introduction to the course ma1016 mathematics i raul ignacio ramos garcia, ph.d

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

MA1016 Mathematics I 2

Syllabus1) 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 Ramos

Bibliography• Text (recommended):

• Haeussler, E.F.; Paul, R.S.; Wood, R.W.; “Matemáticas para Administración y Economía”, 12th Edition, Pearson Education.

• Galván, Cienfuegos, Elizondo, Fabela, Rodríguez, Romero; “Cálculo Diferencial para Administración y Ciencias Sociales”, 2nd Edition, Pearson.

• Other:

• Blackboard.

Dr. Raul Ramos

Grading Policy

Dr. Raul Ramos

Three midterms 60 %*In-class activities 10 %Homework 10 %Final 20 %Course grade 100 %

* 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 Ramos

Why study this course?

Dr. Raul Ramos

• 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 IRaul 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 Ramos

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 Ramos

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 Ramos

We have described a function…

• In both earlier cases we have seen a relationship between two quantities, and that one quantity depends on the other.

• In mathematics, to describe such relationship we use the concept of function.

• Quantity is a function of quantity , if and only if (iff) relates to ONLY ONE value of .

Dr. Raul Ramos

• Example: When you talk in your cell phone, what would the cost of the call depend on?

Dr. Raul Ramos

𝐶𝑜𝑠𝑡𝑜𝑓 𝑐𝑒𝑙𝑙 h𝑝 𝑜𝑛𝑒= 𝑓𝑒𝑒𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒×𝑐𝑎𝑙𝑙 𝑖𝑛𝑚𝑖𝑛𝑢𝑡𝑒𝑠

𝑦 ¿ 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 × 𝑥

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 Ramos

𝑆𝑎𝑙𝑎𝑟𝑦=𝑐𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛×𝑐𝑎𝑟𝑠 𝑠𝑜𝑙𝑑+4000

? ? ? ?

• 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 Ramos

𝑆𝑎𝑙𝑎𝑟𝑦=𝑐𝑜𝑚𝑚𝑖𝑠𝑠𝑖𝑜𝑛×𝑐𝑎𝑟𝑠 𝑠𝑜𝑙𝑑+4000

𝑦 𝑥𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡× +¿¿

Dependent variable Independent variable

Is this a function?....Mapping variables

Dr. Raul Ramos

Independent Variable Dependent Variable

Is this a function?....Mapping variables

Dr. Raul Ramos

Yes!Only one of value of of every value of

Is this a function?

Dr. Raul Ramos

Is this a function?

Dr. Raul Ramos

No!There are two values of ( and ) for

Vertical Line Test

Dr. Raul Ramos

𝑥Independent variable

𝑦 Dependent variable

MA1016 Mathematics I 20Dr. Raul Ramos

The value of intersects with the read line in two values of ( and ). Hence, this is NOT a function.

𝑦 2

𝑦 1

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.

.𝑥 𝑖...𝑥𝑁

− 𝑦𝑘..𝑦 𝑗

.𝑦𝑀

Is this a function?

Dr. Raul Ramos

For every value of there is only one value of

Yes!It’s a function

.𝑥 𝑖...𝑥𝑁

.𝑦 𝑗

.𝑦𝑀

Is it a function? (from tables)

• Table 1:

• Table 2:

Dr. Raul Ramos

(independent variable ) 1 2 3 4 5

(dependent variable) 1 4 9 16 25

(independent variable ) -7 5 0 6 0

Is it a function? (from tables)

• Table 1:

• Table 2:

Dr. Raul Ramos

(independent variable ) 1 2 3 4 5

(independent variable ) -7 5 0 6 0

h𝑇 𝑒𝑟𝑒 𝑎𝑟𝑒𝑡𝑤𝑜𝑣𝑎𝑙𝑢𝑒𝑠𝑜𝑓 𝑦 (9𝑎𝑛𝑑100 ) 𝑓𝑜𝑟 𝑥=0 ,

Function representation

• Table with data:

