−1
Practice Final Exam
Discrete math
1.) Find
4
X
(i2 − 1)
i=1
2.) Find
∞
X
2
6i
i=1
3.) What is the 61st term of the sequence 7, 11, 15, 19, ...?
4.) What’s the 57th term of −3, 6, −12, 24, ...?
5.) What’s the sum of the first 60 terms of the sequence 3, 5, 7, 9, . . .?
√
2
31.)
4(x
+
1)
+
2
32.)
2
25.) −3(x
2x − +
3 2) − 1
36.)
39.) f−x
(x)+ 1
1
−1
6.) Suppose a set A contains 243 objects. How many 92 object subsets of
A are there?
7.) How many ways are there to choose and order 49 objects from a collection of 304 objects?
8.) How many different ways are there to order 93 different objects?
9.) You’re decorating a room by choosing a color to paint the walls with
and a color of carpet to use for the floor. You have 6 different colors of paint
to choose from for the walls, and 11 different colors of carpet to choose from
for the floor. How many different wall and floor color combinations could you
create?
10.) Write
9
3
as an integer in standard form.
√
32.)
39.) f−x
(x)+ 1
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
2
x
5.)24.)
ex e
************************************************
Algebra
2
−3
11.) Write ( 27
as a rational number in standard form.
8)
√
1
2
12.) Write 7−4 79 (7 2 )3 as a rational number in standard form.
13.) If a �= 0, then what is a0 as a rational number in standard form?
(Notice that this is asking for the y-intercept of the graph of ax .)
14.) If a > 0, what is loga (1) as a rational number in standard form? (The
answer has something to do with the x-intercept of the graph of loga (x).)
15.) Write log3
��
5
1
81
�
as a rational number in standard form.
26.)25.)
logelog
(x)e (x)
3
−1
3
16.) Find x where x3 ( 12 x + 3) = 8.
17.) Find x where 2
e2x
ex+3
+ 5 = 7.
18.) Find x where 4 loge (x) + loge (x3 ) + 8 = 11.
19.) Find g ◦ f (x) if f (x) = x + 2 and g(x) = x2 .
20.) Find the inverse of g(x) = 7 loge (x + 3).
√
32.)
39.) f−x
(x)+ 1
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
4
−1
21.) What is the implied domain of f (x) = x2 − 2x + loge (3 − 7x) ?
22.) What is the implied domain of g(x) =
23.)
Find
x3
2
√
−73x−4 ?
x3 − 3x2 − 5x + 14
x2 − 4
√
32.)
39.) f−x
(x)+ 1
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
5
x
25.)24.)
ex e
24.) Complete the square: Write −2x2 − 4x − 5 in the form α(x + β)2 + γ
where α, β, γ ∈ R.
25.) How many roots does 2x2 − 3x + 4 have?
26.) Find a root of x3 + 2x2 − x + 6.
27.) (2 pts.) Completely factor −2x3 + 2x + 12. (Hint: 2 is a root.)
26.)25.)
logelog
(x)e (x)
6
−1
28.) |x − y| is the distance between which two numbers?
29.) Solve for x if |3x − 2| < 4.
************************************************
Linear algebra
30.) What’s the determinant of the matrix below?
2 −3
1 −5
31.) Find the product
1 0
1 2
3 1
0 1
32.) What’s the inverse of the matrix below?
1 4
2 3
√
32.)
39.) f−x
(x)+ 1
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
7
−1
33.) Write the following system of three linear equations in three variables
as a matrix equation
2x−y + z = 2
y +2z = 1
−x +y − z = 0
34.) Solve for x, y, and z if
   
−2
−1 2 −1
x
−2 2 −1 y  =  2 
1
3 −1 1
z
−1 
1 −1
−1 2 −1
−2 2 −1 = −1 2
−4 5
3 −1 1


and
√
32.)
39.) f−x
(x)+ 1
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
8

0
1
2
x
25.)24.)
ex e
************************************************
Graphs
√ √
35.) Graph the following functions: 3, x, x2 , x3 , 2 x, 3 x, x1 ,
1
x2 ,
ex , loge (x).
