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MATH 112 Class Notes
Department of Mathematics
The University of Arizona
Spring 2014
Table of Contents
Table of Contents ...................................................................................................................... 1
Syllabus Highlights .................................................................................................................... 3
Quick Reference Guide ............................................................................................................. 4
Strategies for Checking Answers............................................................................................... 5
CHAPTER 1 ................................................................................................................................ 6
Section 1.1 ............................................................................................................................ 6
Section 1.2 ............................................................................................................................ 9
Section 1.4 .......................................................................................................................... 15
Section 1.5 .......................................................................................................................... 20
CHAPTER 2 .............................................................................................................................. 25
Section 2.3 .......................................................................................................................... 25
Section 2.4 .......................................................................................................................... 29
CHAPTER 3 .............................................................................................................................. 32
Section 3.1 .......................................................................................................................... 32
Section 3.2 .......................................................................................................................... 40
Section 3.3 .......................................................................................................................... 49
Section 3.4 .......................................................................................................................... 55
Section 3.5 .......................................................................................................................... 66
Section 3.6 .......................................................................................................................... 72
CHAPTER 4 .............................................................................................................................. 79
Section 4.1 .......................................................................................................................... 79
Section 4.2 .......................................................................................................................... 85
Section 4.3 .......................................................................................................................... 90
Section 4.4 ........................................................................................................................ 101
Section 4.6 ........................................................................................................................ 107
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CHAPTER 5 ............................................................................................................................ 117
Section 5.1 ........................................................................................................................ 117
Section 5.2 ........................................................................................................................ 128
Section 5.3 ........................................................................................................................ 134
Section 5.4 ........................................................................................................................ 142
Section 5.5 ........................................................................................................................ 147
Section 5.6 ........................................................................................................................ 155
Appendix – Quadratic Formula Program for Graphing Calculators ...................................... 162
Quadratic Formula - TI 82 ................................................................................................. 162
Quadratic Formula - TI 83, TI 83 Plus, and TI 84 Plus........................................................ 163
Quadratic Formula - TI 85, TI 86 ....................................................................................... 164
Quadratic Formula – Casio models ................................................................................... 165
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Syllabus Highlights
Welcome to Math 112! The following highlights some of the most important parts of the full
syllabus. Students are still expected to know the contents of the full syllabus which is posted in
D2L under content.
Required Materials



MyMathLab access
Math 112 Class Notes
Graphing Calculator
In-Person Class Meetings
An important and mandatory part of the course, the in-person class meetings allow students
time to interact with their instructor and classmates. Written assignments will be collected at
class and constitute a significant portion of the course grade.
Online Work
The book, homework, and online quizzes are all hosted on mymathlab.com (MML). Enrollment
in the mymathlab.com course must be completed by the end of the first day of classes,
Wednesday, January 15. Failure to enroll by Friday, January 17 will result in administrative
drop.
Exams
There will be two midterm exam (each covering two chapters), a Chapter 5 test, and a
cumulative final exam.
Course Grade Break Down





Homework:
MML Tests:
Midterms:
Ch 5 Test:
Final Exam
120 points
100 points
300 points (150 points each)
80 points
200 points
Getting Help


Office hours and appointments by email
Tutoring at ThinkTank
Actions That May Result in an Administrative Drop




Failing to sign up for MML by January 17
Failing to come to class the first day
Missing more than 3 classes
Missing more than 5 assignments
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Quick Reference Guide
Math Department Information
Math Dept Home
Math Dept Tutoring
Math 112 Homepage
Math Common Final Exam Schedule
http://math.arizona.edu
http://math.arizona.edu/academics/tutoring
http://math.arizona.edu/~algebra
http://math.arizona.edu/academics/courseinfo/common
University Information
U of A Home
Final Exam schedule
Important Dates and Deadlines
U of A Computing Homepage
24/7 Computing Support
UAccess
D2L
http://www.arizona.edu
http://registrar.arizona.edu/schedules/finals.htm
http://www.em.arizona.edu/datesdeadlines/DatesDeadlines.aspx
http://uits.arizona.edu/
http://uits.arizona.edu/departments/the247
http://uaccess.arizona.edu/
http://d2l.arizona.edu
Important University Policies
Code of Academic Integrity
http://deanofstudents.arizona.edu/policiesandcodes/codeofacademicintegrity
Student Code of Conduct
http://deanofstudents.arizona.edu/policiesandcodes/studentcodeofconduct
Services for Students
University Tutoring Services
Disability Resource Center
Campus Health
http://thinktank.arizona.edu/
http://drc.arizona.edu
http://www.health.arizona.edu/webfiles/main.htm
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Strategies for Checking Answers
One of the components of the written work in the course involves checking your answers.
