Lecture Notes in Math Analysismyweb.ttu.edu/ramirram/Lecture_Notes_for_Business_Calculus.pdfDe...

Lecture Notes in Math Analysis

Ramiro Ramirez c©2019

October 21, 2019

Contents

Contents i

1 Linear Equations and Functions 3

1.1 Solving Linear Equations and Inequalities in One Variable . . . . . . 3

1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Graphs and Graphing Utilities . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 20

1.6 Applications: Business & Economics . . . . . . . . . . . . . . . . . . 25

2 Quadratic and Other Special Functions 29

2.1 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Business Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Special Functions and Their Graphs . . . . . . . . . . . . . . . . . . . 36

2.5 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Matrices 41

3.1 Matrix Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Leontief Input-Output Models . . . . . . . . . . . . . . . . . . . . . . 51

i

ii CONTENTS

4 Inequalities and Linear Programming 55

4.1 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Exponential and Logarithmic Functions 57

5.1 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Equations and Applications . . . . . . . . . . . . . . . . . . . . . . . 57

6 Mathematics of Finance 59

6.1 Simple Interest & Sequences . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Compound Interest & Sequences . . . . . . . . . . . . . . . . . . . . . 61

6.3 Future Values of Annuities . . . . . . . . . . . . . . . . . . . . . . . . 61

6.4 Present Values of Annuities . . . . . . . . . . . . . . . . . . . . . . . 61

6.5 Loans & Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Introduction to Probability 63

8 Data Description 65

9 Limits and Continuity 67

9.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9.2 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

10 Differentiation 71

10.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

10.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

10.3 Significance of the Derivative . . . . . . . . . . . . . . . . . . . . . . 73

10.4 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

11 Applications of the Derivatives 75

12 Integration 77

CONTENTS 1

12.1 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

12.2 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . 77

13 Techniques of Integration 79

Bibliography 81

2 CONTENTS

Chapter 1

Linear Equations and Functions

The goal of this chapter is learn the basic tools we need to be able to solve problems inBusiness and Economics. We begin our study with linear functions and inequalities,so by the end of the chapter we will be able to apply our new found knowledge toanswer important questions in business.

1.1 Solving Linear Equations and Inequalities in

One Variable

Objectives

• Solve linear equations in one variable

• Solve applied problems by using linear equations

• Solve linear inequalities in one variable

Definition 1.1.1. An algebraic expression (quantity) is a mathematical phrasethat contains numbers, variables (like x or y) and operators (like add, subtract,multiply, and divide).

Example 1. 3x− 2, −4x− 1 and x+ y are all examples of algebraic expressions.

Definition 1.1.2. An equation is a statement that two quantities or algebraicexpressions are equal.

In this section we will deal with equation such as

3x− 2 = 7.

3

4 CHAPTER 1. LINEAR EQUATIONS AND FUNCTIONS

Such a equation is known as a equation in one variable. We will see later equationswith more than one variable, but for now we will deal with one variable equations.

Definition 1.1.3. The set of solutions of an equation is called the solution set ofthe equation.

Recall that set notation is denoted with curly brackets {}. Lets see an example.

Example 2. The solution to the equation x+ 1 = 0 is x = −1. So the solution setis {−1}.

So why introduce the notion of a solution set. Well it turns out that equationsdon’t always just have one solution, sometimes they can have multiple solutions,infinite solutions or no solution. Solution sets give a nice compact way of writingall solutions to an equation. Lets give a couple of more definitions.

Definition 1.1.4. Equations that are true for all values of the variable(s) are calledidentities.

Definition 1.1.5. Equations that are true for only some real values of the variable(s)are called conditional.

Example 3. Solve the following and give the solution set:

1. 2x+ x = 3x

2. 3x+ 1 = 3x

Now when we solve equations, we try to move over all terms and numbers that is notthe variable, to one side of the equation using addition, subtraction, multiplicationor division. When you are doing this, you are obtaining equivalent equations.

Definition 1.1.6. Two equations are equivalent if they have exactly the samesolution set.

Here is a list of some of the properties of equality:

• Substitution Property: Substituting one expression for an equal expressionin an equation yields an equivalent equation.

• Addition Property: Adding the same quantity to both sides of an equationyields an equivalent equation.

• Multiplication Property: Multiplying both sides of an equation by thesame nonzero quantity yields an equivalent equation.

1.1. SOLVING LINEAR EQUATIONS AND INEQUALITIES IN ONE VARIABLE5

It is important to keep these properties of equality in mind, because when solvingequations you might have to violate one of these properties, which in turn, will notyield a equivalent equation.

Definition 1.1.7. A linear equation is a equation containing one variable withfirst degree and the variable is not in the denominator.

Example 4. An example of a linear equation is the equation

2x

5− 7 =

x+ 2

4

Steps to Solve Linear Equations

1. If the equation contains fractions, multiply by the Least CommonDenominator (LCD)

2. Get rid of any parentheses

3. Get all terms with a variable on one side of the equation and allother terms on the other side

4. Divide both sides by the coefficient

5. Check solution in original equation

Example 5. Solve the following:

1. 5a7

= 15

2. 2x−44

= 9− 3x2

Lets give a definition for the next example.

Definition 1.1.8. Depreciation is the loss in value to a building over time due toage, wear and tear, and deterioration.

Example 6. (46 in textbook) When an $810,000 building is depreciated for taxpurposes (by the straight line method), its value, y, after x months of use is givenby y = 810000 − 2250x. How many months will it be before the building is fullydepreciated (that is the value is 0)? How many years is this?

Lets move on and start solving a different type of equation.


Definition 1.1.9. A Rational Equation is an equation that contains a variablein the denominator

The steps for solving these types of equations are exactly the same as the steps frosolving linear equations. The only difference now is that your LCD now is a variableexpression AND you must check your answer. Lets highlight solving these types ofequations with some examples.

Example 7. Solve the following:

1. xx−3 − 2 = 1

x−3

2. 13

+ 1x

= 12

3. 5x+10x+2

= 7

4. 2r−5 = 5

5r−25 + 1

Now lets add one more variable to the equations.

Definition 1.1.10. A Linear Equation in Two Variables is an equation withtwo variable of the first degree.

Again to ”solve” for one of the variables, you apply the exact sames steps that wedid before to the variable the question is asking for.

Example 8. Solve the following for y:

4x+ 2y = 6400

Example 9. (50 in textbook) Dish Systems manufactures satellite systems and hasits monthly profit P in dollars related to the number of satellite systems, x, by4P = 81x−29970. Find the number of systems that Dish Systems needs to produceand sell in order to break even.

Example 10. (52 in textbook) The total price of a used car (including 6% salestax) is $21,041. How much of this is tax?

Now we look at how to solve linear inequalities.

Definition 1.1.11. A linear inequality is a statement that one quantity is greaterthan or less than another quantity.

1.1. SOLVING LINEAR EQUATIONS AND INEQUALITIES IN ONE VARIABLE7

Every property we have used to solve equations (i.e Adding to both sides of anequation ect.) are used to solve inequalities, except for the multiplication/divisionproperty. When your presented a inequality where you must multiply or divide bya number, we must use the following properties:

• Multiplication/Division Property for Positive Numbers: Multiplyingeach side of an inequality by the same positive number yields an equivalentinequality.

• Multiplication/Division Property for Negative Numbers: Multiplyingeach side of an inequality by the same negative number and reversing theinequality sign yields an equivalent inequality.

Example 11. Solve the following linear inequalities:

1. 6− 4(6n+ 7) ≥ 122

2. 23k + 5

7< 4k − 25

7

So now we discuss how to actually represent the solutions set of the inequalities wejust solved above. Solution sets for an inequality contain multiple numbers and areoften graphed on a number line.

• <,> are graphed with open circles

• ≥, ≤ are graphed with closed circles

Graph of solution set for Ex. 11.1

Graph of solution set for Ex. 11.2


Example 12. (62 in textbook) Thrift rents a compact car for $33 per day, andGeneral rents a similar car for $20 per day plus an initial fee of $78. For how manydays would it be cheaper to rent from General? Graph the solution.

