Practice
Second
Midterm
Exam
Second
Midterm
Exam
Conics
C onics
For #1-12, match the numbered quadratic equations in two variables with
For #1-12,
the numbered
equations in two variables with
1
their lettered
setsmatch
of solutions.
Worth quadratic
2 point each.
their lettered sets of solutions. Worth point each.
1.) y = x2
2
1.)y=x
2.) x2 − y 2 = 0
—
2.)x
=
2
O
y
3.) x2 = 0
4.) xy = 1
4.) xy
5.) x2 + y 2 = 0
2+y
5.) x
0
2
6.) x2 + y 2 = −1
2+y
6.) x
2 = —1
8.) x2 = 1
2 = 1
8.) x
9.) x2 − y 2 = 1
= 1
9.) x
2
1
7.) x2 = −1
2 = —1
7) x
2
—
2
10.) y − x = 1
10.) y
= 1
2
11.) xy = −1
11.) xy
—1
=O
2
3.)x
2
—
12.) x + y 2 = 1
12.) x
2+y
2 = 1
::__
QUr_
1
1
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Trigonometry
13.) What is the distance between the points (4, −1) and (−3, 5)?
(J
(J
14.) Find the length of the unlabled side of the triangle below.
LA)
(Fi
(Fi
15.) Find sin(θ), cos(θ), and tan(θ) for the angle θ given below. (3 points.)
2
(Fi
(J
LA)
16.) Find sin(θ) for the angle θ given below.
(J
(Fi
17.) Find cos(θ) for the angle θ given below.
(Fi
18.) Find the length c shown below.
3
19.) Write the vector (−2, 5) in polar coordinates.
20.) Rotate the point 2 cos(4), sin(4) counterclockwise by an angle of 5.
21.) Write the matrix that rotates the plane clockwise by an angle of
Simplify your answer so that it does not contain the letters sin or cos.
2π
3 .
22.) Use your answer from #21 to rotate the vector (2, 4) clockwise by an
angle of 2π
3 . Write your answer as a row vector.
4
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Transformations of Solutions of Equations in Two Variables
The “Pear Shaped Quartic” is the set of solutions, S, of the polynomial
equation x4 − x3 + y 2 = 0.
V•)
V•)
23.) Give an equation for A(−1,2) (S), the Pear Shaped Quartic shifted left
1 and up 2.
4
S
_
4
S
_
_
2 0
. Give an equation for D(S), the Pear Shaped Quartic
24.) Let D =
0 13
scaled by 2 in the x-coordinate and 31 in the y-coordinate.
_
5
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Planar Transformations
Shown below is a set S in the plane. (The x- and y-axes are not part of S.
They are just drawn for perspective.)
25.) Draw A(2,−3) (S)
I.
‘
I
I
I
I
I
-4 0%
ItIIO
—
‘4
4
0%
U’
S
27.) Draw R −π
(S)
4
h
26.) Draw
3 0
(S)
0 21
S.
‘- 0%
‘4
4
I
I
I
ItIIO
I
I
-4 0%
I
I
I
I
-4 0%
I
ItIIO
—
—
U’
0%
‘4
4
U’
4,I—
6
U’
‘4
4
0%
U’
—
ItIIO
I
I
-4 0%
I
I
I
Conies
Conies
Conies
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For #26-28, Draw the set of For
given
in two of
solutions
variables.
of the
Draw
the set
of solutio
Draw
the equation
in
set ofFor
solutions
the given
equation
#26-28,
#26-28,
Conics
(Label at least one point precisely
(Label atinleast
one point precisely i
(Label in
at #26.)
least one point precisely
#26.)
For #28-30, Draw the set of solutions of the given equation in two
variables.
(Label
at least one point precisely
in #28.)
26.)
xy=2
27.) xy=2
+
x
=
2
y
9
26.)
28.) +4y2=1
26.)
