Download Math Analysis

Math Analysis
Midterm Review Packet
1)
2)
3)
4)
5)
6)
Name__________________________
Date____________Period______
How to use this review packet:
Attempt to do this entire review packet without using the book.
Put an asterisk next to the problems which you are unable to answer
Research the information which you have not memorized
Try again. Put a star next to those problems you are still unable to do
Ask the teacher for the answers to any problem you are unable to do
Study and memorize; especially those problems which have asterisks or stars next to them
Answer each of the following as thoroughly as possible:
1. Use inductive reasoning to determine the next three numbers in the following patterns:
a. 5, 7, 10, 14, 19, . . .
b. 1, 1, 2, 3, 5, 8, 13, . . .
2. Pick a number, multiply the number by 3, add 12, divide by 3, and subtract 4. What did you get? Use
deductive reasoning to prove that you always get a number with the same relationship to the original
number as the one specific example you did.
3. Can you differentiate between?:
a. deductive and inductive reasoning
b. roster form and set-builder notation
c. is an element of, is a subset of, is a proper subset of , ,  
d. conjunction and disjunction
e. conditional, converse, inverse, and contrapositive statements
f. a finite and an infinite set
g. conditional and biconditional statements
Math Analysis Midterm Review packet (12/11/08)
page 1 of 18
4. Can you define each of the following words or phrases?:
a. counterexample
b. cardinal number
c. element
d. logically equivalent
e. natural numbers
f. integers
g. rational numbers
h. irrational numbers
i.
union of sets
j.
intersection of sets
k. complement of a set
l.
tautology
m. self contradiction
n. implication
o. closed set
p. DeMorgan’s Laws
q. divisibility rules
r. arithmetic sequence
s. geometric sequence
t. Fibonacci sequence
u. % of Change Formula
Math Analysis Midterm Review packet (12/11/08)
page 2 of 18
5. Estimate each of the following. Show your work:
a. A six pack of cola costs $3.45. A carton of 4 six packs costs $12.60. How much will be saved by
purchasing the carton rather than 4 individual six packs?
b. What is the cost of eight pairs of socks if each pair costs $1.09?
c. What is 6% sales tax on an item that costs $248?
6. List all of the subsets of the following:
should be?
**do you remember how to determine how many subsets there
a. A={5, 6, 7, 8}
b. B= x | x  N and x  4
7. Which of the subsets in #6 are not proper subsets?
8. Determine the intersection of the following pairs of sets:
a. Sets A and B in #6:
b. C = x | x  N and x  15
D = {2, 4, 6, 8}
c. E=the set of integers between 5 and 10
d. G = x | x  5
F = the set of rational numbers less than 20
H = {the whole numbers}
Math Analysis Midterm Review packet (12/11/08)
page 3 of 18
9. Determine the union of the following pairs of sets:
a. I = set of integers between 5 and 10
b. K = {3, 5, 7, 9, 11}
c.

J = set of rational numbers between 5 and 10, inclusive
L = {2, 4, 6, 8, 10}
M = {4, 5, 6, 7}
10. What is the cardinal number of:
a. Set L in problem #9?
b.  ?
c. Set C in problem #8?
11. Determine whether the following are true or false:
a.
6 2, 4, 6, 8
b. 6 2, 4, 6, 8
c.
6  2, 4, 6, 8
d.
2, 4, 6, 8  2, 4, 6, 8
e.
2, 4, 6, 8  2, 4, 6, 8
12. Construct a Venn diagram depicting the response to the following survey:
500 people at a shopping mall were asked to indicate which sporting evens they enjoyed watching on
television. The responses were as follows:
F
B
224 enjoyed watching football
196 enjoyed watching baseball
73 enjoyed watching hockey
155 enjoyed watching football and baseball
59 enjoyed watching football and hockey
27 enjoyed watching baseball and hockey
23 enjoyed watching all three sports
a.
b.
c.
d.
H
How many people enjoyed watching either football or hockey?
How many people enjoyed watching exactly 2 sports?
How many people enjoyed watching at least one sport?
How many people enjoyed watching baseball and hockey, but not football?
Math Analysis Midterm Review packet (12/11/08)
page 4 of 18
13. Determine if the following statements are logically equivalent by producing truth tables, or using
DeMorgan’s Laws:
Statement 1: The temperature is not over 80 or the air conditioner will come on.
Statement 2: It is false that the temperature is over 80 and the air conditioner will not come on.
p
T
T
F
F
q
T
F
T
F
p
T
T
F
F
q
T
F
T
F
14. Let p represent the statement, "Heather likes CSI", and let q represent "Joan watches Jeopardy". Convert
the compound statements into symbols.
Statement: It is false that Heather likes CSI and Joan doesn’t watch Jeopardy.
15.Given p is true, q is true, and r is false, find the truth value of the statement. (You do not need to do a
table for this!)
Statement:
q  r    p  r 
16. Determine whether each of the following is sometimes, always, or never true:
a. The product of two irrational numbers is an irrational number.
b. The sum of two irrational numbers is irrational.
c. The difference of two natural numbers is a natural number.
d. The quotient of two rational numbers is a rational number.
17. Determine whether the following are true or false:
a. The set of natural numbers is closed over addition.
b. The set of irrational numbers is closed over multiplication.
c. The set of integers is closed over division.
18. Determine the following using the numbers 4440 and 432:
a. Greatest Common Factor
b. Least Common Multiple
Math Analysis Midterm Review packet (12/11/08)
page 5 of 18
19. Determine whether each of the following sequences are geometric, arithmetic, or neither. Then, find “d”
or “r” for each of the geometric and arithmetic sequences.
a. 8, 4, 2, 1, .5, . . .
b. 3, 6, 9, 12, 15, . . .
c. 4, 16, 64, 256, . . .
d. 3, 0, -3, -6, . . .
e. 15, -10, 3.333, -2.222, . . .
20. Find the requested term, given the following information:
a. Find a 10 if a 1  6 and d  
b. Find a 5 if a 1  6 and r  
1
2
1
2
21. Find the sum of the arithmetic sequence 3, 6, 9, 12, . . .30 n = 10
22. Find the sum of the first n terms of the geometric sequence if:
n = 5 a1  2 r = 3
23. Find the sum of the first fifty positive integers.
24. Simplify the following:
a. 2 3  4 3
b. 4 5 2 15
c.