Example 13. (63 in textbook) Sean can spend at most $900 for a camera and somememory sticks. He plans to buy the camera for $695 and the memory sticks for$5.75 each. Write an equality that could by used to find the number of memorysticks (x) that he could buy. How many sticks could he buy?

1.2 Functions

Objectives

• Define Relations and Functions

• How to determine if a Relation is a Function

• Find Domain and Range of a Function

• Find Operations and Compositions of Functions

In this section we will get comfortable with the idea of a function. Functions areubiquitous through out mathematics (pure and applied), and therefore will also bepresent in business mathematics. Lets start of with defining what a functions is:

Definition 1.2.1. A set is a collection of objects, called elements, where the objectsare to be determined.

When we talked about the solution set of an equation in the last section, that initself was a example of a set. Recall that we use curly brackets {} to denote aset. We can also represent a set using set builder notation. For example, lets sayyou wanted to write the set of all positive even numbers, which would look like thefollowing

{2, 4, 6, 8, ...}

We can write this in set builder notation by the following:

{x : x is even and x > 0}

Remark 1. This is not important to this class, since we won’t implicitly use it,but recall the operations that you can perform on sets, namely the union ∪ and theintersection ∩.

1.2. FUNCTIONS 9

Now we define one more thing to be able to understand what a relation is on abetter level than how the book presents it:

Definition 1.2.2. Let A and B be sets. The Cartesian Product of A and B,written

A×B = {(a, b) : a ∈ A b ∈ B}

Now we can finally define a relation:

Definition 1.2.3. A relation R on a set A is a subset of A× A.

Don’t let this definition scare you, basically it is just a bunch of ordered pairs thatyou pick from a larger collection of ordered pairs using a rule that you pick. Therules can be anything, and in the case of this class those will be equations. Thebooks definition of this goes as follows:

• A Relation is defined by a set of ordered pairs or by a rule that determineshow the ordered pairs are found. Can also be defined by a table, graph,equation, or inequality

As you can see this definition without any context does not make any sense, whichis why I wanted to provide a little background before just defining it. Now lets lookat some examples:

Example 14. Some example of ralations:

1. {(0, 3), (2, 4), (−1, 3), (0, 6), (3,−2)}

2. y = 2x+ 3 (Why is this a relation?)

Definition 1.2.4. The Domain is the set of first components.

Definition 1.2.5. The Range is the set of the second components.

Definition 1.2.6. A function is a relation where each member of the domain mapsto exactly one member of the range. We denote a function as y = f(x) and say thaty is a function of x.

Example 15. Looking at Example 14.1, we see that the domain and range arerepresented as

D = {0, 2,−1, 0, 3}

R = {3, 4, 3, 6,−2}

Looking at the order pairs as mapping in a picture can be seen below:


−1

0

2

3

−2

3

4

6

Remark 2. The variable that represent the numbers in the domain is called theIndependent Variable and the variable that represents the numbers in the range iscalled the Dependent Variable.

Example 16. (1 and 2 from textbook)

x -7 -1 0 3 4.2 9 11 14 18 22y 0 0 1 5 9 11 35 22 22 60

1. Explain why the table defines y as a function of x.

2. If the table expresses y = f(x), find f(0) and f(11).

3. If the function defined by the table is denoted by f , so that y = f(x), is f(9)an input or an output of f?

4. Does the table describe x as a function of y? Explain.

Example 17. Determine if the equations represent y as a function of x.

1. y = 3x3

2. y = 6x2

Next we discuss a graphical way of discerning whether or not a graph is a function:

Theorem 1.2.1. (Vertical Line Test) If no vertical line exists that intersects thegraph at more than one point, then the graph is a function.

1.2. FUNCTIONS 11

Example 18. Determine whether each graph represents y as a function of x.

Again recall that a function is denoted using f(x). Here x is the input argument forthe function. In this class, the values functions will take will be real numbers. Nextwe will get comfortable manipulating functions:

Example 19. If f(x) = x2 + 2x+ 1, find f(2), f(−1), and f(a).

Example 20. The cost of producing x tools by a company is given by C(x) =1200x+ 5500.

1. Calculate C(100).

2. Interpret your answer from part 1.

Example 21. The table below shows the amount of money in an account that earns12% quarterly interest.

t(years) A(t)0 $2001 $225.102 $253.333 $285.15

1. What is A(3)? Explain its meaning.

2. If A(t) = 225.10, what is t?

Example 22. Let f(x) = x2 − x− 1. Find f(x+h)−f(x)h

. Assume h 6= 0.

Now we discuss finding the domain and range of a function, but first a few definitions:

Definition 1.2.7. A real-valued function is a function whose domains and rangescontain only real numbers, denoted as f : R→ R.


Finding the domain of a function is summarized as follows. To find the domain ofa function, you must identity if you function contains variables in the denominatorOR variables in even roots. So look for:

1. Values that result in a denominator of zeroi.e. (12/0)

2. Values that result in an even root of a negative numberi.e. 4√−3,√−10

Once you identify these values, exclude them from the domain. We express thedomain of a function using interval notation and unions as needed. Let us practicethis with some examples:

Example 23. Find the domain of the following functions

1. y = 3x+ 6

2. y = x4

3. y =√

1− x

4. y = 3x+4

Next we see the operations on functions:

Definition 1.2.8. Let f(x) and g(x) be functions. then we define the followingoperations:

• (f + g)(x) = f(x) + g(x) (Addition)

• (f − g)(x) = f(x)− g(x) (Subtraction)

• (f · g)(x) = f(x) · g(x) (Multiplication)

• (fg)(x) = f(x)

g(x), provided g(x) 6= 0 (Division)

Example 24. Let h(x) = 4x + 3 and g(x) = x2 − 7. Find (h + g)(x), (h − g)(x),(h · g)(x), and (h

g)(x).

Now we talk about one last operations we can perform on function:

Definition 1.2.9. If f and g are functions, then the composition of f and g isdenoted by f ◦ g and is defined by

(f ◦ g)(x) = f(g(x))

1.2. FUNCTIONS 13

The domain of f ◦ g is the subset of the domain of g for which f ◦ g is defined, i.e,the domain of f ◦ g is the set of all x such that:

1. x is in the domain of g.

2. g(x) is in the domain of f .

Lets do some examples:

Example 25. Let f(x) =√x and g(x) = x−2. Find f ◦ g and g ◦f and state their

domains.

Example 26. (44 in textbook) Let f(x) = 1x3 and g(x) = 4x+ 1.

Find (f ◦ g)(x), (g ◦ f)(x), f(f(x)), and f 2(x).

Now we end the section by applying everything we learned about functions to wordproblems:

Example 27. (46 in textbook) When a debt is refinanced, sometimes the term ofthe loan (the time it takes to repay the debt) is shortened. Suppose the currentinterest rate is 7%, and a couple’s current debt is $100,000. The monthly paymentR of the refinanced debt is a function of the term of the loan, t, in years. If werepresent this function R = f(t), then the following table defines the function.

t 5 10 12 15 20 25R 1980.12 1161.09 1028.39 898.83 775.30 706.78

1. If they refinance for 20 years, what is the monthly payment? Write this cor-respondence in the form R = f(t).

2. Find f(10) and write a sentence that explains its meaning.

3. Is f(5 + 5) = f(5) + f(5)? Explain.

Example 28. (52 in textbook) The profit from the production of and sale of aproduct is P (x) = 47x − 0.01x2 − 8000, where x represents the number of unitsproduced and sold. Give

1. the profit from the production and sale of 2000 units

2. the value of P (5000).

3. the meaning of P (5000).


1.3 Linear Functions

Objectives

• Find the intercepts of graphs

• Graph linear functions

• Find the slope of a line from its graph and from its equation

• Find the rate of change of a linear function

• Graph a line, given its slope and y-intercept or its slope and one pointon the line

• Write the equation of a line, given information about its graph

We have discussed in section 1.1 how to solve linear equation of one variable. Nowwe will see them in the context of functions and look at all of the properties theyhave. Lets start of with some definitions:

Definition 1.3.1. A linear function is a function of the form y = f(x) = ax+ b,where a, b ∈ R.

Definition 1.3.2. The x-intercept of a function is the point(s) where a graphcrosses the x-axis, i.e, all values of x such that f(x) = 0.

