introduction to the course ma1016 mathematics i raul ignacio ramos garcia, ph.d
TRANSCRIPT
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
MA1016 Mathematics I 3
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
MA1016 Mathematics I 4
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.
MA1016 Mathematics I 5
• 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
MA1016 Mathematics I 6
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.
MA1016 Mathematics I 8
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
MA1016 Mathematics I 9
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
MA1016 Mathematics I 10
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
MA1016 Mathematics I 11
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
MA1016 Mathematics I 12
• 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.
MA1016 Mathematics I 13
• 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
? ? ? ?
MA1016 Mathematics I 14
• 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
MA1016 Mathematics I 15
Is this a function?....Mapping variables
Dr. Raul Ramos
Independent Variable Dependent Variable
MA1016 Mathematics I 16
Is this a function?....Mapping variables
Dr. Raul Ramos
Independent Variable Dependent Variable
Yes!Only one of value of of every value of
MA1016 Mathematics I 17
Is this a function?
Dr. Raul Ramos
Independent Variable Dependent Variable
MA1016 Mathematics I 18
Is this a function?
Dr. Raul Ramos
Independent Variable Dependent Variable
No!There are two values of ( and ) for
MA1016 Mathematics I 19
Vertical Line Test
Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 20Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 21Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
𝑥1
MA1016 Mathematics I 22Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
𝑥1
MA1016 Mathematics I 23Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
The value of intersects with the read line in two values of ( and ). Hence, this is NOT a function.
𝑥1
𝑦 2
𝑦 1
MA1016 Mathematics I 24Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 25Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 26Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 27Dr. Raul Ramos
𝑥Independent variable
𝑦 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.
MA1016 Mathematics I 28Dr. Raul Ramos
.
.
.𝑥 𝑖...𝑥𝑁
− 𝑦𝑘..𝑦 𝑗
.
.
.𝑦𝑀
MA1016 Mathematics I 29
Is this a function?
Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 30Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 31Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 32Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 33Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 34Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 35Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 36Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 37Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
MA1016 Mathematics I 38Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
For every value of there is only one value of
MA1016 Mathematics I 39Dr. Raul Ramos
𝑥Independent variable
𝑦 Dependent variable
Yes!It’s a function
MA1016 Mathematics I 40Dr. Raul Ramos
.
.
.𝑥 𝑖...𝑥𝑁
.
.
.𝑦 𝑗
.
.
.𝑦𝑀
MA1016 Mathematics I 41
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
(dependent variable) 20 57 9 14 100
MA1016 Mathematics I 42
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
(dependent variable) 20 57 9 14 100
Yes!
No!
h𝑇 𝑒𝑟𝑒 𝑎𝑟𝑒𝑡𝑤𝑜𝑣𝑎𝑙𝑢𝑒𝑠𝑜𝑓 𝑦 (9𝑎𝑛𝑑100 ) 𝑓𝑜𝑟 𝑥=0 ,
MA1016 Mathematics I 43
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
MA1016 Mathematics I 44Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2 9
-1 3
0 1
1 3
2 9
3 19
MA1016 Mathematics I 45Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2 9
-1 3
0 1
1 3
2 9
3 19
Rangeor
Image
Domain
MA1016 Mathematics I 46
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
MA1016 Mathematics I 47Dr. Raul Ramos
𝑥Dependent variable
𝑦 Independent variable
DOMAIN DOMAIN
RAN
GE
RAN
GE
MA1016 Mathematics I 48Dr. Raul Ramos
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
MA1016 Mathematics I 49Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2 9
-1 3
0 1
1 3
2 9
3 19
MA1016 Mathematics I 50Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2.5 13.5
-2 9
-1.5 5.5
-1 3
-0.5 1.5
0 1
0.5 1.5
1 3
1.5 5.5
2 9
2.5 13.5
3 19
MA1016 Mathematics I 51Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2.5 13.5
-2 9
-1.5 5.5
-1 3
-0.5 1.5
0 1
0.5 1.5
1 3
1.5 5.5
2 9
2.5 13.5
3 19
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}
MA1016 Mathematics I 52
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
MA1016 Mathematics I 53
• 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
MA1016 Mathematics I 54
• [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
1 5
1 5
1 5
1 5
MA1016 Mathematics I 55Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2.5 13.5
-2 9
-1.5 5.5
-1 3
-0.5 1.5
0 1
0.5 1.5
1 3
1.5 5.5
2 9
2.5 13.5
3 19
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?
Is a continuous or discrete variable?
MA1016 Mathematics I 56
• But, for , what is the actual domain and range of the function?
Dr. Raul Ramos
𝑥
𝑦
MA1016 Mathematics I 57
• But, for , what is the actual domain and range of the function?