• Equation:

• Graph:

Dr. Raul Ramos

x (independent variable) 1 2 3

y (dependent variable) 1 4 9

𝑦=𝑥2

𝑓 (𝑥)=𝑥2

𝑦= 𝑓 (𝑥 )→ 𝑦 𝑖𝑠𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑜𝑓 𝑥

𝑦=2 𝑥2+1

Rangeor

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 Ramos

𝑥Dependent variable

𝑦 Independent variable

DOMAIN DOMAIN

x y-3 19-2 9-1 30 11 32 93 19

𝐷𝑜𝑚𝑎𝑖𝑛 :{−3 ,−2 ,−1,0 ,1 ,2 ,3 }

𝑅𝑎𝑛𝑔𝑒 : {19 ,9 ,3,1 }

To give the domain and range, just list the values without duplication

𝑦=2 𝑥2+1

-2.5 13.5

-1.5 5.5

-0.5 1.5

0.5 1.5

1.5 5.5

2.5 13.5

𝑦=2 𝑥2+1

-2.5 13.5

-1.5 5.5

-0.5 1.5

0.5 1.5

1.5 5.5

2.5 13.5

𝐷𝑜𝑚𝑎𝑖𝑛 :{−3 ,−2.5 ,−2 ,−1.5 ,−1 ,−0.5 ,0,0.5 ,1,1.5 ,2,2.5 ,3 }

𝑅𝑎𝑛𝑔𝑒 : {19 ,13.5,9,5 .5 ,3,1}

Discrete and Continuous Variables• A variable is discrete if the number of possible values it can take is

finite. That is, the values can be listed.

• Example 1: Variable represents the number on a die, i.e. .

• Example 2: Variable represents the performance rate for questions in a survey, i.e. .

Dr. Raul Ramos

• A variable is continuous if the number of possible values it can take is infinite. That is, the values can not be listed.

• Example 1: Variable represents the temperature b/w and , i.e. .

• Example 2: Variable represents al possible values b/w 0 and 1, i.e.

Dr. Raul Ramos

• [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 Ramos

𝑦=2 𝑥2+1

-2.5 13.5

-1.5 5.5

-0.5 1.5

0.5 1.5

1.5 5.5

2.5 13.5

𝐷𝑜𝑚𝑎𝑖𝑛 :{−3 ,−2.5 ,−2 ,−1.5 ,−1 ,−0.5 ,0,0.5 ,1,1.5 ,2,2.5 ,3 }

𝑅𝑎𝑛𝑔𝑒 : {19 ,13.5,9,5 .5 ,3,1}

Is a continuous or discrete variable?

• But, for , what is the actual domain and range of the function?

Dr. Raul Ramos

Domain

𝐷𝑜𝑚𝑎𝑖𝑛 : (−∞ ,∞ )𝑜𝑟 𝐴𝑙𝑙 𝑟𝑒𝑎𝑙(ℝ )𝑛𝑢𝑚𝑏𝑒𝑟𝑠

𝑅𝑎𝑛𝑔𝑒 : [1 ,∞ )𝑜𝑟 𝐴𝑙𝑙𝑟𝑒𝑎𝑙 (ℝ )𝑛𝑢𝑚𝑏𝑒𝑟𝑠 h𝑤𝑖𝑡 𝑒𝑥𝑐𝑒𝑝𝑐𝑖𝑜𝑛𝑜𝑓 (1 ,−∞)

set of numbers formed from all positive and negative integers, zero, and rational numbers

𝑦=2 𝑥2+1

-2.5 13.5

-1.5 5.5

-0.5 1.5

0.5 1.5

1.5 5.5

2.5 13.5

Is a continuous or discrete variable?