Definition 1.3.3. The y-intercept of a function is the point(s) where a graphcrosses the x-axis, i.e, all values of y such that f(0) = y.

To summarize, in order to find a x-intercept of a function y = f(x), set y = 0 andsolve for x and to find a y-intercept set x = 0 and solve for y. Since linear functionsare lines, two points are sufficient to graph one. So to graph a linear equation dothe following:

Steps to Graphs Linear Function

1. Find x-intercept

2. Find y-intercept

3. Plot x-intercept and y-intercept

4. Connect points with line

1.3. LINEAR FUNCTIONS 15

Example 29. Find the intercepts of 2x+ 5y = 10 and graph the function.

Example 30. Find the intercepts of y = 3x and graph the function.

Definition 1.3.4. If a nonvertical line passes through the points (x1, y1) and (x2, y2),then the slope of the line (m) is defined as

m =y2 − y1x2 − x1

.

What the slope of a line tells you how quickly the linear function is changing, i.e,Rate of change. Rate of change is very central part in applied mathematics, as itcan tell you how fast some specific quantity is changing. For example if you havea model that describes your revenue, then rate of change can tell you how fast yougaining money and how long it will take for you to either break even or make profit.Next semester you will deal with calculus, and rate of change is at the center of it.

• The slope of a vertical line is undefined.

• The slope of a horizontal line is zero.


Example 31. Find the slope of the line passing through the points (-7,3), (6,-2).Same for (3,-2) and (4,-1).

Here are a few properties about slopes for lines:

• If m > 0, the function is increasing.

• If m < 0, the function is decreasing.

• If m = 0, the function is constant.

• If m is undefined, the graph is not a function.

Definition 1.3.5. Two non-intersecting lines that lie in the same plane are parallel.

Properties if Parallel Lines

1. If two non-vertical lines are parallel, then they have the same slope.

2. If two distinct non-vertical lines have the same slope then they areparallel.

3. Two distinct vertical lines , both with undefined slopes, are parallel.

Definition 1.3.6. Two lines that intersect at a right angle (90o) are said to beperpendicular.

1.3. LINEAR FUNCTIONS 17

Properties if Perpendicular Lines

1. If two non-vertical lines are perpendicular, then the product of theirslopes is -1.

2. If the product of the slopes of two lines is -1, then the lines areperpendicular.

3. A horizontal line with zero slope and a vertical line with undefinedslope, are perpendicular.

An equivalent was to say the above is if a lines has slope m, then the line perpen-dicular to it has slope

− 1

m

Now lets talk about how to get equations of lines from the properties he have learnedso far:

Definition 1.3.7. The point-slope form of a of a line with slope m passingthrough the point (x1, y1) is

y − y1 = m(x− x1)

Definition 1.3.8. The slope-intercept form of a of a line with slope m passingthrough the y-intercept (0, b) is

y = mx+ b

Definition 1.3.9. The equation of a vertical line passing through the x-intercept(a, 0) has the form

x = a

Definition 1.3.10. The equation of a horizontal line passing through the y-intercept (0, b) has the form

y = b

Example 32. Write the equation in point-slope for line with slope 14

that goesthrough (2,-3).

Example 33. Write the equation for the line with a slope of zero going through(2,7).

Example 34. ‘Write the equation of the line passing through (3,−5) and perpen-dicular to the line whose equations is x+ 4y − 8.


Example 35. Write the equation in point slope form for the line going through(3,6) and (-2,1).

Example 36. Write the slope-intercept form of the line going through (2,5) and(1,9).

Example 37. A weight loss clinic charges a flat fee of $70 and $5 per pound lost.

1. What are the slope and y-intercept?

2. Write an equation to model this scenario.

Example 38. Graph the following equations.

1. y = 12x− 2

2. y = −3x+ 1

3. y = −23x

Example 39. (50 from textbook) Using Social Security Administration data forselected years from 1950 and projected to 2050, the US population (in millions) canbe described by p(t) = 2.53t+ 162.2 where t is the number of years past 1950.

1. Find p(0) and explain its meaning.

2. Find the slope and explain its meaning.

3. Graph this function for t ≥ 0.

1.4. GRAPHS AND GRAPHING UTILITIES 19

Example 40. (58 in textbook) Residential customers who heat their homes withnatural gas have their monthly bills calculated by adding a base service charge of$9.19 per month and an energy charge of 91.91 cents per hundred cubic feet. Writean equation for the monthly charge y in terms of x, the number of hundreds of cubicfeet used.

Example 41. (59 in textbook) The size of the US civilian workforce for the yearsfrom 1950 and projected to 2050 can be approximated by a linear equation deter-mined by the line connecting the points (1950, 62.2) and (2050, 191.8), where thex-coordinate is the year and the y-coordinate is the number in the civilian workforce(in millions) in year x.

1. Write the equation of the line connecting the two points.

2. Interpret the slope of this line as a rate of change.

1.4 Graphs and Graphing Utilities

We will skip this section, due to the nature of it. You can view a slide show on howto use the graphing calculators on my website. Get familiar with the different toolsthat the graphing calculator has, as it will be invaluable skill to know for the restof you math classes.


1.5 Systems of Linear Equations

Objectives

• Know Types of Systems and Equivalent Systems.

• Solve Systems of Linear Equations using the Graphing, Substitution andElimination Method.

• Solve Systems of Linear Inequalities.

• Solve Application Problems.

Back in section 1.1, we learned how to solve linear equations of one variable.We also had a glimpse of a equation of two variables, however when we ”solved”for one of the variables we did not get a numerical value. Rather we got one of thevariables in terms of the other [See Example 8]. So why could we not get a numericalvalue? Well it turns out that in order to get a numerical solution to a equation oftwo variables, we need at least 2 equations. Similarly, if we have a equation of say nvariables, we need at least n equations, however we need not worry about this case,as it is out of the scope of this class [ If you are curious of how to approach problemslike this, a Linear Algebra class is for you! ]. In this section we discuss methods onhow to solve systems of two equations.

Definition 1.5.1. A System of Equations is when you are given two or moreequations. If all the equations are linear, then they are called System of LinearEquations.

Definition 1.5.2. A Solution to Systems of Equations are all the ordered pairs(x, y) that satisfy all equations in a system.

We will look three common methods to solve systems of linear equations:

• Graphing

• Substitution

• Elimination

Steps for the Graphing Method (Graphing Calculator)

1. Graph both linear equations.

2. The solution is their intersection point(s).

1.5. SYSTEMS OF LINEAR EQUATIONS 21

Example 42. Solve the system by graphing:

y =1

2x+ 5

y =−5

2x− 1


y = 3x+ 1

y = 3x− 2


x+ y = −1

2x+ 2y = −2


We saw from the examples above that we got three situations:

• The lines intersected at one point. (One Solution)

• The lines never intersected and where parallel. (No Solution)

• The lines intersected and where parallel. (Infinitely Many Solutions)

We will now classify these three situations.

Definition 1.5.3. A system of equations with no solution is called an InconsistentSystem.

Definition 1.5.4. A system of equations with finitely many solution is called anConsistent System.

Definition 1.5.5. A system of equations with infinitely many solution is called anDependent System.

Recall that in section 1.1, we talked about the idea of equivalent expression. Remem-ber this was how we could manipulate an equation without changing it’s solutionset. This idea extends to systems of equations, and the technical word for it isEquivalent Systems. Equivalent systems result when

1. One expression is replaced by an equivalent expression

2. Two equations are interchanged

3. A multiple of one equation is added to another equation

4. An equation is multiplied by a nonzero constant

Again this is just how you manipulate the system of equations to be able to find thesolutions without changing the solution set [Again if you ever take Linear Algebra,you are applying elementary matrices from the left on your augmented matrix topreserve the kernal/null space]. Now we are ready to move on to the next method

1.5. SYSTEMS OF LINEAR EQUATIONS 23

Steps for the Substitution Method

1. Solve one of the equations for a single variable.

2. Substitute this expression into the other equation.

3. Solve the new equation for the variable.

4. Substitute this value into either of the original equations to solvefor the other variable.

5. Check the solution in both of the original equations.

Example 45. Solve the system using the substitution method.

3x− 2y = 6

4y = 8

Example 46. Solve the system using the substitution method.

