Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
Students also learn to graph linear equations using x and y intercepts.
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
PROJECT (PPT) ON PAIR OF LINEAR EQUATIONS IN TWO VARIABLES - CLASS 10mayank78610
THIS A PROJECT BEING MADE BY INFORMATION COLLECTED FROM CLASS 10 MATHS NCERT BOOK.
THANK YOU FOR SEEING MY PROJECT ... I THINK THIS MIGHT HELP YOU IN YOUR HOLIDAY HOMEWORK PROJECTS .
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
Students also learn to graph linear equations using x and y intercepts.
En este archivo se muestran las consideraciones preliminares para entender limites, tal como factorización, racionalización y valor absoluto. El tema es iniciado con la definición intuitiva, los diferentes teoremas que se aplican en límites, la indeterminación 0/0 y los diversos ejemplos al respecto
PROJECT (PPT) ON PAIR OF LINEAR EQUATIONS IN TWO VARIABLES - CLASS 10mayank78610
THIS A PROJECT BEING MADE BY INFORMATION COLLECTED FROM CLASS 10 MATHS NCERT BOOK.
THANK YOU FOR SEEING MY PROJECT ... I THINK THIS MIGHT HELP YOU IN YOUR HOLIDAY HOMEWORK PROJECTS .
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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3. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
𝑹𝒆𝒄𝒐𝒏𝒐𝒄𝒆𝒓 𝒍𝒂 𝒊𝒅𝒆𝒂
𝒅𝒆 𝒍í𝒎𝒊𝒕𝒆
𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒓 𝒆𝒍 𝒍í𝒎𝒊𝒕𝒆
𝒅𝒆 𝒖𝒏𝒂 𝒇𝒖𝒏𝒄𝒊ó𝒏
𝑨𝒑𝒍𝒊𝒄𝒂𝒓 𝒆𝒍 𝒎𝒂𝒓𝒄𝒐
𝒕𝒆𝒐𝒓𝒊𝒄𝒐 𝒆𝒏 𝒍𝒂 𝒓𝒆𝒔𝒐𝒍𝒖𝒄𝒊ó𝒏
𝒅𝒆 𝒑𝒓𝒐𝒃𝒍𝒆𝒎𝒂𝒔.
4. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
SARS-CoV-2 y las matemáticas
Modelo SIR :Es un modelo matemático que nos
permite predecir el comportamiento de una
enfermedad infecciosa a partir de ciertas
condiciones iniciales.
El modelo SIR clasifica a una población en tres
grupos distintos S(Susceptible), I(infectados) y
R(recuperados).
La principal propiedad del modelo SIR, aplicado a
SARS-CoV-2 o a cualquier otro patógeno, es que la
propagación de la enfermedad termina con el paso
del tiempo (es decir, la enfermedad es no
endémica), dicho de otra forma:
𝐼∞ = lim
𝑡→∞
𝐼 𝑡 = 0
𝑆(𝑡) 𝐼(𝑡) 𝑅(𝑡)
5. C R E E M O S E N L A E X I G E N C I A
𝑥
𝑓(𝑥)
Noción del límite
C U R S O D E Á L G E B R A
Sea la función: 𝑓 𝑥 = 𝑥 + 4
Ahora en ves de decir que 𝑥 se
aproxima al cero, podemos decir
también que 𝑥 tiende al cero y en ves de
decir 𝑓 se aproxima al 4, podemos decir
que 𝑓 tiende al 4 .
Observación
Tratemos de acercarnos al cero por la derecha
0,5 1
0,001
−1
…
4,1
0,1
0
0,01
4,01 4,001 …
Notamos que mientras 𝑥 se aproxima al cero por la
derecha la función se aproxima al número 4.
𝑥
𝑓(𝑥)
Tratemos de acercarnos al cero por la izquierda
−0,001 …
3,9
−0,1 −0,01
3,99 3,999 …
Notamos que mientras 𝑥 se aproxima al cero por la
izquierda la función se aproxima al número 4.
Veamos el siguiente ejemplo:
𝑓 𝑥 =
𝑥2
− 1
𝑥 − 1
, 𝑥 ≠ 1
𝑓 𝑥 =
(𝑥 + 1)(𝑥 − 1)
𝑥 − 1
, 𝑥 ≠ 1
𝑓 𝑥 = 𝑥 + 1, 𝑥 ≠ 1
Analicemos la siguiente función
−0,5
6. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Ahora graficamos la función:
Matemáticamente se expresa:
Pero también se observa que si 𝑥 se
aproxima al 1 por la derecha la función se
aproxima al 2, dicho de otra manera, si 𝑥
tiende al 1 por la derecha (1 < 𝑥), la
función tiende al 2.