36.) Graph f : (−2, 0] → R where f (x) = x2 .
37.) Graph g : {−4, −2, 2} → R where g(x) = 21 x − 1.
38.) Graph −2e−x and label its x- and y-intercepts (if there are any).
39.) Graph 4(x + 2)2 + 1 and label its vertex.
√
40.) Graph − 3 x + 2 and label its x- and y-intercepts (if there are any).
26.)25.)
logelog
(x)e (x)
9
−1
41.) Graph p(x). (Label all x-intercepts.)
p(x) = −2(x + 1)(x + 1)(x − 2)(x2 + 1)
42.) Graph r(x) (Label all x-intercepts and all vertical asymptotes.)
−3(x − 1)(x − 1)
r(x) =
4(x + 2)(x2 + 1)
43.) Graph
(
ex
h(x) =
−3
if x 6= 1;
if x = 1.
44.) Graph


if x ∈ (−∞, 1);
1
m(x) = 3
if x = 1;

x2 if x ∈ (1, 2].
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
10
√
32.)
39.) f−x
(x)+ 1
−1
Last Name:
First Name:
1.)
16.)
2.)
17.)
3.)
18.)
4.)
19.)
5.)
20.)
6.)
21.)
7.)
22.)
8.)
23.)
9.)
24.)
10.)
25.)
11.)
26.)
12.)
27.)
13.)
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
28.)
14.)
29.)
11
15.)
30.)
√
32.)
39.) f−x
(x)+ 1
−1
31.)
33.)
32.)
34.)
√
32.)
39.) f−x
(x)+ 1
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
12
x
3
S
************************************************
3
x
x2
Graphs
2
5 2 C2
5
C
73
8 It
2
3
It
5 3 C2
7 3 8 It
x
x
√
√
1 1
2
3
x
2
3
27.) 3Graph the—lfollowing functions:
3, x,
x, x , x2 , —l
e , loge (x).x3
—l x , x , xx,
35.)
2
x 3
3
27.)
x
x3
x2
27.)35.) S
27.)
√
3 xGraph
28.)
27.) f : (−1, 1] → xR2 xwhere f (x) = x3 .
3
2
x
x
√
27.)
x 27.)
2
3
x2
x
x
2
2
2
3
√
√
x xGraph g : {2, 3, 5} →xRx where g(x) = 3x − 10.
29.)
3
2
x
x
g(x)
30.)
−
log
(x
+
2)
g(x)
30.)
−
log
(x
+
2)
√
2
√
3
ee
3
x
2
x
2
x
30.)
x
3
x3
31.)
x
√
3
3
2 x
xGraph
− loge (x + 2) and
x label its x-intercept.√
3
√
x
√
2
3
x
x
√ 2 7 3 8 It
√
34(x +It 1)2 +
52 √
22
3 2Cx
x
Graph
and
its vertex. 5 1
x label
√
3
1x
x
—l
x
√
√
√
3
3
x its y-intercept.
x
Graph
−x + 1 and x1label
√
2
x
√
2
x
32.)
√
1
3
x
x
√
3
1 1x-intercepts.)
x x1Graph p(x). (Label all
33.)
x2 x
1
x
1
x
C
7
ex
ex 2 + 1)
p(x) = −2(xx2+101)(x
+ 1)(x − 2)(x − 2)(x
10
1
1
x2
8 It
5
x
1
12
x
x2
1
73
e x x2
8
1
x1
x
28
1
x12
x2
exx
e
loge (x)
loge (x)
FILLING BOUNDARIES
FILLING BOUNDARIES
SEMISIMPLE AND SOLV
SEMISIMPLE AND SOLV
loge(x)
(x)
log
e
√
x
FILLING BOUNDARIES
BOUNDARIES
OF
COARSE
MANIFO
3loge (x)
ex
2
FILLING
OF MLADEN
COARSE
MANIF
e− 1
BESTVINA,
A
23.)