−7x− 2y = −13

x− 2y = 11

Steps for the Elimination Method

1. If necessary, multiply one or both equations by a nonzero numberthat will make the coefficients of one of the variables be the samenumber, but with opposite signs. (i.e. 4x, −4x)

2. Add the equations to eliminate one of the variables.

3. Solve for the remaining variable.

4. Substitute the solution into one of the original equations and solvefor the other variable.

5. Check the solution in both of the original equations.

Example 47. Solve the system using the elimination method.

−4x− 2y = −12

4x+ 8y = −24



−3x+ 3y = 4

−x+ y = 3


2x+ 8y = 6

−5x− 20y = −15

Now we finish up the section by applying what we learned to some word problems:

Example 50. Tickets to a musical are being sold. On the first day, they sold threeadult tickets and nine youth tickets for a total of $75. On the second day, they soldeight adult tickets and five youth tickets for a total of $67. How much does eachtype of ticket cost?

Example 51. A chemist has a 25% and a 50% acid solution. How much of eachsolution should be used to form 200 mL of a 35% acid solution?

Example 52. Donald has $8000 to invest in two accounts. One account earns 6%annual interest and the other account earns 9% annual interest. If he wants to earn$600 in one year, how much should he invest in each account?

Example 53. (40 in textbook) A bank lent $237,000 to a company for the devel-opment of two products. If the loan for product A was for $69,000 more than theloan for product B, how much was lent for each product?

1.6. APPLICATIONS: BUSINESS & ECONOMICS 25

1.6 Applications: Business & Economics

Objectives

• Profit, Revenue and Cost

• Marginals

• Break Even Point

• Supply and Demand

• Taxation

In this section, we will learn new vocabulary and functions from the businessworld. Once we do that, we will then solve word problems using everything youhave learned from the previous sections.

Definition 1.6.1. The Revenue is the income generated from sale of goods orservices, or any other use of capital or assets.

Definition 1.6.2. The Profit is the difference between the amount a companyreceives from sales and its cost. If x units are produced and sold, then

P (x) = R(x)− C(x)

where P (x) = profit from sale of x units, R(x) = total revenue from sale of x units,and C(x) = total cost of production and sale of x units.

In general, revenue is found using

Revenue = (price per unit)(number of units sold).

While Cost is composed of two parts, variable costs and fixed costs.

Cost = variable costs+ fixed costs

Here variable cost is those directly related to the number of units produced (i.e. costof materials) and fixed cost is the cost that remain constant (i.e. rent, depreciation).

Example 54. It costs a T-shirt company $250 per week to operate all the equipmentand $5 to make each shirt. The company then sells the T-shirts for $15 each. Writeequations for C(x), R(x), and P (x).

Example 55. (2 in textbook) Suppose a stereo receiver manufacturer has the totalcost function C(x) = 210x+ 3300 and the total revenue function R(x) = 430x.


1. What is the equation of the profit function for this commodity?

2. What is the profit on 500 items?

Definition 1.6.3. The Marginal Profit is the rate of change in profit with respectto the number of units produced and sold.

Definition 1.6.4. The Marginal Cost is the rate of change in cost with respectto the number of units produced and sold.

Definition 1.6.5. The Marginal Revenue is the rate of change in revenue withrespect to the number of units produced and sold.

If any of the functions are linear, R(x), C(x) and P (x), the rate of change is simplythe slope of the function. When your functions are non-linear, you start to needCalculus.

Example 56. (7 in textbook) A linear revenue functions is R = 27x.

1. What is the slope?

2. What is the marginal revenue, and what does it mean?

3. What is the revenue received from selling one more item if 50 are currentlybeing sold? If 100 are being sold?

Example 57. (13 in textbook) A company manufactures helmets for skiing. Thefixed costs for one model of helmet ar $6600 per month. Materials and labor foreach helmet of this model are $35, and the company sells this helmet to dealers for$60 each.

1. Write the functions for monthly total costs and total revenue.

2. Write the profit function for this helmet.

3. Find C(200), R(200), and P (200). Interpret each answer.

4. Find the marginal profit and explain its meaning.

Now we discuss a concept in economic theory, break-even analysis.

Definition 1.6.6. The Break-Even Point is the point where cost and revenue areequal.

Example 58. (16 in textbook)A manufacturer of shower-surrounds has a revenuefunction of R(x) = 81.50x and a cost function of C(x) = 63x + 1850. Find thenumber of units that must be sold to break even.

1.6. APPLICATIONS: BUSINESS & ECONOMICS 27

Example 59. (20 in textbook) A manufacturer sells watches for $50 per unit. Thefixed cost related to this product are $10,000 per month, and the variable cost perunit is $30.

1. Write the functions for revenue and cost.

2. How many watches must be sold to break even?

Example 60. (24 in textbook) Financial Paper, Inc. is a printer of checks and formsfor financial institutions. For individual accounts, boxes of 200 checks cost $0.80 perbox to print and package and sell for $4.95 each. Financial Paper’s monthly fixedcosts for printing and packaging these checks for individuals are $1245.

1. Write the function for Financial Paper’s monthly total costs.

2. Write the function for Financial Paper’s monthly total revenue.

3. Write the function for Financial Paper’s monthly total profit.

4. Find the number of orders for boxes of checks for individual accounts thatFinancial Paper must receive and fill each month to break even.

Definition 1.6.7. The Law of Demand states that the quantity demanded willincrease as price decreases and decrease as price increases. (negative slope)

Definition 1.6.8. The Law of Supply states that the quantity supplied for salewill increase as the price of the product increases. (positive slope)

Definition 1.6.9. The Market Equilibrium it the point where the quantity of aproduct demanded equals the quantity supplied

Eventhough supply and demand depend on price, p, we will graph p on they-axis.

Example 61. The supply and demand function for a product are given below. Findthe market equilibrium point.

Demand: p = 600− 0.004xSupply: p = 480 + 0.001x

Example 62. (48 in textbook) Retailers will buy 45 cordless phones from a whole-saler if the price is $10 each but only 20 if the price is $60. The wholesaler willsupply 35 phones at $30 each and 75 at $50 each. Assuming the supply and demandfunctions are linear, find the market equilibrium point.


Now we will discuss taxation: If a supplier is taxed $K per unit sold, the tax isoften passed on to the consumer. This tax is then added to the supply functionso p = f(q) becomes p = f(q) + K. The demand function will remain the same.Adding a tax will cause the market equilibrium to change.

Example 63. The demand for computers is given by p = 1200 − 0.0002q and thesupply is given by p = 980 + 0.0006q. If the wholesaler is taxed $123 per unit soldand decides to pass this tax along to the consumer, find the equilibrium.

Chapter 2

Quadratic and Other SpecialFunctions

Last chapter we only talked about linear functions/equations (and some rationalfunctions/equations), their properties and their applications to business and eco-nomics. This chapter we discuss functions that are not linear and some of theirproperties.

2.1 Quadratic Equations

Objectives

• Factoring

• Square Root Method

• Quadratic Formula

• Applications

In this section we discuss how to how to solve for quadratic equations:

Definition 2.1.1. A Quadratic Equation in One Variable is an equation of seconddegree that can be written in the general form ax2 + bx + c = 0, where a, b, and care all constants and a 6= 0.

Some examples of these types are 3x2 + 4x + 1 = 0 and 2x2 + 1 = x2 − x. We willonly consider real solutions to these equations and will cover three ways to solve

29

30 CHAPTER 2. QUADRATIC AND OTHER SPECIAL FUNCTIONS

them.

Theorem 2.1.1. For real numbers a and b, ab = 0 if and only if a = 0, b = 0, orboth a and b equal zero.

To solve a quadratic equation by factoring, we must get them in the general form.The most common way to factor is the AC method. This is preformed as follows:If a Trinomial of the form ax2 + bx+ c is factorable, it can be done by

1. Make sure the trinomial is in standard form

2. Factor out a GCF (Greatest Common Factor) if applicable.

3. Begin listing factor pairs of ac Continue until you find the pair of numbersthat multiply to equal ac, but add up to equal b.

4. Use the two numbers found in Step 3 to rewrite the trinomial as a 4 termpolynomial by breaking up the middle term into two parts.