27.) x
=
+
2
27.)xy=2
2
+
x
=
2
y
9
************************************************
************************************************
******************************
2
2
2
=1
28.)
xyc
=2
29.) xc
+ y 2 = 9 Functions
30.) x4 + 4y
i
4
Trigonorn
FunctionsTrigonorn
i
4
ic
4
Trigonorn
Functions
given functions
For !Iraw the graphs ofFor
the!Iraw
given functions
the graphs For
of the
!Iraw
the graphs of the g
cos(6)
29.)30.)
sin(O)
tan(O)
31.)
cos(6)
29.) 30.)
sin(O)
30.)
I
I
I
29.) sin(O)
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******************************
************************************************
I
I
I
I
I
I
I
I
-4 0%
ItIIO
—
‘4
4
************************************************
******************************
************************************************
Trigonorn
i
4
c FunctionsTrigonorn
ic Functions
4
Trigonorn
ic Functions
4
************************************************
given functions
For !Iraw the graphs ofFor
the!Iraw
!Iraw
the graphs of the g
given functions
the graphs For
of the
Trigonometric Functions
32.)
csc(O)
33.)
sec(&)
34.)
32.)30.)
csc(O)
cot(O)
33.)
sec(&)
32.) 30.)
csc(O)
33.)
For sin(O)
#31-36, draw the graphs
ofsin(O)
the
given functions. 29.)
29.)
tan(O)
cos(6)
31.)
sin(O)
30.)
29.)
cos(6)
31.) sin(θ)
a
32.) csc(O)
34.) csc(θ)
a
32.) cos(θ)
a
33.) tan(θ)
a
sec(&)
32.)33.)
csc(O)
7
34.)
cot(O)
32.) 33.)
csc(O)
sec(&)
7
35.) sec(θ)
36.) cot(θ)
a
a
7
7
7
33.)
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Equations in One Variable
The remaining questions are each worth 2 points. For #37-42, find the
solutions of the given equations, and show your work. If an equation has no
solution, explain why.
37.) log3 (x − 7) = 4
38.) (2x − 5)2 = 16
39.)
√
3x2 − 2 = −3
8
40.) loge (x) + loge (x + 1) = loge (6)
41.) (ex )2 ex+4 = 5
42.)
x
x+1 +x
x−2
=1
9
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*****************************
Conies
Last Name: Conies
For #26-28, Draw the set of For
given
in
solutions
of the
Draw
the equation
set of soluti
#26-28,
(Label at least one point precisely
in least
(Label at
#26.)one point precisely
First Name:
27.) xy=2
26.)
+9
x
=
2
y
2
27.) 2
x
=
+
2.)
15.) sin(θ) =*****************************
5.)
Conies
************************************************
cos(θ) =For #26-28, Draw the set of soluti
Conics
(Label at least one point precisely
For #26-28, Draw the set of solutions of the given equation in
(Label at least one point precisely
in #26.)
26.) xy=2
27.) x
=
+
2
tan(θ) =
************************************************
*****************************
2
26.)
xyc
=2
27.) xc
+ y 2 = 9 Functions
i
4
Trigonorn
FunctionsTrigonorn
i
4
given functions
For !Iraw16.)
the graphs ofFor
the!Iraw
the graphs of the
29.) sin(O)
cos(6)
29.)30.)
sin(O)
17.)
7.)
18.)
I
6.)
*****************************
9.)
I
I
8.)
30
I
I
I
4.)
I
3.)
ItIIO
—
‘4
4
0%
U’
14.)
U’
1.)
I
I
-4 0%
ItIIO
I
I
-4 0%
I
I
I
—
4
26.) xy=2
ic Functions
4
Trigonorn
19.)
************************************************
For !Iraw the graphs of the
Trigonometric Functions
32.)
32.)33.)
csc(O)
33
For csc(O)
#29-34, draw
ofsin(O)
thesec(&)
given functions
29.)
30
20.) the graphs
29.) sin(✓)
30.) cos(✓)
10.)
11.)
a
a
12.)
21.)
32.) csc(O)
7
13.)
32.) csc(✓)
22.)
33.) sec(✓)
10
a
33