Introduction to the course MA1016 Mathematics I Raul Ignacio Ramos Garcia, Ph.D.

Syllabus 1) Numeric analysis, algebraic and geometric of functions and its applications Functions and their representation Linear function Nonlinear function Exponential function Logarithmic function Sine and cosine functions Compound functions 2) Limits and continuity Rational functions Intuitive idea of the limit of a function Infinite limits, horizontal asymptotic Infinite limits, vertical asymptotic Continuity of a function 3) Derivatives Derivative ad the rate of change through the geometric and numeric form Derivative of a function and its practical interpretation Derivative theorems Higher-order derivatives Problem solving for business and/or social sciences 4) Function Optimization Maximum and minimum theorems First derivative criterion Second derivative criterion Concave analysis and point of inflexion Problem solving for business and/or social sciences Dr. Raul RamosMA1016 Mathematics I2

Bibliography Text (recommended): Haeussler, E.F.; Paul, R.S.; Wood, R.W.; “Matemáticas para Administración y Economía”, 12 th Edition, Pearson Education. Galván, Cienfuegos, Elizondo, Fabela, Rodríguez, Romero; “Cálculo Diferencial para Administración y Ciencias Sociales”, 2 nd Edition, Pearson. Other: Blackboard. Dr. Raul RamosMA1016 Mathematics I3

Grading Policy Dr. Raul RamosMA1016 Mathematics I4 Three midterms60 % *In-class activities10 % Homework10 % Final20 % Course grade100 % * In-class activities: Set of exercises to work in class individually or in groups, must be turn in by the end of the class, a grade (0- 100) will be assign to the activity. The lowest grade from the set of activities will be dropped. No make-up in-class activities are allowed.

Policies Check list starts at the beginning of class. An absence is considered if a student arrives 10 minutes after the class has started; student may enter to take the class and work in the activities. The use of cell phones, iPads, and tables are only allowed if it is necessary for a class, otherwise these devices must be stored away. Penalty: student will be given an absence on that day. Regarding in-class activities, homework, and exams, the student must show all his or her work to get a full grade. Providing only the answer without showing the procedure will not be acceptable. Dr. Raul RamosMA1016 Mathematics I5

Why study this course? Dr. Raul RamosMA1016 Mathematics I6 Basis for future courses in business, social sciences, and engineering. Math surround us in every aspect of our lives. We can use math to create models and get an idea of the behavior of our model.

Lecture 1: Numeric analysis, algebraic and geometric of functions and its applications MA1016 Mathematics I Raul Ignacio Ramos Garcia, Ph.D.

Outline Functions Variables: independent and dependent Mapping of variables values Vertical line test Representation of functions Domain and range (or image) of a function Dr. Raul RamosMA1016 Mathematics I8

Functions and their representations Daily we encounter situations where two or more quantities are related using a rule or pattern. For example: The cost to publish a note in the news paper depends on the number of words of the note. The salary of an employee is given the number of hours he or she worked in a week. etc.. All these can be represented with a mathematical model! Dr. Raul RamosMA1016 Mathematics I9

Small exercise Example 1: When you talk in your cell phone, what would the cost of the call depend on? Example 2: A car salesman has a fix biweekly salary of $4000. Additionally, it receives a commission for every sold car. What would his salary depend on in his next paycheck? Dr. Raul RamosMA1016 Mathematics I10

We have described a function… Dr. Raul RamosMA1016 Mathematics I11

Example: When you talk in your cell phone, what would the cost of the call depend on? Dr. Raul RamosMA1016 Mathematics I12 Dependent variable Independent variable Variable: is a value that varies or change. Independent variable: is a quantity that you have control over, what you can choose and manipulate in a function. Dependent variable: represents the outcome or effect that the independent variable has in the function.

Example: A car salesman has a fix biweekly salary of $4000. Additionally, it receives a commission for every sold car. What would his salary depend on in his next paycheck? Dr. Raul RamosMA1016 Mathematics I13 ? ? ? ?

Example: A car salesman has a fix biweekly salary of $4000. Additionally, it receives a commission for every sold car. What would his salary depend on in his next paycheck? Dr. Raul RamosMA1016 Mathematics I14 Dependent variableIndependent variable

Is this a function?....Mapping variables Dr. Raul RamosMA1016 Mathematics I15 Independent VariableDependent Variable

Is this a function?....Mapping variables Dr. Raul RamosMA1016 Mathematics I16 Independent VariableDependent Variable

Is this a function? Dr. Raul RamosMA1016 Mathematics I17 Independent VariableDependent Variable

Is this a function? Dr. Raul RamosMA1016 Mathematics I18 Independent VariableDependent Variable

Vertical Line Test Dr. Raul RamosMA1016 Mathematics I19 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I20 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I21 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I22 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I23 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I24 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I25 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I26 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I27 Independent variable Dependent variable Vertical Line Test: If at least one value of x (independent variable) intersects with two values of y (dependent variable at the same time, then, it is not a function.