5. Factor the resulting polynomial by grouping

6. . Factor out a GCF from each of the paired factors if applicable.

7. The remaining terms inside the two sets of parenthesis should be identical.This is one factor of the trinomial. The other factor is formed by combiningthe GCFs into a second set of parenthesis.

Example 64. Solve x2 + 4x+ 3 = 0 by factoring.

Example 65. Solve x2 − 5x+ 4 = 0 by factoring.

Example 66. Solve x2 − 4x = 12 by factoring.

Example 67. Solve x2 = 11x− 10 by factoring.

Example 68. Solve 5p2 − p− 18 = 0 by factoring.

Example 69. Solve 5x2 = 10x by factoring.

Example 70. Solve x2 − 11x+ 19 = −5 by factoring.

Example 71. Solve xx−2 − 1 = 3

x+1.

Example 72. Solve 2xx+2

= 5x2−x−6 −

1x−3 .

2.1. QUADRATIC EQUATIONS 31

We don’t always have to do the AC method. There are situations were we can solvefor the solution quickly.

Square Root Method

If b = 0 in the general form of a quadratic equation, we can solve theequation by getting the equation in the form x2 = k and taking thesquare root of both sides of the equation.

Example 73. Solve x2 − 16 = 0.

Example 74. Solve 9x2 = 15.

Example 75. Solve (x+ 3)2 = 16.

While factoring and the square root method can only be used on some quadraticequations, the quadratic formula can be used to solve ANY quadratic equation

Definition 2.1.2. If ax2 + bx+ c = 0 and a 6= 0, then the quadratic formula is

x =−b±

√b2 − 4ac

2a

The expression b2− 4ac is called the discriminant. The discriminant tells how manysolutions a quadratic equation will have.

• b2 − 4ac < 0, no real solutions

• b2 − 4ac = 0, one real solution

• b2 − 4ac > 0, two distinct real solutions

Example 76. Solve 10x2 − 6 = 9x using the quadratic formula.

Example 77. Solve x2 + 4x+ 5 = 0 using the quadratic formula.

Example 78. The profits of a company can be represented by y = −4x2 + 19x−12where y is in hundred thousand dollars and x is the number of years. When will thecompany start and stop making money?

Example 79. If the profit from the sale of x units of a product is P = 16x−0.1x2−100, what level(s) of production will yield a profit of $180?


2.2 Parabolas

Now we will look at quadratic equations as functions and explore some of theirproperties.

Definition 2.2.1. A quadratic function is function with the general form y =f(x) = ax2 + bx+ c, where a, b, and c are real numbers and a 6= 0.

These are also called second degree functions and has the shape of a parabola.

Some things to keep in mind are:

• If a > 0, the parabola opens up

• If a < 0, the parabola opens down

• The vertex is the point where the parabola turns. The vertex gives the optimalvalue.

• Depending on the value of a, the vertex is either a maximum or a minimum

• The vertical line through the vertex is called the axis of symmetry

2.2. PARABOLAS 33

Example 80. For each graph below, answer the following questions.

1. Is a < 0 or is a > 0?

2. What is the vertex?

3. Is the vertex a maximum or a minimum value?

4. What is the axis of symmetry?

Some more properties to keep in mind are the following:

Properties of Quadratic Functions

Given the functionf(x) = ax2 + bx+ c

• The y-intercept is (0, c).

• The vertex point is (−b2a, f(−b

2a)).

• The axis of symmetry is x = −b2a

.

• The zeros of the function are the x-intercepts.

– To find the zeros, you may factor, use the square root method,or use the quadratic formula.

Example 81. Find the vertex, axis of symmetry, y-intercept, and zeros of

y = −x2 − 4x+ 12


Example 82. (1 in textbook) Given the function y = 12x2 + x, determine the

following.

1. Find the vertex of the graph.

2. Determine if the vertex is a maximum or a minimum.

3. Determine what value of x gives the optimal value of the function.

4. Determine the optimal value of the function.

5. What are the zeros of the function?

In Chapter 1, we discussed how to graph a line. A line is uniquely determined bytwo points. However, we need more information to graph a parabola. We can use:

• One must be the vertex.

• Use symmetry to help you find the other two.

Example 83. Graph f(x) = 2x2 + 4x+ 2.

Example 84. A manufacturer has daily costs of C(x) = 200 − 10x + 0.01x2 toproduce x items. How many items should be produced to minimize costs?

Example 85. The value of a stock portfolio is given by V (t) = 50+72t−3t2, whereV is the value in hundreds of dollars and t is time in months.

1. How much money did the investor start with?

2. When will the value reach the maximum amount?

3. What will be the maximum value?

2.3. BUSINESS APPLICATIONS 35

Example 86. When priced at $30 each, a shirt has annual sales of 4000 units. Themanufacturer has determined that each $1 increase in cost will decrease sales by 100units. Find the unit price that will maximize total revenue. (Recall Revenue=(priceper unit)(number sold))

Similar to Chapter 1, we will take about rate of change, however since we haveparabola, the rate of change is not exact. The rate of change of a quadratic functionis not constant. We can estimate the rate of change by calculating the average rateof change:

Definition 2.2.2. The average rate of change of f(x) with respect to x over theinterval from x = a to x = b (where a < b) is given by

f(b)− f(a)

b− a

This gives the slope of the secant line connecting the two points.

Example 87. The number of horsepower needed to overcome a wind drag is givenby N(s) = 0.005s2 + 0.007s− 0.031, where s is speed in miles per hour. What is theaverage rate of change of horsepower from 35 mph to 55 mph?

2.3 Business Applications

In this section, we will extend the ideas learned in section 1.6 to quadratics.

Recall that we can find the break even point by finding the quantity x that makes

C(x) = R(x).

We considered C and R as linear equations, however we can consider them asquadratics now.

Example 88. (2 in textbook) If a firm has the following cost and revenue functions,find the break-even points.

C(x) = 3600 + 25x+1

2x2 R(x) = (175− 1

2x)x

Example 89. (5 in textbook)Given that

P (x) = 11.5x− 0.1x2 − 150

and that production is restricted to fewer than 75 units, find the break-even points.


Recall that our revenue function in section 1.6, did not factor in demand. If youknow factor in demand, then we can have Market Monopoly. This is when therevenue of a company is restricted by the demand for the product. So then ourrevenue function changes to

R(x) = px = demand ∗ x

Example 90. (10 in textbook) If, in a monopoly market, the demand for a productis p = 1600− x, what price will maximize revenue?

When quadratic equations are used to represent supply and demand curves, we cansolve the equations simultaneously to find the market equilibrium. Sometimes, wemay need to manipulate the equations we are given.

Example 91. (26 in textbook) If the supply function for a commodity is p =q2 + 8q + 20 and the demand function is p = 100 − 4q − q2, find the equilibriumquantity and the equilibrium price.

Example 92. (28 in textbook) If the supply and demand functions for a commodityare given by

4p− q = 42

(p+ 2)q = 2100,

respectively, find the price that will result in market equilibrium.

2.4 Special Functions and Their Graphs

Definition 2.4.1. A Polynomial Function is a function that has the form

f(x) = anxn + an−1x

n−1 + · · ·+ a1x+ a0

Note that

• A linear function is a polynomial function of degree one

• A quadratic function is a polynomial function of degree two

some examples include y = x, y = x2 − 3x + 6, y = x100, y = x5 − x + 3. You candetermine if the power of a polynomial is odd or even by looking at its graph. Youcan do this by looking at the end behavior of the graph.

2.4. SPECIAL FUNCTIONS AND THEIR GRAPHS 37

• If the ends of the function are going in the same direction, the polynomial hasan even power.

• If the ends of the function are going in the opposite direction, the polynomialhas an odd power.

Example 93. Determine if the polynomials below have odd or even degree.

Definition 2.4.2. A rational function is a function of the form

y =f(x)

g(x)

with g(x) 6= 0 and f(x) and g(x) are both polynomials. Its domain is the set of allreal numbers for which g(x) 6= 0.


Example 94. (53 in textbook) The demand function for a product is given by

p =200

2 + 0.1x

where x is the number of units and p is the price in dollars.