3 2 4 6 5

Math Analysis Midterm Review packet (12/11/08)
page 6 of 18
25. Convert the following numbers into the given mod system:
a. 7  ? (mod 3)
b. 18  ? (mod 4)
c. -13  ? (mod 4)
26. Perform the modular arithmetic operation.
a. 34 + 16  ? (mod 7)
b. 32 – 43  ? (mod 3)
27. Find all replacements (less than the modulus) for the question mark that make the statement true. If
there are no numbers that satisfy the ?, then say “No solution”.
7 ?  5(mod 6)
? 4  3(mod 5)
28. Convert to a logarithmic equation:
a. 105 = 100,000
b. 161/4 = 2
c. e 2  7.389
29. Convert to exponential equations.
a. log66 = 1
b. log 0.1 = -1
c.
ln4  1.386
Math Analysis Midterm Review packet (12/11/08)
page 7 of 18
30. Solve for the variable in the following equations.
a. log3x = 2
b. log4x = 3
c. logx16 = 2
d. log2x = -1
e. log8x = 1/3
f. log264 = x
g. log464 = x
h. log 100 = x
i.
log420 = x
j.
ln 7 = x
31. Find the value of x 3  7 , when x = -2
32. Find the value of 3x 2  5x  2 , when x = -4
33. Find the value of 5x 2  6y , when x = 3 and y = -7
For 34—35, combine like terms:
34. 5c - 8c +4
35. 9  x  4   3  11  2x 
Math Analysis Midterm Review packet (12/11/08)
page 8 of 18
Solve the following equations. Show your work to get full credit!:
36. 6t  5  12  13t
37.
2x  1 x  4

7
6
1
38. The directions on the box for making Spooky Snacks calls for 4 cups of rice crispies to make 14 servings.
3
How much rice crispies should I use to make 5 servings?
2
39. If V  2 R 2 r 2 , find V when R = 4 and r = 8 .
5
Solve for the variable indicated:
40. A=4F-Q , for F
41.
1
a

1
b

1
c
, for c
Write the phrase in mathematical terms:
42.
7 decreased by 4 times a number
43.
11 more than the product of a number and 7
Write an equation that can be used to solve the problem. Show your work and solve the problem:
44.
Three times the sum of a number and 6 is 34
Math Analysis Midterm Review packet (12/11/08)
page 9 of 18
45.
6 less than 3 times a number is 2 times the sum of the number and 9
46.
A town recycles 35 tons of newspaper and cardboard each week. The amount of newspaper is four
times the amount of cardboard. Determine the number of tons of newspaper and the number of
tons of cardboard recycled each week. (Make sure you label your answers as to what item the
tonnage represents.)
47.
Mr. Thomas worked a 52 hour week last week. He is paid 1 ½ times his regular hourly rate for all
hours over a 40 hour week. His pay last week was $548.10. What is his hourly rate?
Write the variation equation, and find the quantity indicated:
48.
y varies inversely as the square of x. Find y when x = -5 and k = 210.
49.
y varies jointly as x and z. If y is 3630 when x = 110 and z = 12, find y when x = 27 and z = 32.
50.
The gravitational force of attraction, F, in newtons, between an object and the Earth is
directly proportional to the mass, m, of the object in kilograms. If the force of attraction is
448 newtons when the object’s mass is 14 kg, find the force of attraction when the object’s
mass is 11 kg.
Math Analysis Midterm Review packet (12/11/08)
page 10 of 18
Solve the inequality, then graph the solution on the number line.
51. 4a  1  2
52. 12n  12  15n  6
53. 3  3y  13   12 y  12
54. 14  4y  2  2
55. Plot the points given by the pairs.
A(6, 2), B(2, -3)
56. Give the coordinates of the points shown on
the graph.
G ( , ), H( , )














G















        












        
















H




1
58. Graph the equation: y  x  2
3


57. Graph the equation: x = -8
























        













        
