𝑓 𝑥 = 𝑥 + 1, 𝑥 ≠ 1
1 2
−2 −1 3 4
−3 0
−1
𝑋
𝑌
1
2
3
Observamos que si 𝑥 se aproxima al 1
por la izquierda la función se aproxima
al 2, dicho de otra manera, si 𝑥 tiende al
1 por la izquierda (𝑥 < 1), la función
tiende al 2.
𝐿 = lim
(1<𝑥)
𝑓(𝑥) = 2
𝐿 = lim
(𝑥<1)
𝑓(𝑥) = 2
Matemáticamente se expresa:
7. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Límite laterales
𝑋
𝑌
𝑥0
𝐿
𝑏
𝑓(𝑏)
lim
𝑥→𝑥0
−
𝑓(𝑥) = 𝐿
Limite por la izquierda
Se da cuando la variable 𝑥 se aproxima o tiende a 𝑥0 por la
izquierda ( es decir 𝑥 < 𝑥0) y se denota:
Limite por la derecha
Se da cuando la variable 𝑥 se aproxima o tiende a 𝑥0 por la
derecha ( es decir 𝑥0 < 𝑥) y se denota:
lim
𝑥→𝑥0
−
𝑓(𝑥) = 𝐿
𝑋
𝑌
𝑥0
𝐿
𝑏
𝑓(𝑏)
8. C R E E M O S E N L A E X I G E N C I A
Ejercicios
↔
Según la grafica determine los siguientes
limites
C U R S O D E Á L G E B R A
𝑋
𝑌
3
4
2
−1
−4
lim
𝑥→2−
𝑓(𝑥) = lim
𝑥→2+
𝑓(𝑥) =
lim
𝑥→−4+
𝑓(𝑥) =
1
lim
𝑥→0−
𝑓(𝑥) =
lim
𝑥→0+
𝑓(𝑥) =
0
Teorema (unicidad del límite)
El límite de una función existe y es único cuando 𝑥 tiende a
𝑥0 si y solo si existen los limites laterelas y además son
iguales
𝑓
lim
𝑥→𝑥0
𝑓(𝑥) = 𝐿 lim
𝑥→𝑥0
+
𝑓(𝑥) = lim
𝑥→𝑥0
−
𝑓(𝑥)
Ejemplo
De la ejercicio anterior
lim
𝑥→0+
𝑓(𝑥)= lim
𝑥→0−
𝑓(𝑥) = 1 → lim
𝑥→0
𝑓(𝑥) = 1
como.
Observación
lim
𝑥→2−
𝑓(𝑥) ≠ lim
𝑥→2+
𝑓(𝑥)
Entonces diremos que el limite de 𝑓
cuando 𝑥 tiende a 2 no existe .
3 4
1 1
−1
= 𝐿
9. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Límite de una función
Dada la función 𝑓: 𝐴 → ℝ; 𝐿 ∈ ℝ y 𝑥0 ∈ ℝ
Se dice que 𝐿 es el límite de la función 𝑓(𝑥) cuando 𝑥 tiende
al 𝑥0 y se denotado por
lim
𝑥→𝑥0
𝑓(𝑥) = 𝐿
Ejemplos
Si 𝑓 𝑥 = 2𝑥 − 1
Calculo del límites
Para el calculo del límite se presentan los siguientes
casos:
lim
𝑥→5
(2𝑥 − 1) = 𝐿1
Si lim
𝑥→2
(
𝑥
𝑥 − 1
) = 𝐿2
𝑔 𝑥 =
𝑥
𝑥 − 1
Límites determinados Límites indeterminados
Formas indeterminadas:
Es cuando al evaluar la
función en el valor al cual
tiende "𝑥" este límite existe
o es un número real
Es cuando al evaluar la
función en el valor al cual
tiende "x" el límite no
existe, es decir da una
indeterminación
0
0
;
∞
∞
; 1∞
Ejemplos
1) lim
𝑥→5
(2𝑥 − 1)
lim
𝑥→2
(
𝑥
𝑥 − 1
)
lim
𝑥→−1
(
𝑥
3 − 𝑥 + 1
) =
−1
3 − (−1) + 1
=
1
3
Ejemplos
2)
3)
lim
𝑥→1
𝑥2
− 1
𝑥 − 1
lim
𝑥→4
𝑥 − 2
𝑥 − 4
NOTA
Se puede evitar la
indeterminación mediante
procesos algebraicos como
la factorización y la
racionalización.