−2
x
25.)
2x
−
3
29.)
g(x)
30
36.)
−3(x
+
2)
−
1
3
√
MLADEN
BESTVINA,
A
34.)
Graph
r(x)
(Label
all
x-intercepts
and
all
vertical
asymptotes.)
SEMISIMPLE
AND
SOLVABLE
ARITHMETIC
3
SEMISIMPLE
AND
SOLVABLE
ARITHMETIC
2
log
(x)
x
e
23.) −2 x − 1 FILLING
25.)
2x
−+
3 2) MANIFOLDS
BOUNDARIES36.)
OF −3(x
COARSE
IN
−1
39.) f (
e
36.)
f (x)
−3(x
− 1)
Abstract.
pa
SEMISIMPLE
AND
SOLVABLE
ARITHMETIC
GROUPS
ex loge (x)
loge (x) BOUNDARIES
FILLING
OF
MANIFOLDS
IN
Abstract. We
We provide
provide 28
pa
28.)
f (x)
36.)
28.)
fCOARSE
(x)
r(x)
=
MLADEN
BESTVINA, ALEX
ALEX ESKIN,
&
KEVIN
WORTM
eralization
of
a
theorem
of
L
MLADEN
BESTVINA,
ESKIN,
&
KEVIN
WORTM
eralization of a theorem of L
SEMISIMPLE
SOLVABLE
GROUPS
28.) f OF
(x) ARITHMETIC
4(x
+AND
2)(xMANIFOLDS
+ 2)
FILLING
BOUNDARIES
COARSE
MANIFOLDS
IN
FILLING
BOUNDARIES
OF
COARSE
IN
metic
groups
(over
number
loge (x)
28.)
f (x)
MLADEN BESTVINA, ALEX
ESKIN,
& KEVIN WORTMAN metic groups (over number
SEMISIMPLE
AND
SOLVABLE
ARITHMETIC
GROUPS
low
both
polyn
AND
SOLVABLE
ARITHMETIC
GROUPS
28.)
f
(x)
logSEMISIMPLE
(x)
Abstract.
towards
low dimensions,
dimensions,
both
polyn
Abstract. We provide partial results
towards aa conjectura
conjectura
e
MLADEN BESTVINA,
ALEX ESKIN,
& KEVIN
WORTMAN
x ex
FILLING
BOUNDARIES
OF
COARSE
MANIFOLDS
IN
24.)
25.)
log
(x)
inequalities
p
25.) e OF Abstract.
26.)of alog
inequalities and
and finiteness
finiteness
eralization
theorem
Lubotzky-Mozes-Raghunathan
for
eralization
forp
e (x)eof Lubotzky-Mozes-Raghunathan
FILLING BOUNDARIES
COARSE
MANIFOLDS
IN towards a conjectural
We
provide GROUPS
partial results
genSEMISIMPLE
AND
SOLVABLE
ARITHMETIC
As
a
tool
in
our
proof,
we
As
a
tool
in
our
proof,
we
metic
groups
(over
number
fields
or
function
fields)
that
impl
groups
fields) that impl
MLADEN
BESTVINA,
ALEX metic
ESKIN,
& KEVIN WORTMAN
MLADEN BESTVINA, ALEX
ESKIN,
&
KEVIN
WORTMAN
eralization
of
theorem
of GROUPS
Lubotzky-Mozes-Raghunathan
forgenarith- for
SEMISIMPLE
AND SOLVABLE
ARITHMETIC
isoperimetric
inequalitie
Abstract.
Wea provide
partial
results
a conjectural
for
isoperimetric
inequalitie
lowtowards
dimensions,
upper
bounds
isoperim
low
dimensions,
both polynomial
bounds for
for
isoperim
ex
1
x2
1
x2
35.) Graph
metic
(over number
fields or function fields) thatfor
implies,
in able
� groups
as ss
eralization
of a theorem
of Lubotzky-Mozes-Raghunathan
arithable groups
groups that
that appear
appear as
inequalities
and finiteness
properties.
inequalities
MLADEN
BESTVINA,
ALEX
ESKIN,
& 3KEVIN
WORTMAN
Abstract.
