’o aicI .z-1.>O
Io€
CQi
a;
5c
Il
q
N ‘u
1
O
N)N)
$
N
)‘
4
V.,
5
p.
II
I
‘JI)
1
1
I)
J)
‘1
U)
)
(A)
J
I’
I
)
0
NI
I)
JVi\
N1
0
N
0
—
+
0
I.
+
H
V
Jp
4—I
(f4
1
I
))
I,
—
-
,)-i..
c.,J
‘I
(__••‘%
Ni,
0
L-.0
Final Exam
Equations in One Variable
How many real number solutions are there for the following equations?
Your answers will either be 0, 1, 2, or oc, if there are infinitely many solutions.
Worth point each.
1.) eX
4.)
=
1
2
=
—2
Q
7.) x
+x+4=O
2
—9-(s)’t: —15<0
1
10.) eX = 0
Q
Q
13.) tan(x)
=
2.) x
2
=
0
1
3.) sin(x)
=
5.) x
2
=
3
2
6.) cos(x)
=
8.)
Q
2=_4
11.) x
2
—
2x + 1
=
0
(_2)2._
—5 c
14.) cos(x)
1.
=
5
1
12.) sin(x)
=
Q
15.)
4
=
Linear Algebra
(3
_2) (_2)
/-\
/--z\
(-8,7)
J
17.) Find the product
( )(
3)
o
2•1
2(-3)
0
9.) log(x)=—1
************************************************
16.) Give the vector written as a ROW vector
o
—
5
—2
1
Conics and other solutions of equations in two variables
18.) Give the eqilation for a line of slope —3 that passes through the origin.
-3x
19.) Give the equation for a line of slope —3 that passes through
the point (1,5).
5.
20.) Give the equation of the unit circle.
(5
21.) Give the equation of the circle of radius 3 centered at the origin.
4
22.) Give the equation of the circle of radius 3 centered at the point (6, 4).
7
‘14.
0
(2
‘3
23.) Give the equation of the ellipse obtained by starting with the unit
circle, and then scaling the x-axis by 2, and the y-axis by 4.
I
LI-
24.) Give the equation of the ellipse from #23, shifted right by 3 and
upby4.
8Q
(Z
Xf—’x--3
25.) Draw the set of solutions of the equation y
2
26.) Draw the set of solutions of the equation y
=
27.) Draw the set of solutions of the equation xy
28.) Draw the set of solutions of the equation x
29.) Draw the set of solutions of the equation
=
1.
=
1.
—
x2.
=
=
x2
—
1.
.
2
y
30.) R
=
()
is the rotation of the plane by angle
is the set of solutions of y
=
.
If P
cR
2 from #26, then draw R(P).
x
31.) Let H be the set of solutiois of xy 1 from #27. What’s the equation
for H shifted right by 3 and up by 5?
32.) Name the three types nondegenerate conies.
************************************************
Trigonometry
33.) Wh
is the distance between the points (3, 4) and (5, 1)?
(3Z()a
34.) Find the length of the unlabeled side of the triangle below.
2
3:Ia
zQ
1
35.) Find sin(6), cos(O), and tan(O) for the angle 0 given below. (3 points.)
5
I2
12
Match the functions with their graphs.
36.)
39.)
42.)
45.)
A
cos(x)
cos()4
cos(x) + 1
37.)
40.)
43.)
46.)
3
cos(x+)B
cos(x) C
sin(x)
cos(x—)E
cos(2x) G
38.)
41.)
44.)
47.)
B)
A.)
A
cos(—x)
cos(x) 1 0
2cos(x)F
—cos(x)
—
C.)
—a
D.)
F.)
E.)
a
-7r-%
/
7T
-2
H.)
G)
I.)
2
“\/\ f/
-z
J.)
cI
rr
-‘it-!
z
—I
-2.
2.
Match the functions with their graphs.
48.) f(x)
=
{:: B
if x
ifx
A.)
<
0;
0.
49.) g(x)
=
{: /4
if x <0;
if x 0.
B.)
]
<1
Find the values for #50-52. All answers will be either or or
50.) arccos():4
51.) arcsin()r
52.) arctan()
‘vce
SHICL
s;nc€.
53.) What’s the Vythagorean Identity?
For #54-62, graph sin(x), cos(x), csc(x), sec(x), cot(x), tan(x),
arccos(x), and arctan(x).
Complex numbers
63.) Find (3 + i4) + (2 + i4).
+
5i8
64.) Find (3 + i4)(2 + i).
j
_17L
z
+2;)!
a
65.) What’s the
norm
of 3 + i5.
J3-i5/
66.) Write 3 + i
9-i-zS
in
7
f-i
polar coordinates.
j3+jj
so
(-, +-)
67.) Find 3(cos(2) + isin(2))5(cos(4) + isin(4)).
3.
aycL
5j5
2+q-z, so
68.) Find
2.
5
2.
69.)
1
(
70.) Draw a dot on the number 3 cos
(7r)
+ i sin (yr))
(—)
O’Cw
3,
(—
clocw”- /
71.) Draw an X on the number 2(cos
+ isin
(The number z is drawn on the answer sheet.)
5cle ,y 2. , ta4
ctrjl€
7h