= 2 5 − 1 = 9
=
2
2 − 1
= 2
=
12 − 1
1 − 1
=
0
0
=
4 − 2
4 − 4
=
0
0
10. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Ejercicio 1
Determine el valor de 𝑚 si el lim
𝑥→3
log𝑚 𝑥 + 𝑒𝑥−3
𝑥 − 1
= 1
Resolución
lim
𝑥→3
log𝑚 𝑥 + 𝑒𝑥−3
𝑥 − 1
=
log𝑚 3 + 𝑒3−3
3 − 1
= 1
=
log𝑚 3 + 1
2
=
log𝑚 3 + 1
2
log𝑚 3 = 1
∴ 𝑚 = 3
Ejercicio 2
Determine el siguiente limite lim
𝑥→4
𝑥 − 2
𝑥 − 4
Resolución
lim
𝑥→4
𝑥 − 2
𝑥 − 4
= lim
𝑥→4
𝑥 − 2
𝑥 − 4
𝑥 + 2
𝑥 + 2
= lim
𝑥→4
𝑥 − 4
𝑥 − 4 𝑥 + 2
= lim
𝑥→4
1
𝑥 + 2
=
1
4 + 2
=
1
4
∴ lim
𝑥→4
𝑥 − 2
𝑥 − 4
Como el limite es
indeterminado de la
forma 0
0
Evitaremos la
indeterminación
racionalizando la
función
11. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Teoremas
1)
2)
3)
Sean 𝑓, 𝑔 dos funciones y 𝐿, 𝑀 ∈ ℝ tales que
=
lim
𝑥→−1
(𝑥 + 10)
lim
𝑥→−1
(𝑥4)
4)
lim
𝑥→𝑥0
𝑓(𝑥) = 𝐿; lim
𝑥→𝑥0
𝑔(𝑥) = 𝑀
lim
𝑥→𝑥0
(𝑓 ± 𝑔) = lim
𝑥→𝑥0
𝑓 𝑥 ± lim
𝑥→𝑥0
𝑔 𝑥 = 𝐿 ± 𝑀
1) lim
𝑥→2
𝑥2
+ 𝑥4
𝑥
lim
𝑥→𝑥0
(𝑓. 𝑔) = lim
𝑥→𝑥0
𝑓 𝑥 . lim
𝑥→𝑥0
𝑔 𝑥 = 𝐿. 𝑀
lim
𝑥→𝑥0
𝑓(𝑥)
𝑔(𝑥)
=
lim
𝑥→𝑥0
𝑓 𝑥
lim
𝑥→𝑥0
𝑔 𝑥
=
𝐿
𝑀
; 𝑀 ≠ 0
lim
𝑥→𝑥0
𝑘 = 𝑘; 𝑘 ∈ ℝ
Ejemplos
= lim
𝑥→2
𝑥2
+ lim
𝑥→2
𝑥4
2
= lim
𝑥→2
𝑥2
+ 𝑥4
lim
𝑥→2
(𝑥)
= 22 + 24 2 = 40
2) lim
𝑥→−1
𝑥 + 10
𝑥4
= 9
=
lim
𝑥→−1
𝑥 + lim
𝑥→−1
10
(−1)4
=
−1 + 10
(−1)4
∴ lim
𝑥→2
𝑥2 + 𝑥4 𝑥 = 40
∴ lim
𝑥→−1
𝑥 + 10
𝑥4
= 9
12. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Limites al infinito
Analicemos las siguientes gráfica
Notamos:
Observación
Notamos
𝑋
𝑌
𝑓
𝑓 𝑥 =
1
𝑥
0
lim
𝑥→0+
𝑓 𝑥
lim
𝑥→0−
𝑓 𝑥
Veamos otro ejemplo, sea la función
0
lim
𝑥→0+
1
𝑥
lim
𝑥→0−
1
𝑥
lim
𝑥→+∞
1
𝑥
lim
𝑥→−∞
1
𝑥
lim
𝑥→±∞
𝑘
𝑥
= 0 ; 𝑘 ∈ ℝ lim
𝑥→±∞
𝑘
𝑥𝑛
= 0 ; 𝑘 ∈ ℝ; 𝑛 ∈ ℤ+
= +∞
= +∞
= +∞
= −∞
= 0
= 0
13. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Ejercicio:
Resolución
Dividiremos entre 𝑥2
al numerador y
denominador.