There are many ways to check your answer to a problem. These are some strategies you can
try:
1. Use the answer you found and substitute it back into the original equation. When you
are asked to solve an equation, this is the simplest and most straight-forward way to
check your results.
2. When you are solving a word problem, check to see whether your solution yields the
desired outcome. For example, if you are solving for the dimensions of a rectangle that
give a certain area, use the length and width you found in your solution to calculate the
area, and see if you get what you expect.
3. Solve the problem using a different technique or algorithm. For example, if you solved a
quadratic equation by factoring, you could check it by using the quadratic formula.
4. Use a different approach to solve the problem. For example, if you solved a problem
strictly algebraically, you can try graphing on your calculator to check your solution.
(This can work the other way around as well!)
5. Check to see whether your answer makes sense. For example, if you are solving a
problem to find the speed of a car driving on the highway, 350 mph is not a reasonable
solution.
6. To take the previous strategy one step farther, you can refine what kind of answer is
reasonable. For example, if the problem states that a person is given one 100 mg
dosage of a drug, and you need to calculate the amount of the drug left in the person’s
system after a certain amount of time, then your answer must be greater than or equal
to 0 mg, and less than or equal to 100 mg. So if you get an answer of 120 mg or -5 mg,
you know you’ve done something incorrectly.
7. Build a concrete example of the more abstract question. For example, if you are asked
how the graph of
is transformed from the graph of
, you can pick
a specific function, say
the graph of
√ , and take a look at the graph of
√ compared to
√ .
Note: Sometimes students try to check by re-doing the problem in the same way they
approached it the first time. If you made a mistake the first time, it is very likely to make the
same mistake the second time. This is not recommended as a way to check your work (though it
may help in a pinch on an exam).
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CHAPTER 1
Section 1.1
Objectives
• Determine if a given equation is linear.
• Solve linear equations of all types.
• Solve equations that may be rewritten to be linear.
Preliminaries
A linear equation in one variable is an equation that can be written in the form
where and are constants and
.
Warm-up
Solve the equation for :
1.
2.
Class Notes and Examples
How can you tell whether an equation is linear?
1.1.1
Determine if the given equation is linear or non-linear.
(A)
(B) √
(C)
(D)
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In general, how do you solve a linear equation?
1.1.2
Solve each equation. Check your solution(s).
(A)
(B)
(C)
(D)
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1.1.3
Determine the values of x and y that solve the system of equations.
1.1.4
Determine the values of A and B that solve the system of equations.
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Section 1.2
Objectives
• Write mathematical statements given a verbal description.
• Set up a linear equation from a verbal description and solve.
• Set up and solve mixing problems.
• Set up and solve problems involving distance, rate, and time.
• Set up and solve shared work problems.
Preliminaries
What is the equation involving distance, rate and time?
How much pure salt is in 5 gallons of 20% salt solution?
Warm-up
Rewrite each description as a mathematical statement:
1. 3 more than twice a number.
2. The sum of a number and 16 is three times the number.
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Class Notes and Examples
What strategies should you use for solving word problems?
1. Read the problem carefully. Identify what you are trying to find or solve for.
2. Draw a picture or diagram if applicable.
3. Define variables.
4. Write down known relationships.
5. Use all known information to find equations to solve.
6. Solve. Check solutions. Does your answer make sense? Can you check whether it is
correct?
7. Describe solution in the context of the problem. Be sure to include units.
For each question below, solve the problem algebraically. Clearly define all variables used. The
columns are set up so you can check your solution on the right-hand side.
1.2.1
The sum of three consecutive integers is 78. What is the smallest of the three integers?
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1.2.2
Tina has $6.30 in nickels and quarters in her coin purse. She has a total of 54 coins. How
many of each coin does she have?
1.2.3
A company produces a pair of skates for $43.53 and sells them for $89.95. If the fixed
costs are $742.72, how many pairs must the company produce and sell in order to break
even?
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What strategies can you use for solving mixture problems?
1.2.4
A premium mix of nuts costs $12.99 per pound, while almonds cost $6.99 per pound. A
shop owner adds almonds into the premium mix to get 90 pounds of nuts that costs
$10.99 per pound. How many pounds of almonds did she add?
1.2.5
A chemist wants to strengthen her 40L stock of 10% solution of acid to 20%. How much
24% solution does she have to add to the 40L of 10% solution in order to obtain a
mixture that is 20% acid?