59. Find the x and y-intercepts, and then
graph the equation: y  21 x  4
x-int:______________ y-int:____________

Math Analysis Midterm Review packet (12/11/08)











60. Graph the linear inequality: 4x  y  2

page 11 of 18

















        












        



















61. Graph the linear inequality: 3x  4y  12



















        




















62. Factor completely: x 2  2x  15
Find all solutions by factoring:
64. 15x 2  16x  4  0
63. Factor completely: 9x 2  12x  4 
65. x 2  8x  0
66. Find the slope of the line through the points: (4, 5) and (9, -5)
Math Analysis Midterm Review packet (12/11/08)
page 12 of 18
67. Find the slope of the line through the points: (9, 8) and (9, -1)
68. Find the equation from the graph:
8
7
6
5
4
3
2
1
-8 -7 -6 -5 -4 -3 -2 -1-1
1
2
3
4
5
6
7
8
9
-2
-3
-4
-5
-6
-7
-8
69. Kafka Inc. imports hard drives and sells them on the internet. The profit is given by the equation
P  145n  8000 where n is the number of hard drives sold. Graph P  145n  8000 for n<150. How many
hard drives must be sold for the company to break even? Round your answer to the units place, in other
words, to the nearest number of hard drives.
8000
6000
P
4000
2000
n
0
-2000 0
-4000
-50
50
100
150
-6000
-8000
-10000
-12000
Solve the following problems by using the quadratic formula: x 
70. 3x  6x  2  0
b 
b  4ac
2
2a
71. 2 x  5x  3
2
2
Determine whether each relation is a function:
72.
7
6
5
4
3 (12/11/08)
Math Analysis Midterm Review packet
2
1
73.
7
6
5
4
3
2
1
page 13 of 18
74.
 1, 1  ,  3, 9  ,  4, 3  ,  7, 4  ,  11, 4 
75.
 4, 8  ,  1, 5  ,  2, 4  ,  2, 1 
Find the given value of the function:
76. f (x )  2x  6, f  6   ?
77. f (x )  2x 2  5x  4, f (3)  ?
78. Find the vertex of the parabola given by the equation: y  2x 2  16x  34
Determine whether the graphs open up or down:
79. y  2x 2  20x  46
80. y  3x 2  16x  36
Graph the following functions, and find the range and domain:
81. y  x 2  3


10
9
8
Math Analysis Midterm Review
packet (12/11/08)
7
6
5
82. y   1 2 x 2  3 
10
9
8
7
6
5
page 14 of 18
















83. y  3x


















10
9
8
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1 2 3 4 5 6 7 8 9 10 11
84. Bob owns a watch repair shop. He has found that the cost of operating his shop is given by
C (x )  3x 2  150x  71 , where C is the cost and x is the number of watches repaired. How many watches
must he repair to have the lowest cost?
85. The population of a small country increases according to the function B  1300000e 0.02t , where t is
measured in years. How many people will the country have after 7 years? (Rounded to the nearest person.)
Math Analysis Midterm Review packet (12/11/08)
page 15 of 18
86. A car rental company has two rental rates. Rate 1 is $48 per day plus $0.16 per mile. Rate 2 is $96 per
day plus $0.08 per mile. If you plan to rent for one week, how many miles would you need to drive to pay
less by taking Rate 2?
A.
B.
C.
D.
more than 14,700 miles
more than 4200 miles
more than 8400 miles
more than 4300 miles
87. Indicate whether the following is a simple or compound statement. If it is compound, indicate whether it
is a negation, conjunction, disjunction, conditional, or biconditional by using both the word and its
appropriate symbol.
It is false that whales are fish and bats are birds.
88. Write a negation of the statement: Not all people like football.
89. Let p represent the statement “Jim plays football”, and let q represent “Michael plays basketball”.
Convert the compound statements into symbols.
a. Neither Jim plays football nor Michael plays basketball.
b. Jim plays football but Michael doesn’t play basketball.
c. Michael plays basketball if and only if Jim doesn’t play football.
90. Let p = The puppy is well-trained, q = The puppy behaves well, and r = His owners are happy. Write
the compound statements in words.
a. p  r
b. r  (q  p )
c.
q p
91. Select letters to represent the simple statements and write each statement symbolically by using
parentheses then indicate whether the statement is a negation, conjunction, disjunction, conditional, or
biconditional. If people drive small cars, then people will use less fuel and the ozone hole will not expand.
Math Analysis Midterm Review packet (12/11/08)
page 16 of 18
92. Construct a truth table for the following:
p q  
p
p
T
T
F
F
q
T
F
T
F
p
q  r 
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
MATH ANALYSIS
MIDTERM FORMULA SHEET
a n  a 1  (n  1)d
a n  a 1r n  1
sn 
n a 1  a n 
2
a 1 1  r n 
sn 
1r
Math Analysis Midterm Review packet (12/11/08)
page 17 of 18
loga b 
A P
P
log b
log a

r
1 
 n
nt
A
nt


r
1 
 n
I  PRT
p q  
p q
p q  
p q
x
b  b 2  4ac
2a
Math Analysis Midterm Review packet (12/11/08)
page 18 of 18