We
provide
partial
results
towards
a
conjectural
genlow
dimensions,
both
polynomial
upper
bounds
for
isoperimetric
Abstract.
We provide
partial
results
towards
a
conjectural
genple
groups,
thus
generalizin
metic
groups
(over
number
fields
or
function
fields)
that
implies,
in
x
if
x
=
�
−1;
As
we also provide polynomial
polynomial upper
upper bb
As aa tool
tool in our proof,for
MLADEN BESTVINA, ALEX ESKIN,
& KEVIN
WORTMAN
eralization
ofboth
afiniteness
theorem
of
Lubotzky-Mozes-Raghunathan
arithinequalities
and
properties.
eralization of a theoremh(x)
of Lubotzky-Mozes-Raghunathan
for
arith=
low
dimensions,
polynomial
upper
bounds for isoperimetric
for
isoperimetric
inequalities and finiteness results
results for
for certain
certain
for
isoperimetric
metic
groups
(over
number
fields
fields) upper
that implies,
As
our
proof,
we
also
provide
polynomial
boundsin
metic groups
(over number
fields
or
function
that
implies,
inor function
2a tool
iffields)
x
=
−1.
Abstract.
We provide
partial results
towards
ain
conjectural
geninequalities
and
finiteness
properties.
able
groups
that appear as subgroups of parabolic
parabolic groups
groups in
in se
se
able
groups
dimensions,
both
polynomial
upper
bounds
forcertain
isoperimetric
for
isoperimetric
inequalities
and
finiteness
results upper
for
solvAbstract.
provide
results
a conjectural
genlow dimensions,
both
polynomial
upper
bounds
for
isoperimetric
eralization
of xaWe
theorem
of partial
Lubotzky-Mozes-Raghunathan
for
arithAslow
atowards
tool
in
our
proof,
we
also
provide
polynomial
bounds
10
ple
groups,
thus
generalizing
a
theorem
of
Bux.
Our
main
result
is
Theorem
x
ple
groups,
thus
generalizing
a
theorem
of
Bux.
x
xappear
24.)
log
(x)
inequalities
and
finiteness
24.)
e foras25.)
25.)
able
groups
that
subgroups
parabolic
groups
in semisimeralization
a theorem
of Lubotzky-Mozes-Raghunathan
arithinequalities
and
finiteness
25.)
egroupseof(over
26.)
log
(x)eof
25.)
ethat
26.)
logelog
(x)e (x)
metic
number
fieldsproperties.
orfor
function
fields)
implies,
inproperties.
isoperimetric
inequalities
and
results for
certain
solvefiniteness
some
background.
As
a
tool
in
our
proof,
we
also
provide
polynomial
upper
bounds
ple
groups,
thus
generalizing
a
theorem
of
Bux.
metic
groups
(over
number
fields
or
function
fields)
that
implies,
in
As
a
tool
in
our
proof,
we
also
provide
polynomial
upper
bounds
√ able
dimensions, both polynomial
upper
bounds
isoperimetric
groups
thatfor
appear
as subgroups of parabolic
groups in semisim36.) low
Graph
−x2x
2 3x−
2
−x
for
isoperimetric
inequalities
and
finiteness
results
for 2)
certain
x +

low
dimensions,
both
polynomial
upper
bounds
for
isoperimetric
for
isoperimetric
inequalities
and
finiteness
results
for certain
solv25.)
2x
−
3
23.)
−2
1
25.)
−
3Theorem
37.)
g(x)
38.)
−2e
x
29.)
g(x)
30.)
−
log
(x
+
29.)
g(x)
30
36.)
−3(x
+
2)
−
1
39.)
f
(x)
36.)
−3(x
− solv15 below. Before stating
ple
groups,
thus
generalizing
a
theorem
of
Bux.
and
finiteness
properties.