Determine el siguiente limite
lim
𝑥→+∞
𝑥2
− 3
2𝑥2 + 𝑥
lim
𝑥→+∞
𝑥2
− 3
2𝑥2 + 𝑥
= lim
𝑥→+∞
𝑥2 − 3
𝑥2
2𝑥2 + 𝑥
𝑥2
= lim
𝑥→+∞
1 −
3
𝑥2
2 +
1
𝑥
=
1
2
Teoremas
lim
𝑥→+∞
𝑎𝑥𝑛
+ 𝑎1𝑥𝑛−1
+ ⋯ + 𝑎𝑛
𝑏𝑥𝑚 + 𝑏1𝑥𝑚−1 + ⋯ + 𝑏𝑛
=
𝑎
𝑏
; 𝑠𝑖 𝑛 = 𝑚
0; 𝑠𝑖 𝑛 < 𝑚
𝑎
𝑏
∞; 𝑠𝑖 𝑛 > 𝑚
Ejemplo:
lim
𝑥→+∞
2 𝑥3 + 2𝑥 − 3
3 𝑥3 + 𝑥
Determine los siguientes limites:
=
2
3
lim
𝑥→+∞
𝑥2
+ 2𝑥 − 3
3𝑥5 + 𝑥
= 0
lim
𝑥→+∞
𝑥6
+ 2𝑥 − 3
3𝑥2 + 𝑥
= ∞
lim
𝑥→+∞
𝑥2
+2𝑥 − 3
3 𝑥2 + 𝑥
=
1
3
lim
𝑥→+∞
2𝑥 − 3
𝑥2 + 𝑥
= 0
0
0
1
Como el limite es
indeterminado
de la forma ∞
∞
Trasformaremos la
función para evitar
la indeterminación
14. C R E E M O S E N L A E X I G E N C I A
C U R S O D E Á L G E B R A
Limites notables
1)
2) lim
𝑥→±∞
1 +
𝑘
𝑥
𝑥
= 𝑒𝑥
;
lim
𝑛→+∞
𝑟𝑛 = 0; 𝑟 < 1
3)
lim
𝑥→0
1 + 𝑘𝑥
1
𝑥 = 𝑒𝑥
4)
lim
𝑥→0
𝑠𝑒𝑛𝑥 = 0 lim
𝑥→0
𝑐𝑜𝑠𝑥 = 1
5)
lim
𝑥→±∞
𝑠𝑒𝑛𝑥
𝑥
= 0 ; lim
𝑥→0
𝑠𝑒𝑛𝑥
𝑥
= 1
Ejercicios:
Determine los siguientes limites
lim
𝑥→+∞
1
2
𝑥
lim
𝑛→0
𝑠𝑒𝑛 3𝑛
𝑛
= 0
= lim
𝑛→0
3𝑠𝑒𝑛 3𝑛
3𝑛
= 3 lim
3𝑛→0
𝑠𝑒𝑛 3𝑛
3𝑛
1
= 3
lim
𝑥→+∞
1 +
10
𝑥
𝑥
= 𝑒10
lim
𝑥→+∞
𝑥 − 5
𝑥
𝑥
= 𝑒−5
= lim
𝑥→+∞
1 +
−5
𝑥
𝑥
lim
𝑥→+∞
𝑥
𝑥 + 1
𝑥+1
= 𝑒
lim
𝑥→+∞
𝑥+1
𝑥
𝑥+1
−1
= 𝑒
lim
𝑥→+∞
𝑥+1
−1
𝑥+1 = 𝑒
lim
𝑥→+∞
−1
= 𝑒−1
−5
10
lim
𝑥→+∞
𝑓 𝑥 𝑔 𝑥 = 𝑒
lim
𝑥→+∞
𝑔 𝑥 𝑓 𝑥 −1
15. w w w . a c a d e m i a c e s a r v a l l e j o . e d u . p e