1. Graph this demand function for 0 ≤ x ≤ 250, with x on the horizontal axis.

2. Does the demand ever reach 0?

Definition 2.4.3. The absolute value function can be written as

f(x) = |x| =

{x if x ≥ 0

−x if x < 0

Because the absolute value function is defined by two equations, we say it is apiecewise defined function.

Definition 2.4.4. A Piecewise Function is a function that is defined by multipleequations for different subintervals of the domain.

2.5. MODELING 39

Example 95. Determine the piecewise function for the following graph.

Example 96. (32 in textbook) If

k(x) =

{4− 2x if x < 0

|x− 4| if 0 < x < 4

find the following.

1. k(−0.1)

2. k(0.1)

3. k(3.9)

4. k(4.1)

Example 97. The 2012 monthly charge in dollars for x kilowatt hours (kWh) ofenergy used by a residential customer of Excelsior Electric Membership Corporationduring the months of November through June is given by the function

C(x) =

10 + 0.094x if 0 ≤ x ≤ 100

19.40 + 0.075(x− 100) if 100 < x ≤ 500

49.40 + 0.05(x− 500) if x > 500

1. What is the monthly charge if 1100 kWh of electricity are consumed in amonth?

2. What is the monthly charge if 450 kWh of electricity are consumed in a month?

2.5 Modeling

We will skip this section, due to the nature of it. You can view a slide show on howto use the graphing calculators on my website. Get familiar with the different toolsthat the graphing calculator has, as it will be invaluable skill to know for the restof you math classes.

Chapter 3

Matrices

3.1 Matrix Addition

Objectives

• Definitions

• Adding Matrices

• Scalar Multiplication

• Solving for Missing Values in Matrix Equations

• Applications

Definition 3.1.1. A Matrix (matrices) is a rectangular/square array of numbers,symbols, or expressions, arranged in rows and columns.

Definition 3.1.2. The Order of a Matrix is said to be m × n if the matrix hasm rows and n columns.

Definition 3.1.3. The Elements of a Matrix are the numbers in a matrix. Weuse subscripts to reference different elements. aij would be the entry in matrix A inthe ith row and the jth column.

Definition 3.1.4. A Square Matrix is a matrix with the same number of rowsand columns.

41

42 CHAPTER 3. MATRICES

Example 98. Given Matrix A below, answer the following questions.

A =

[1 2 3 4 53 4 5 6 7

]1. What is the order of matrix A?

2. Is matrix A a square matrix?

3. What is the element a23?

4. Let Matrix B be given below, do matrix A and matrix B have the same order?

B =

[11 21 32 −44 8 15 0

]Definition 3.1.5. A Row Matrix is a matrix with one row.

Definition 3.1.6. A Columns Matrix/ Vector with a matrix is one column.

Definition 3.1.7. The Zero Matrix is a matrix where every entry is 0.

Definition 3.1.8. The Transpose of a Matrix is formed by interchanging therows and columns of a matrix. The transpose of matrix A is denoted by AT

Definition 3.1.9. The Identity Matrix is a square matrix with ones on the di-agonal and 0’s everywhere else. The identity matrix is often denoted by I.

As a example the following is a 3× 3 identity matrix:

I =

1 0 00 1 00 0 1

3.1. MATRIX ADDITION 43

Example 99. Determine the transpose of matrix A.

A =

4 3 62 −1 0−3 6 9−7 −2 11

Do A and AT have the same order?

Example 100. Use the matrices below to answer the question.

1. What is the order of matrix E?

2. Which of the matrices are square?

3. Write the transpose of matrix F .

4. In matrix A has element a3j = 0, what is j?

Definition 3.1.10. If two matrices have the same order, you can add/substractthem by adding/subtracting the entries in the same position.

Example 101. Find A+B and A−B.

A =

5 −2 32 −1 7−4 5 8

B =

−1 2 34 2 −50 7 10

We can also define Scalar Multiplication of a Matrix by multiplying a matrix by a

real number (called a scalar) which results in a matrix in which each entry ismultiplied by the real number.

Example 102. Let A be given below. Find 5A and -3A.

A=

[1 3 2 1020 5 6 3

]


Example 103. Given the following matrix equation below, solve for x and y.[−4 x3 y − 1

]=

[−4 23 −5

]Example 104. (32 in textbook) Given the following matrix equation, solve for eachvariable. [

x y (x+ 3)z 4 4y

]=

[(2x− 1) −1 w

x (5 + y) −4

]Example 105. Solve for the variables.

3

[x 44y w

]− 2

[4x 2z−3 −2w

]=

[20 206 14

]

Entering a Matrix in a Calculator

• Press 2nd, and then x−1 (MATRX).

• Scroll over to EDIT.

• Hit Enter on the matrix you want to edit.

• Input the order of your matrix.

• Input the entries of your matrix.

To perform calculations with your matrix, you press 2nd, and then x−1 (MATRX).Scroll down to the matrix you want to use and press enter.

Example 106. Use the calculator to find the following sum 4C + 2D.

C =

[5 −31 2

]D =

[−4 23 −5

]Example 107. (Ex. 5 in textbook) The table below summarizes the actual andprojected US dollar value of US exports and imports of pipeline natural gas withCanada and Mexico and of liquefied natural gas with other countries for selectedyears. Using matrices, find the balance of trade in natural gas with the variouscountries in the selected years. (Balance of Trade=Exports - Imports)

3.1. MATRIX ADDITION 45

Example 108. (Ex. 8 in textbook) Suppose that in a government agency, paper-work is constantly flowing among offices according to the diagram.

1. Construct matrix A with elements

aij =

{1, if paperwork flows directly from i to j

0, if paperwork does not flow directly from i to j

2. Construct matrix B with elements

bij =

1, if paperwork can flow from i to j through at mostone intermediary, when i is not equal to j

0, if this is not true

3. The person in office i has the most power to influence others if the sum of theelements of row i in the matrix A+B is the largest. What is the office numberof this person?

Example 109. (36 in textbook) The following tables show important demographicsfor China, Bangladesh, and the Philippines for 2012 and projected for 2062.

1. Find a matrix that shows the changes of demographics from 2012 to 2062 forthese countries.

2. Which negative entries in the matrix definitely indicate a positive change forthese countries?


3.2 Matrix Multiplication

Definition 3.2.1. The Product of two m× n matrix A and an n× p matrix B isan m× p matrix C, with the ij entry of C given by the formula

cij = ai1b1j + ai2b2j + · · ·+ ainbnj

Example 110. Answer the following questions:

1. If A is a 3 × 5 matrix, and B is a 5 × 2 matrix, what order will AB be?

2. Can you multiply a matrix that is 3 × 2 with a matrix that is 4 × 1?

Multiplication of two matrices A and B can be illustrated by the figure below

Example 111. Given matrix A and B below, find AB.

A =

[1 2 34 5 6

]B =

7 89 1011 12

Is multiplying matrices commutative? (AB = BA)

Example 112. Given matrix A and B below, find AB and BA.

A =

[1 23 4

]B =

[2 01 2

]MULTIPLYING MATRICES IS NOT COMMUTATIVE!

Example 113. Suppose an electronics retailer has an inventory represented by thetable below. If the value of each plasma TV is $800, the value of each LCD TV is$750, and the value of each home theater system is $500. Represent the value of

3.2. MATRIX MULTIPLICATION 47

each item in a matrix, and find the value of the inventory of each store using matrixmultiplication.

Example 114. Pentico Industries must choose a supplier for the raw materials thatit uses in its two manufacturing divisions at Clarion and Brooks. Each division usedifferent amounts of steel, wood, and plastic as shown in the first table below. Thetwo supply companies being considered, Western and Coastal, can each supply allof these materials, but at different prices per unit, as described in the second tablebelow. Use matrix multiplication to decide which supplier should be chosen to sup-ply each division.


Example 115. Suppose that a bank has three main sources of income: businessloans, auto loans, and home mortgages. Suppose the income from the sources foreach of three years is given in the table below. The bank uses its income to providestart up loans for small businesses. The bank uses 45% of the income from businessloans, 20% of the income from auto loans, and 30% of the income from mortgagesto finance the start up loans. Find the available money for the start up loans foreach of the three years.