37.) inequalities
g(x)
38.)
−2e
24.)
e2)
e result
29.)
g(x)
30.)
−
log
(x
+
2)
25.)
e
26.)
Our
main
is
it3
e
able
groups
that
appear
as
subgroups
of
parabolic
groups
in
semisimOur
main
result
is
Theorem
5 below. Before stating
it,
inequalities
and
finiteness
properties.
able
groups
that
appear
as
subgroups
of
parabolic
groups
in
semisim
√
x
if
x
∈
[−2,
1);
g(x)
30.) − loge37.)
(x +p(x)
2)
As 29.)
a tool in
our proof, we also provide
polynomial upper bounds

4
some
ple polynomial
groups,
thus
generalizing
abackground.
theorem
of−3(x
Bux.
24.)
−x
−finiteness
+ 2)2it,−we
1 provide
Asple
a tool
in our
proof,
we
also
provide
bounds
groups,
thus
generalizing
a1
theorem
of Bux.
Our
main
result
is upper
Theorem
530.)
below.
Before
stating
29.)
g(x)
−26.)
log
(x
+
2)
some
background.
for isoperimetric
inequalities
and
results
for
certain
solve
m(x)
=
3 results
if xfor=certain
1; solv√ appear
for
isoperimetric
inequalities
and finiteness
11
29.)
g(x)
30.)
−
loge−3(x
(xBefore
++
2) 2)
some
2
able
groups
that −x
of √
parabolic
groups
in semisim37.)
p(x)
40.) g(
Our background.
main
result is
Theorem
526.)
below.
stating
it, we provide

24.)
−asas1subgroups
−
1

able groups
that appear
subgroups
of parabolic groups in semisimple groups, thus generalizing
a theorem
ofxBux.if x ∈ (1, ∞). 1
some
background.
ple groups,
generalizing
of Bux.
main
result
Theorem
5 below.
stating
it, we provide
x5 below.
Our
main thus
result
is Theorem
Beforeisstating
it, 1we
provideBefore
x a theorem
24.)
eOur
25.)
log
25.)
e
26.)
log
(x)
e (x)
e
some background.
some background.
√
2
1
1
31.)
4(x
+
1)
+
2
32.)
2
− 1Our
25.)
2x
−8+
3 stating
below.
Before
−3(x
2)
− it,
1it,we
f−x
ex1 −
1Before
log
++
2) 1
Our main
main result
result isis Theorem
Theorem
below.
stating
weprovide
provide 28.)27.)
e (x
27.)26.)
e5x536.)
−
loge39.)
(x +
2)(x)
some
some background.
background.
x
27.)26.)
ex −e 1 − 1
11
√
37.) −3(x
p(x)24.)
26.)
+ 2)2−x
−−
1 1
13
10
10 13
10
x
10
10
e 1 −27.)
1 elog
e (x
log
(x +
2)+ 2)
27.)26.)
ex −28.)
10
e (x
28.)27.)
logelog
(x +
2)+ 2)
x
26.)
40.) g(x)
37.)
p(x)
26.)
+12)2 − 1
27.)
ex−3(x
−e 1 −
38.) q(x)
4
28.)
5.)
16.)
x
25.)24.)
ex e
26.)25.)
logelog
(x)e (x)
6.)
17.)
6.)
17.)
17.)
6.)
17.)
x
x
x
e 25.) log (x)
25.)24.)
e 26.)
26.)25.)
logelog
(x)e (x)
25.)24.)
ex e
loge (x)e
√
√
7.)
18.)
√ √
√
2
21)
3−x
2+
3+ +
21)
31.)
4(x
32.)
+
31.)
4(x
2−121
32.)
1221− 1
2+
x
39.)
4(x
+
2)
+
40.)
−
x+
31.)
4(x
+
1)
+
2
32.)
−x
+
1+2)
25.)
2x
−
3
23.)
−2
x
−
1
25.)
2x
−
3+
x−3(x
36.)
−3(x
+
2)
39.)
fe−x
(x)
36.)