𝐷𝑜𝑚𝑎𝑖𝑛 : (−∞ ,∞ )𝑜𝑟 𝐴𝑙𝑙 𝑟𝑒𝑎𝑙(ℝ )𝑛𝑢𝑚𝑏𝑒𝑟𝑠

Given the domain is a continuous variable

𝑅𝑎𝑛𝑔𝑒 : [1 ,∞ )𝑜𝑟 𝐴𝑙𝑙𝑟𝑒𝑎𝑙 (ℝ )𝑛𝑢𝑚𝑏𝑒𝑟𝑠 h𝑤𝑖𝑡 𝑒𝑥𝑐𝑒𝑝𝑐𝑖𝑜𝑛𝑜𝑓 (1 ,−∞ )

Given the range is a continuous variable

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 Ramos

Something to keep in mind…

• The domain and range of a function are not necessary numeric values, i.e. it could be also a set of words.

• Example 1: Variable represents a day of the week, i.e. .

• Example 2: Variable represents the performance rate for questions in a survey, i.e. .

Dr. Raul Ramos

• Example 1: Cars sold per month

Dr. Raul Ramos

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Which one is the independent variable?

Which one is the dependent variable?

Dr. Raul Ramos

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Which one is the independent variable?

Which one is the dependent variable?

h𝑀𝑜𝑛𝑡 =𝑥

𝐶𝑎𝑟𝑠 𝑠𝑜𝑙𝑑=𝑦

Dr. Raul Ramos

(independent

variable)

(dependent

variable)January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Is it a function?

Dr. Raul Ramos

Is it a function? (independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

Is it a function?

Yes, it’s a function.

(independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

What is the domain? (independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

What is the domain?

Is the domain discrete or continuous?

(independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

What is the domain?

𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒

What is the range?

(independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

What is the domain?

What is the range?

𝑅𝑎𝑛𝑔𝑒 : {0 ,1 ,2 ,3 ,4 ,…}

(independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

What is the domain?

What is the range?

𝑅𝑎𝑛𝑔𝑒 : {0 ,1 ,2 ,3 ,4 ,…}

Is the range discrete or continuous?

(independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

Dr. Raul Ramos

What is the domain?

What is the range?

𝑅𝑎𝑛𝑔𝑒 : {0 ,1 ,2 ,3 ,4 ,…}

Is the range discrete or continuous?

(independent

variable)

(dependent

variable)

January 50

February 0

March 2

April 16

June 79

July 44

August 0

September 1

October 3

November 5

December 20

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 Ramos

Which one is the independent and dependent variable?

Dr. Raul Ramos

* The cost is a function of time* The cost depends on time

Dr. Raul Ramos

Is it a function?

𝑐=100 𝑡40

=2.5 𝑡

Dr. Raul Ramos

Is it a function?

𝑐=100 𝑡40

=2.5 𝑡

Dr. Raul Ramos

Is it a function?

𝑐=100 𝑡40

=2.5 𝑡Yes, it’s a function

Dr. Raul Ramos

What is the domain of the function?

Dr. Raul Ramos

𝑡∈[0 ,40 ]

Dr. Raul Ramos

𝑡∈[0 ,40 ]

Is it continuous or discrete?

Dr. Raul Ramos

𝑡∈[0 ,40 ]

𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠

Dr. Raul Ramos

What is the range of the function?

Dr. Raul Ramos

𝑐∈[0 ,100 ]

Dr. Raul Ramos

𝑐∈[0 ,100 ]

Dr. Raul Ramos

𝑐∈[0 ,100 ]

𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠

Example 3:• represents a set of points

Dr. Raul Ramos

Is it a function? Yes

What is the domain? Is it continuous or discrete? {1 ,2.5 ,3 ,4.1 },𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒

What is the range? Is it continuous or discrete? {0 ,1 ,4 ,6 },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 Ramos

introduction to the course ma1016 mathematics i raul ignacio ramos garcia, ph.d

class activities10

beginning of class

set of activities

geometric of functions

business andor social

function optimizationmaximum

1syllabus1 numeric analysis

functioninfinite limits

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