Dr. Raul RamosMA1016 Mathematics I28

Is this a function? Dr. Raul RamosMA1016 Mathematics I29 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I30 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I31 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I32 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I33 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I34 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I35 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I36 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I37 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I38 Independent variable Dependent variable

Dr. Raul RamosMA1016 Mathematics I39 Independent variable Dependent variable Yes! It’s a function

Dr. Raul RamosMA1016 Mathematics I40

Is it a function? (from tables) Table 1: Table 2: Dr. Raul RamosMA1016 Mathematics I

Is it a function? (from tables) Table 1: Table 2: Dr. Raul RamosMA1016 Mathematics I Yes! No!

Function representation Table with data: Equation: Graph: Dr. Raul RamosMA1016 Mathematics I43 x (independent variable)123 y (dependent variable)149

Dr. Raul RamosMA1016 Mathematics I44 xy

Dr. Raul RamosMA1016 Mathematics I45 xy Range or Image Domain

Domain and Range of a Function Domain: the set of possible values for the independent variable Range or Image: the set of values that the dependent variable can take. Dr. Raul RamosMA1016 Mathematics I46

Dr. Raul RamosMA1016 Mathematics I47 Dependent variable Independent variable DOMAIN RANGE

Dr. Raul RamosMA1016 Mathematics I48 xy To give the domain and range, just list the values without duplication

Dr. Raul RamosMA1016 Mathematics I49 xy

Dr. Raul RamosMA1016 Mathematics I50 xy

Dr. Raul RamosMA1016 Mathematics I51 xy

Discrete and Continuous Variables Dr. Raul RamosMA1016 Mathematics I52

Dr. Raul RamosMA1016 Mathematics I53

[1,5] = All values between (b/w) 1 and 5, including 1 and 5. (1,5] = All values b/w 1 and 5, excluding 1. [1,5) = All values b/w 1 and 5, excluding 5. (1,5) = All values b/w 1 and 5, excluding 1 and 5. Dr. Raul RamosMA1016 Mathematics I

Dr. Raul RamosMA1016 Mathematics I55 xy

Dr. Raul RamosMA1016 Mathematics I56

Dr. Raul RamosMA1016 Mathematics I57

Dr. Raul RamosMA1016 Mathematics I58

Dr. Raul RamosMA1016 Mathematics I59 Domain Range 1

Dr. Raul RamosMA1016 Mathematics I60

Dr. Raul RamosMA1016 Mathematics I61 xy

Review So, in a relationship of quantities we can ask: Is it a function? Which one is de independent and dependent variable? What is the domain of the function? Is the domain continuous or discrete? What is the range of a function? Is the range continuous or discrete? Dr. Raul RamosMA1016 Mathematics I62

Something to keep in mind… Dr. Raul RamosMA1016 Mathematics I63

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I64 January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I65 January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20 Which one is the independent variable? Which one is the dependent variable?

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I66 January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20 Which one is the independent variable? Which one is the dependent variable?

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I67 January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20 Is it a function?

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I68 Is it a function? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I69 Is it a function? Yes, it’s a function. January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I70 What is the domain? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I71 What is the domain? Is the domain discrete or continuous? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I72 What is the domain? Is the domain discrete or continuous? What is the range? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I73 What is the domain? Is the domain discrete or continuous? What is the range? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I74 What is the domain? Is the domain discrete or continuous? What is the range? Is the range discrete or continuous? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 1: Cars sold per month Dr. Raul RamosMA1016 Mathematics I75 What is the domain? Is the domain discrete or continuous? What is the range? Is the range discrete or continuous? January50 February0 March2 April16 May8 June79 July44 August0 September1 October3 November5 December20

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I76 Which one is the independent and dependent variable?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I77 Which one is the independent and dependent variable?

Example 1: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I78 Which one is the independent and dependent variable?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I79 Which one is the independent and dependent variable?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I80 Is it a function?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I81 Is it a function?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I82 Is it a function? Yes, it’s a function

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I83 What is the domain of the function?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I84 What is the domain of the function?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I85 What is the domain of the function? Is it continuous or discrete?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I86 What is the domain of the function? Is it continuous or discrete?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I87 What is the range of the function?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I88 What is the range of the function?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I89 What is the range of the function? Is it continuous or discrete?

Example 2: The cost of a cell phone bill depends on the time, in minutes, the cell phone is used to make a call. Suppose your cell phone has a credit of $100, which is equivalent to 40 minutes: Dr. Raul RamosMA1016 Mathematics I90 What is the range of the function? Is it continuous or discrete?

Example 3: Dr. Raul RamosMA1016 Mathematics I91 Is it a function? Yes What is the domain? Is it continuous or discrete? What is the range? Is it continuous or discrete?

Conclusion A function is a relationship, a set of rules, a pattern between quantities. A functions is a model than can help us to analyze a situation, make predictions and make better decisions. In a function we have variables: independent and dependent. In a function there is always a one to one correspondence between the independent and dependent variables. We can test a function using a map diagram, the vertical line test, looking at the values of a table. Variables, depending on the set of values, could be either discrete or continuous. The domain and range gives us the set of values independent and dependent variable can take, respectively. Dr. Raul RamosMA1016 Mathematics I92