24.)
25.)
e
7.)
18.)
18.)
7.)
18.)
___________________________
___________________________
___________________________
______________
______________
___________________________
___________________________
___________________________
___________________________
______________
8.)
8.)
19.)
8.)
x ex
24.)
x 25.) e
27.)26.)
ex −e 1 − 1
19.)
___________________________
19.)
19.)
25.)
log (x)
26.)
log
e (x)e
27.)
log
(x
+
2)
e
28.)
log
(x
+
2)
9.)
e
9.)
20.)
10.)
10.)
x ex − 1
26.)
27.) e − 1
10.)
21.)
______________
___________________________
20.)
___________________________
9.)
______________
25.)
26.)39.)
logfel((
___________________________
___________________________
___________________________
______________
______________
______________
20.)
20.)
21.)
21.)
27.)
log
(x
e
28.) loge (x + 2)+ 2)
21.)
___________________________
___________________________
___________________________
11.)x
22.)
x
26.)
e
−
1
27.)
e (x
27.) e −28.)
1 27.) log
28.) logelog
(x +
2)+ 2)
1
(x2)+ 2)
e*22.)
27.)26.)
ex −e 1 −11.)
e (x +
* * ** ** log
* *****
* * *** *** *
*** *** ** ** * ***** ** ** ** ** ** ******
11.)
22.)
22.)
√
33.)
p(x)
34.)
r(x)
33.)
p(x)
34.)
r(x)
x
41.)
p(x)
42.)
33.)
p(x)
34.)
r(x)
xp(x)
2
26.)
er(x)
27.)
l
37.)
p(x)
40.)
g(x)
37.)
40.)
27.)
e
−
1 −+1**
logg(
26.)
−3(x
+
2)
−
1
24.)
−x
−
1
26.)
−3(x
2)2**−**1** ** ******
23.)
p(x)
24.) *rQc)
e(
* ***
****
****
* *****
* * * *******
***
** ** ** *****
* * * * * 28.)
* ** *****
*** * *** ************************************************************************
** ** * *****
** ** ******
* * ******
* * **
*****
23.) p(x)24.)
24.) r(x) 24.) rQc)
23.)rQc)
p(x)
x
x ex − 1
26.)
27.)
e
−1
28.)
f
(x)
29.) f (x)
-~
I
I
-3
-l
I
I
a
-,
27.) loge (x2)+ 2)I
30.)29.)
g(x)g(x)28.) loge (x +
-3
I
I
I
I
a
i
~
1
I,
-~
1
I
I
-3
-l
I
I
a
-,
1
I
-l
-~
I
i
I
a
I
I
a
-,
I
~
i
I,
1
29.)28.)
f (x)f (x)
30.)29.)
g(x)g(x)
f (x)29.) g(x)
29.)28.)
f (x)30.)
30.)29.)
g(x)g(x)
29.)28.)
f (x)f (x)
g(x)
√
35.)
h(x)
36.)
m(x)
43.)
h(x)
44.)
35.)
h(x)
36.)
m(x)
35.)
h(x)
36.)
m(x)
3
28.)
fm(x)
(x)
23.)
−2
12 −1
25.)
2x
−
3 2)2 − 1
38.)
38.)
q(x)
29.)
f−3(x
(x)
25.)
2xq(x)
−x +
3−2)
36.)
+
36.)
−3(x
39.)
f (x)
29.)28.)
f (x)f (x)
2
2
2
√
37.) −3(x
p(x)
24.)
−x −
26.)
+12)2 − 1
29.)
30.)
g(x)
39.)
f (xg
30.)29.)
g(x)g(x)
2
2
31.)
+2 2− 1
25.)4(x
2x +
− 1)
3 2)
36.)
−3(x
+
−1
I
a
2
√
32.)
39.) f−x
(x)+ 1
2
2
2
14
11 11
2 11
12
2
37.)
p(x)
40.)−3(x
g(x)+ 2)2 − 1
26.)
2
2 12 2
40.) g(x