3.3 Gauss-Jordan Elimination

Given a system of equations, one can organize a system of equations into a matrix.This is called an augmented matrix.

Definition 3.3.1. Given a system of equations:

a11x1 + a12x2 + ...+ a1nxn = b1

a21x1 + a22x2 + ...+ a2nxn = b2

....

am1x1 + am2x2 + ...+ amnxn = bm

then the augmented matrix isa11 a11 ... a1n b1a21 a22 ... a2n b2

...am1 am2 ... amn bn

Definition 3.3.2. The identity matrix is a square matrix with ones on the diag-onal and zeros everywhere else.

We can ”solve” the system of equations via the Gauss-Jordan Elimination. Themethod requires you to perform row operations on the matrix to get the left side ofthe matrix (left of the bar) reduced to the identity matrix. This is called the reducedrow echelon form of the matrix. There are three types of row operations one cando:

3.3. GAUSS-JORDAN ELIMINATION 49

1. Row switching: A row within the matrix can be switched with another row.

2. Row multiplication: Each element in a row can be multiplied by a non-zeroconstant.

3. Row addition: A row can be replaced by the sum of that row and a multipleof another row.

One can also use the TI Graphing calculator to get a augmented matrix into reducedrow echelon form as follows:

Using the Calculator to Perform Gauss-Jordan Elimination

• Input your augmented matrix into matrix A.

• Press second, x−1 (MATRX).

• Scroll to MATH.

• Choose option B: rref.

• Place matrix A in the parentheses. (2nd, x−1, ENTER)

Example 116. The matrix below is a solution of a system of equations. What isthe solution of the system? 1 0 0 −8

0 1 0 10 0 1 1/3

Example 117. Solve the system of equations using Gauss-Jordan Elimination.

x+ 4y − 2z = 9

x+ 5y + 2x = −2

x+ 4y − 28z = 22

Example 118. (31 in textbook) Solve the system of equations using Gauss-JordanElimination.

2x− 5y + z = −9

x+ 4y − 6z = 2

3x− 4y − 2z = −10


Example 119. Solve the system of equations using Gauss-Jordan Elimination.

x1 + 2x2 − x3 + x4 = 3

x1 + 3x2 + 4x3 + x4 = −2

2x1 + 5x2 + 2x3 + 2x4 = 1

2x1 + 3x2 − 6x3 + 2x4 = 3

Example 120. (55 in textbook) The following table gives the calories, fat, andcarbohydrates per ounce for three brands of cereal. How many ounces of each brandshould be combined to get 443 calories, 5.7g of fat, and 113.4g of carbohydrates?

Example 121. (58 in textbook) Each ounce of substance A supplies 5% of therequired nutrition a patients needs. Substance B supplies 15% of the required nutri-tion per ounce, and substance C supplies 12% of the required nutrition per ounce.If digestive restrictions require that substances A and C be given in equal amountsand that the amount of substance B be one fifth of these other amounts, find thenumber of ounces of each substance that should be in the meal to provide 100%nutrition.

Example 122. A trust account manager has $220,000 to be invested. The invest-ment choices have current yields of 8%, 7%, and 10%. Suppose that the investmentgoal is to earn interest of $16,600.

1. Find a general description for the amounts invested at the three rates.

2. If $10,000 is invested at 10%, how much will be invested at each of the otherrates?

3. What is the minimum amount that will be invested at 7%? How much will beinvested at the other rates?

4. What is the maximum amount that will be invested at 7%? How much willbe invested at the other rates?

3.4. LEONTIEF INPUT-OUTPUT MODELS 51

3.4 Leontief Input-Output Models

In this section, we will start to model real-world problems using matrices. First wewill discuss the analouge of division for matrices:

Definition 3.4.1. Two square matrices, A and B, are called inverses of each otherif AB = I and BA = I. In this case, B = A−1 and A = B−1.

Example 123. Determine if the two matrices are inverses of each other.

A =

1 2 10 0 31 0 1

B =

0 −1/3 11/2 0 −1/20 1/3 0

There are many ways to find the inverse of a matrix, but here we will only discusshow to calculator the inverse using the graphing calculator:

Finding the Inverse of a Matrix using Gaussian Elimination

• Input your augmented matrix into matrix A.

• Augment the matrix A with the identity matrix of the same size [A|I].

• Perform the Gauss-Jordan Elimination for the augmented matrix. (rref)

• The matrix on the right is the inverse of matrix A. If there is a row ofzeros in the matrix, then A has no inverse.

Example 124. Find the inverse of matrix C.

C =

3 1 21 2 31 1 1

A second way to find the inverse is to do the following:

Finding the Inverse of a Matrix in a Calculator

• Input the matrix.

• Type [A]−1 in the calculator.

If your matrix has decimals in it and you want to convert it to fractions, pressMATH and choose . Frac.


Example 125. Find the inverse of matrix B.

B =

3 4 −14 2 22 6 −4

Now consider the following scenario: Suppose a simple economy is based on threeindustries: agricultural products, manufactured goods, and fuels. In this economyeach industry uses outputs from the other industries as inputs for its own production.The table below summarizes this interdependency for each industry in terms of thevarious inputs required for 1 unit of production.

Example 126. Use the table to answer the questions.

1. How many units of agricultural products and of fuels are required to produce100 units of manufactured goods?

2. Production of which commodity is least dependent on the other two?

3. If fuel costs rise, which two industries will be most affected?

Lets look at the Leontief model. The Leontief model was developed by WassilyLeontief and is useful in predicting the effects on the economy when there are pricechanges or shifts in government spending. There are two types of Leontief models:open and closed. We will use several types of matrices in these models.

Definition 3.4.2. A Technology Matrix A that shows the interdependency ofvarious commodities and industries.

Definition 3.4.3. A Gross Production Matrix is column matrix X that givesthe gross production of each industry.

Multiplying AX gives the amounts of the gross productions used in the economy.

Definition 3.4.4. A Final Demands Matrix D is one that represents the unitsof gross production not used by these industries.

3.4. LEONTIEF INPUT-OUTPUT MODELS 53

Definition 3.4.5. The Open Leontief Model is a model where some of the goodsfrom the economy are available to those outside of the economy (there are surpluses).The technological equation is (I −A)X = D. The solution may be found by calcu-lating X = (I − A)−1D.

Definition 3.4.6. The Closed Leontief Model is a model where all inputs andoutputs are used within the economy. The technological equation is (I − A)X = 0.To solve this equation, we use an augmented matrix [I − A|0]

Example 127. Use the technology matrix below (from example 3). If we wish tohave surpluses of 85 units of agricultural products, 65 units of manufactured goods,and 0 units of fuel, what should the gross outputs be?

A =

0.5 0.1 0.10.2 0.5 0.30.1 0.3 0.4

Example 128. The following closed Leontief model with technology matrix A mightdescribe the enconomy of the entire country. Find the total budgets for each indus-try.

Example 129. (40 in textbook) Card tables are made by joining 4 legs and a topusing 4 bolts. The legs are each made from a steel rod. The top has a frame madefrom 4 steel rods. A cover and 4 special clamps that brace the top and hold the legsare joined to the frame using a total of 8 bolts. The parts-listing matrix is givenbelow. If an order is received for 10 card tables, 4 legs, 1 top, 1 cover, 6 clamps, and12 bolts, how many of each primary assembly item are required to fill the order?


Example 130. (37 in textbook) A closed model for an economy has a manufacturingindustry, utilities industry, and households industry. Each unit of manufacturingoutput uses 0.5 unit of manufacturing input, 0.4 unit of utilities input, and 0.1 unitof households input. Each unit of utilities output uses 0.4 unit of manufacturinginput, 0.5 unit of utilities input, and 0.1 unit of households input. Each unit ohhousehold output uses 0.3 unit of manufacturing input, 0.3 unit of utilities input,and 0.4 unit of households input. Find the gross production for each industry.

Example 131. (27 in textbook) Suppose that an economy has three industries:fishing, agriculture, and mining. The technology matrix is given below. If surplusesof 110 units of fishing output and 50 units each of agricultural and mining goodsare desired, find the gross production of each industry.