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What strategies can you use for solving distance-rate-time problems?
1.2.6
A boat travels down a river with a current. Travelling with the current, a trip of 66 miles
takes 3 hours while the return trip travelling against the current takes 4 hours. How fast
is the current?
1.2.7
Two motorcycles travel toward each other from Chicago and Indianapolis (about 350km
apart). One is travelling 110 km/hr, the other 90 km/hr. If they started at the same
time, when will they meet?
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What strategies can you use for solving work time alone/together problems?
1.2.8
Suppose it takes Mike 3 hours to grade one set of homework and it takes Jenny 2 hours
to grade one set of homework. If they grade together, how long will it take to complete
one set of homework?
1.2.9
Suppose a journeyman and apprentice carpenter are working on making cabinets. The
journeyman is twice as fast as his apprentice. If they complete one cabinet in 14 hours,
how many hours does it take for the journeyman working alone to make one cabinet?
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Section 1.4
Objectives
• Use the Zero Product Property.
• Solve a quadratic equation by factoring.
• Solve a quadratic equation by completing the square.
• Solve a quadratic equation by using the quadratic formula.
• Use the sign of the discriminant to determine the number of solutions for a quadratic
equation.
Preliminaries
A quadratic equation is an equation that is of the form
.
The zero product property says if
, then either
, where
or
.
Warm-up
1. Rewrite
in the form
.
3.
(A) Factor
.
(B) Solve
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Class Notes and Examples
In general, how do you solve a quadratic equation?
Strategy 1: Determine the nature of solutions by using the discriminant
Strategy 2: Factor
Strategy 3: Complete the square
Strategy 4: Use the quadratic formula
1.4.1
Calculate the discriminant for
. How many solutions does the equation
have? Solve by factoring.
What is the method of completing the square? How do you complete the square?
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1.4.2
Solve
1.4.3
Solve
by completing the square. Check your solution by factoring.
by completing the square.
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1.4.4
Solve
1.4.5
Solve
by completing the square.
. Check your solution by using a different strategy.
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1.4.5
Solve
.
1.4.6
Solve
.
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Section 1.5
Objectives
• Use the four-step problem solving strategy to solve application problems involving
quadratic equations.
• Solve quadratic application problems involving unknown numerical quantities.
• Solve quadratic application problems involving geometric shapes.
• Solve quadratic application problems involving projectile motion.
• Solve quadratic application problems involving distance, rate, and time.
• Solve quadratic application problems involving two people working together.
Preliminaries
The four-step problem solving strategy can be quickly summarized by the following:
1.
2.
3.
4.
Warm-up
Determine a mathematical expression that relays the information described in the following
problems. Remember to define your variables.
1. A car is traveling M mph for H hours. Represent the number of miles traveled.
2. The speed of a boat in still water is S. The current of the water is C. Represent the
boat’s speed going upstream.
3. An alloy contains 40% gold. Represent the number of grams of gold present in G grams
of the alloy.
4. An item originally priced at P dollars will be discounted 15%. Represent the new price.
5. The width of a rectangle is half of its length. Express the perimeter of the rectangle in
terms of width only.
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Class Notes and Examples
For each question below, solve the problem algebraically. Clearly define all variables used.
Check your solution. You may want to refer to the strategies listed in the front of these notes.
The columns are set up so you can check your solution on the right-hand side.
1.5.1
The sum of the square of a number and the square of 7 more than the number is 169.
What is/are the number(s)?
1.5.2
A stone thrown downward from a height of 274.4 meters. The distance it travels in t
seconds is given by the function
. How long will it take the stone to
hit the ground?
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1.5.3
Amy travels 450 miles in her car at a certain speed. If the car had gone 15 mph faster,
the trip would have taken 1 hour less. Determine the speed of Amy’s car.
1.5.4
The area of a square is numerically 60 more than the perimeter. Determine the length
of the side of the square.
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1.5.5
The length of a rectangle is 7 centimeters longer than the width. If the diagonal of the
rectangle is 17 centimeters, determine the length and width.
1.5.6
It takes Julia 16 minutes longer to chop vegetables for soup than it takes Bob. Working
together, they are able chop the vegetables in 15 minutes. How long will it take each of
them if they work by themselves?
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1.5.7
Maria traveled upstream along a river in a boat a distance of 39 miles and then came
right back. If the speed of the current was 1.3 mph and the total trip took 16 hours,
determine the speed of the boat relative to the water.
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