Dr. Raul Ramos
𝑥
𝑦
MA1016 Mathematics I 58
• But, for , what is the actual domain and range of the function?
Dr. Raul Ramos
𝑥
𝑦
MA1016 Mathematics I 59
• But, for , what is the actual domain and range of the function?
Dr. Raul Ramos
𝑥
𝑦
Domain
Range
1
MA1016 Mathematics I 60Dr. Raul Ramos
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
MA1016 Mathematics I 61Dr. Raul Ramos
𝑦=2 𝑥2+1
x y
-3 19
-2.5 13.5
-2 9
-1.5 5.5
-1 3
-0.5 1.5
0 1
0.5 1.5
1 3
1.5 5.5
2 9
2.5 13.5
3 19
Is a continuous or discrete variable?
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
MA1016 Mathematics I 62
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
MA1016 Mathematics I 63
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
MA1016 Mathematics I 64
• Example 1: Cars sold per month
Dr. Raul Ramos
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 65
• Example 1: Cars sold per month
Dr. Raul Ramos
January 50
February 0
March 2
April 16
May 8
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?
MA1016 Mathematics I 66
• Example 1: Cars sold per month
Dr. Raul Ramos
January 50
February 0
March 2
April 16
May 8
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
MA1016 Mathematics I 67
• Example 1: Cars sold per month
Dr. Raul Ramos
(independent
variable)
(dependent
variable)January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
Is it a function?
MA1016 Mathematics I 68
• Example 1: Cars sold per month
Dr. Raul Ramos
Is it a function? (independent
variable)
(dependent
variable)
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 69
• Example 1: Cars sold per month
Dr. Raul Ramos
Is it a function?
Yes, it’s a function.
(independent
variable)
(dependent
variable)
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 70
• Example 1: Cars sold per month
Dr. Raul Ramos
What is the domain? (independent
variable)
(dependent
variable)
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 71
• Example 1: Cars sold per month
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
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 72
• Example 1: Cars sold per month
Dr. Raul Ramos
What is the domain?
Is the domain discrete or continuous?
𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒
What is the range?
(independent
variable)
(dependent
variable)
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 73
• Example 1: Cars sold per month
Dr. Raul Ramos
What is the domain?
Is the domain discrete or continuous?
𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒
What is the range?
𝑅𝑎𝑛𝑔𝑒 : {0 ,1 ,2 ,3 ,4 ,…}
(independent
variable)
(dependent
variable)
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 74
• Example 1: Cars sold per month
Dr. Raul Ramos
What is the domain?
Is the domain discrete or continuous?
𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒
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
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 75
• Example 1: Cars sold per month
Dr. Raul Ramos
What is the domain?
Is the domain discrete or continuous?
𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒
What is the range?
𝑅𝑎𝑛𝑔𝑒 : {0 ,1 ,2 ,3 ,4 ,…}
Is the range discrete or continuous?
𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒
(independent
variable)
(dependent
variable)
January 50
February 0
March 2
April 16
May 8
June 79
July 44
August 0
September 1
October 3
November 5
December 20
MA1016 Mathematics I 76
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?
MA1016 Mathematics I 77
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?
MA1016 Mathematics I 78
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 Ramos
Which one is the independent and dependent variable?
MA1016 Mathematics I 79
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?
* The cost is a function of time* The cost depends on time
MA1016 Mathematics I 80
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
Is it a function?
𝑐=100 𝑡40
=2.5 𝑡
MA1016 Mathematics I 81
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
Is it a function?
𝑐=100 𝑡40
=2.5 𝑡
MA1016 Mathematics I 82
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
Is it a function?
𝑐=100 𝑡40
=2.5 𝑡Yes, it’s a function
MA1016 Mathematics I 83
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
What is the domain of the function?
MA1016 Mathematics I 84
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
What is the domain of the function?
𝑡∈[0 ,40 ]
MA1016 Mathematics I 85
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
What is the domain of the function?
𝑡∈[0 ,40 ]
Is it continuous or discrete?
MA1016 Mathematics I 86
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
What is the domain of the function?
𝑡∈[0 ,40 ]
Is it continuous or discrete?
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
MA1016 Mathematics I 87
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
What is the range of the function?
MA1016 Mathematics I 88
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
What is the range of the function?
𝑐∈[0 ,100 ]
MA1016 Mathematics I 89
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
What is the range of the function?
𝑐∈[0 ,100 ]
Is it continuous or discrete?
MA1016 Mathematics I 90
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
What is the range of the function?
𝑐∈[0 ,100 ]
Is it continuous or discrete?
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
MA1016 Mathematics I 91
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
MA1016 Mathematics I 92
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