Chapter 4

Inequalities and LinearProgramming

4.1 Linear Inequalities

4.2 Linear Programming

55

56 CHAPTER 4. INEQUALITIES AND LINEAR PROGRAMMING

Chapter 5

Exponential and LogarithmicFunctions

5.1 Exponential Functions

5.2 Logarithmic Functions

5.3 Equations and Applications

57

58 CHAPTER 5. EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Chapter 6

Mathematics of Finance

6.1 Simple Interest & Sequences

Simple interest is paid on investments involving time certificates issued by banks andon certain types of bonds. The interest for a given period is paid to the investor,and the principal remains the same.

This is often how you would loan money to friends, but banks and financial in-stitutions will use a different method that we will learn later.

Definition 6.1.1. The simple interest I is given by

I = prt

where I=interest in dollars, P=principal in dollars, r=interest rate, and t=time.

Example 132. (6 in textbook) $800 is invested for 5 years at an annual simpleinterest rate of 14%. How much interest will be earned?

Definition 6.1.2. The future amount of an investment can be found by addingthe principal and the interest. That is,

S = P + I

where S is the future value, P is the principal, and I is the interest earned.

Remark 3. The principle P of an investment is also called the present value of theinvestment. The present value of a loan is the original loan amount, which is alsocalled the face value of the loan.

59

60 CHAPTER 6. MATHEMATICS OF FINANCE

Notice that the future amount if an investment can be rewritten as follows:

S = P + I = P + Prt = P (1 + rt)

Example 133. (8 in textbook) $1800 is invested for 9 months at an annual simpleinterest rate of 15%. What is the future value of the investment after 9 months?

Example 134. (10 in textbook) If you borrow $1600 for 2 years at 14% annualsimple interest, how much must you repay at the end of the 2 years?

Example 135. To buy a Treasury Bill that matures to $20,000 in 2 years, you mustpay $17,000.

1. What annual simple interest rate do you earn? Round your answer to threedecimal places.

2. If the bank charges a $50 fee to buy the bill, what is the actual interest rateyou earn?

Example 136. (20 in textbook) What is the present value of an investment at 6%annual simple interest if it is worth $832 in 8 months?

Example 137. (22 in textbook) How long does it take for $8500 invested at 11%annual simple interest to be worth $13,000?

Example 138. (24 in textbook) An investor owns several apartment buildings. Thetaxes on these buildings total $30,000 per year and are due before April 1. The latefee is 1/2% per month up to six months, at which time the buildings are seizedby the authorities and sold for back taxes. If the investor has $30,000 available onMarch 31, will he save money by paying the taxes at that time or by investing themoney at 8% and paying the taxes and the penalty on September 30?

Example 139. (26 in textbook) Suppose you lent $5000 to friend 1 for 18 months atan annual simple interest rate of 9%. After 1 year, you need money for an emergencyand decide to sell the note to friend 2.

1. How much does friend 1 owe when the loan is due?

2. If your agreement with friend 2 means she earns simple annual interest at anannual rate of 12%, how much did friend 2 pay you for the note?

6.2. COMPOUND INTEREST & SEQUENCES 61

6.2 Compound Interest & Sequences

6.3 Future Values of Annuities

6.4 Present Values of Annuities

6.5 Loans & Amortization

62 CHAPTER 6. MATHEMATICS OF FINANCE

Chapter 7

Introduction to Probability

63

64 CHAPTER 7. INTRODUCTION TO PROBABILITY

Chapter 8

Data Description

65

66 CHAPTER 8. DATA DESCRIPTION

Chapter 9

Limits and Continuity

In this chapter we finally arrive to the important tools that we need to construct inorder to ”do” calculus. We will first intuitively address what a limit of a functionis of f(x) in R2.

9.1 Limits

We start of by giving a provisional definition:

Limit

A function f : R → R approaches the limit l (on the y-axis) near a (on thex-axis), if we can make f(x) as ”close” as we like to to the value l by requiringx be sufficiently close to, but unequal to a.

Consider the following graphs of the three function f(x), g(x) and h(x) below:

67

68 CHAPTER 9. LIMITS AND CONTINUITY

Graphs.JPG

Notice that for f(x), g(x) and h(x), if x approaches a from either the ”left” or ”right”of the x-axis, then f(x), g(x) and h(x) approaches l on the y-axis, even though g isnot defined at a [g(a) =DNE] and h at a does not take the value l [h(a) 6= l]. Sothis leads us to the remark:

Remark 4. The limit l of a function f near a does not care if f is defined at a orif f(a) = l. The only thing that matters is if f approaches l as x approaches a fromthe left and right side.

Limit

A function f : R→ R is said to have limit l at a, if for any ε > 0, there existsa δ > 0 such that for any x ∈ R with |x− a| < δ implies that

|f(x)− l| < ε

9.2. CONTINUOUS FUNCTIONS 69

Remark 5. This definition is so important, that it is worth memorizing. If you willdo any form of advanced mathematics, you will encounter this definition. Every-thing we do from now one will implicitly use this definition.

9.2 Continuous Functions

Continuous

A function f : R → R is said to be continuous if the graph contains nobreaks, jumps or wild oscillations.

Continuous

A function f : R → R is said to be continuous at a if for any ε > 0, thereexists δ > 0 such that for any x ∈ R with |x− c| < δ implies that

|f(x)− f(c)| < ε

In a more compact form, f is said to be continuous at a if

limx→a

f(x) = f(a)

70 CHAPTER 9. LIMITS AND CONTINUITY

Chapter 10

Differentiation

10.1 Derivatives

Derivative

A function f : R→ R is said to be differentiable at a if

limh→0

f(a+ h)− f(a)

h

exists. In this case the limit is denoted as f ′(a) and is called the derivative off at a.

10.2 Differentiation

Theorem 10.2.1. If f : R→ R is a constant function, f(x) = c, then

f ′(a) = 0

for every a ∈ R.

Proof.

Theorem 10.2.2. If f : R→ R is a identity function, f(x) = x, then

f ′(a) = 1

for every a ∈ R.

71

72 CHAPTER 10. DIFFERENTIATION

Proof.

Theorem 10.2.3. If f, g : R → R are differentiable at a, then f + g is also differ-entiable at a, and

(f + g)′(a) = f ′(a) + g′(a)

Proof.

Theorem 10.2.4. (Product Rule) If f, g : R → R are differentiable at a, thenf · g is also differentiable at a, and

(f · g)′(a) = f ′(a) · g(a) + f(a) · g′(a)

Proof.

Theorem 10.2.5. (Power Rule) If f(x) = xn for n ∈ N, then

f ′(a) = nan−1

for all a ∈ R.

Proof.

Lemma 10.2.6. If g : R→ R are differentiable at a and g(a) 6= 0, then 1/g is alsodifferentiable at a, and

(1/g)′(a) =−g′(a)

[g(a)]2

Proof.

Theorem 10.2.7. (Quotient Rule) If f, g : R → R are differentiable at a andg(a) 6= 0, then f/g is also differentiable at a, and

(f/g)′(a) =g(a) · f ′(a)− f(a) · g′(a)

[g(a)]2

Proof.

Theorem 10.2.8. (Chain Rule) If g : R→ R is differentiable at a, and f : R→ Rat g(a), then f ◦ g differentiable at a, and

(f ◦ g)′(a) = f ′(g(a)) · g′(a)

for every a ∈ R.

Proof.

10.3. SIGNIFICANCE OF THE DERIVATIVE 73

10.3 Significance of the Derivative

Maximum Value

A function f :⊂ R→ R be a function. A point x ∈ A is a maximum pointfor f on A if

f(x) ≥ f(y)

for every y ∈ A. The number f(x) itself is called the maximum value of fon A.

10.4 Inverse Functions

74 CHAPTER 10. DIFFERENTIATION

Chapter 11

Applications of the Derivatives

75

76 CHAPTER 11. APPLICATIONS OF THE DERIVATIVES

Chapter 12

Integration

12.1 Riemann Sums

12.2 The Fundamental Theorem of Calculus

77

78 CHAPTER 12. INTEGRATION

Chapter 13

Techniques of Integration

79

80 CHAPTER 13. TECHNIQUES OF INTEGRATION

Bibliography

[1] L. Lamport. LATEX A Document Preparation System Addison-Wesley,California 1986.

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