Math 1050
Review
Final Exam
Answer all questions below. Every question is worth the same number of
points. You may use the available space for work, but must write ALL
answers on the answer sheet. No cell phones, calculators, or notes are
allowed during the exam. These rules will be strictly enforced.
Sets and Numbers
1. Consider the sets R, N, Q, 1 What is the smallest of these sets that contains
2. Is it true that [0,17)
c
0
(0,17]?
3. Isy’~Q?
Sequences, Sums and Series
4. Is the sequence 4,8,12,16,... arithmetic, geometric, or neither?
5. Is the sequence 3
—~
~
2’4’
—~
8
arithmetic, geometric, or neither?
6. Find the 54th term in the arithmetic sequence 1,6,11,16,...
4
7. Find the sum
Z(~2 —3).
~
264
~
is
ii
500
8. FindthesumZ4.
#/5bo)r?O0~
21
9. Find the sum of the first 101 terms of the sequence 5,7,9,11,13,...
Ibo
—
10. Find the value of the series 100+25+~ + ~
+
16
(33,3
2
~O
tot \
ç~
= Io~os
/
—
•‘‘
Counting
i~.q
11. You are designing a house foyourself to live in. You can choose the house to be made
out of
d brick or metal. The roof can be wood shin les, as halt shin I
tin.3
~ You
can paint the house rown, red, yellow, or greeni You can choose to have tw
e
four, or five bedrooms. How m~7~assthilities are there for the design of your house?
12. You are in an
sundae with a
third different
in the middle,
3~L
ice cream shop. There are 31 flavors of ice cream. You are building a
scoop of ice cream on the bottom, a different flavor on top of that, and a
flavor on top of that. If it matters to you which flavor is on the bottom,
and on the top, then how many different sundaes can you make?
13. You are on a boat with 200 people. The boat is sinking and there is only one life raft
with 8 seats. How many different options are there for which 8 people to put in the raft?
(~to-%~
V
14. Your baseball team is choosing a batting order. There are 27 players on the team, and
you must determine an order in which the players will get to bat. How many different
batting orders are possible? ~~7- I
8
15. Write out (x — 2y)5 so that your answer doesn’t include any numbers that look like (“)
K~_lo.~q’~ ~ ~
k~R~4 —sec4~
k
16. Write
(~) as a natural number in stan’dard form.
Functions & Graphs
17. What is the implied domain of f (z)
~
18. What is the implied domain of g(x)
=
log5(6
19. What is the implied domain of
=
~/3z
h(x)
—
—
2x)?
15?
N25
20. Below is the graph of a function. Answer the following questions:
(a) What is f(2)?
(b) What is the domain of
f?
(c) What is the range off?
(d) What are the x-intercepts of the graph?
(e) What are the y-intercepts of the graph?
\I I3 S
3
—
21. Is the following picture the graph of a function?
Page 2
22. Below is the graph of an invertible function h(x). Graph the inverse function, Ii ‘(x).
23. Below is the graph of a function f(x). Is it invertible?
—
I .0
24. Find the inverse of g(x)
25. If f(x)
jdo
2.3
=
&x2 + 1 and g(x)
3ccj
~.
z+2 find fog(x).
26. Is it sometimes, always, or never true that
f o g(z)
Page 3
=
go f(x)?
S~ie1e4~Mtâ
Polynomials
27. Find a root of the polynomial 2x3 + 3z2
28. Completely factor 2x3
—
3z
—
—
j.,M.i
2.
8z2 + 2z + 12 given that —1 is a root.
29. Find the quotient: 5x3
~—
+7
—
I
A (~3) (~ ~ (~ i)
÷
30. What is the leading term of the polynomial 12x(2x2
31. True or false: the polynomial 13x7 + 11x5
KC
—
3x + 11)(—x7 + 4x5
—
2z +
6x2 + 1 has 13 roots.
32. Complete the square: Write —2x2 + 4x + 1 in the form a(x + ~)2 + ~ where a, fi, 7 E It
33. Completely factor —2x2 + 4x + 1.
-
ffv’) (‘s
34. How many roots does —z2 + 3x —5 have?
—
4,
i!~1)
~J~e
—~5— C)
+
Exponentials and Logarithms
(X)
35. Rewrite log4(x) using logarithms with base 10. ~
Lj ij
36. Rewrite log4(8) as a rational number. (Hint: Use the change of base formula with an
appropriate base.)
j 5
—
‘3
Log (~)
37. What is the greatest integer less than 1og10(9999)?
38. Rewrite
3~2
=
17 as a logarithmic equation.
39. Rewrite 1og10(x2 + 7x + 1)
=
~
-
4 as an exponential equation. ~b
q
~
40. You owe the bank $1000. If the bank charges 12% annual interest, how much will you
owe after 3 years?
\O~o i. 4
(
Solving Equations: Solve for x.
41. (6—x)3+5=32
42. log5(x)
X
=3
x = 5%
3
43. log5(x+1)+log5(2z)=31og5(2)+1
Xz 1t~.z (s2) =
44. (Wz32
45• 8’°~~
4
46. Jx —5
7
x—
(—s,
~9,
—2.5
i7~
Linear Algebra
Page 4
—5~
47. Solve the system of equations:
—
2z+3y=8
48. Is z
=
1, y
=
2, z
3 a. solution to the following system of equations?
Iz + y + z
=
2x + 4y z
—z=0
—
49. Find the inverseof
6
=
7
( ~ ~). ~ ( ~
50. Compute the following matrices:
/1 3 4N/1\
(a)
7
21
2
~
ILl
(U~) (~ ~)
~ /47 3\
(c)3(2 —21 )+(o —18)
4 1,1
\0 1 1)
51. Calculate det
~
(/
\
?
~_J~l1
‘t
)].
/
52. Given that
~
(:1
solve the system of equations
I~
y 3z 5
—3x 4y + lOz = 0
2x + 2y — 7z = —1
—
Graphing
53. Graph the following functions.
Page 5
~
1(1
(a) icl(z)
(b) ~
(c)~
(ci) x2
(e) x~
(f) f(z)
=
5
(g)~
(h) ~
(i)
(1)z
(j) log3(x)
54. Graph the following functions and label all z- and y-intercepts, vertical and horizontal
asymptotes (if there are any).
(a) f(z)
=
-2~x + 4
12z_1 ifxe(—oo,—1]
(b) g(z)
=
I
—3
if z€ (—1,3]
(x~4)~
if x C (3, oo)
(c) 1i~~
=
log3(x + 1) —2
(d) p(z)
=
-3(z- 1)(z+2)(x+2)(z+~)
e~ r(zNI
—
—2(z
4)(z + 2)(x2 + 6)
3(z—1)(x2+z+2)
55. Graph p(x)
—
=
—2(x + 1)2 + 2 and label the vertex.
56. What is the slope of the straight line that passes through the points (—1,3) and (7,5)?
Probability
I
57. If a bag has 17 red marbles, 12 blue marbles, and 3 yellow marbles.
probability of selecting a yellow marble at random?
I
Page 6
What is the
5-41
~a5
b)
c)
__
—i
S
4
4!)
& ~
(
(
‘7
54)