After watching this ppt you will get answers of the questions like...
1) What does it mean?
2) What we study in calculus?
3) Who invented it?
4) What was the need to invent it?
and many more...
You will also learn about the basic difference between discrete and continuous.
And many real life and cool applications of calculus....
critical points/ stationary points , turning points,Increasing, decreasing functions, absolute maxima & Minima, Local Maxima & Minima , convex upward & convex downward - first & second derivative tests.
After watching this ppt you will get answers of the questions like...
1) What does it mean?
2) What we study in calculus?
3) Who invented it?
4) What was the need to invent it?
and many more...
You will also learn about the basic difference between discrete and continuous.
And many real life and cool applications of calculus....
critical points/ stationary points , turning points,Increasing, decreasing functions, absolute maxima & Minima, Local Maxima & Minima , convex upward & convex downward - first & second derivative tests.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
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!
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
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.
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.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
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.
2. 1. COMPLEX NUMBERS.
Some second degree equations have no solution, or at least not in the real number’s
group. These are the square roots of negative numbers.
푥2 + 푥 + 1 = 0 푥 =
−1 ± −3
2
So we have a bigger number’s group called the COMPLEX NUMBER’s group.
3. 1. COMPLEX NUMBERS.
We represent the imaginary unit as “i”. And the value of “i” is 푖 = −1
−9 = 9 ∙ −1 = 9 ∙i = 3i
−25 = 25 ∙ −1 = 25 ∙i = 5i
We represent a complex number as a combination of two numbers. It is called the
binomic form of the complex number.
푧 = 푎 + 푏푖 푤ℎ푒푟푒 푎, 푏 ∈ ℝ
푎 푖푠 푡ℎ푒 푟푒푎푙 푝푎푟푡 표푓 푡ℎ푒 푐표푚푝푙푒푥 푛푢푚푏푒푟
푏 푖푠 푡ℎ푒 푖푚푎푔푖푛푎푟푦 푝푎푟푡 표푓 푡ℎ푒 푐표푚푝푙푒푥 푛푢푚푏푒푟
5. 1. COMPLEX NUMBERS.
The opposite complex number; is the complex number that has the real and the
imaginary members with the signs changed.
푧 = 푎 + 푏푖 ; −푧 = −푎 − 푏푖 푧 = 2 − 3푖; −푧 = −2 + 3푖
푧 = −1 + 푖; −푧 = 1 − 푖
Conjugate of a complex number:
푧 = 푎 + 푏푖 ; 푧 = 푎 − 푏푖 푧 = 2 − 3푖; 푧 = 2 + 3푖
푧 = −1 + 푖; 푧 = −1 − 푖
6. 1. COMPLEX NUMBERS.
2. Graphic representation of the complex numbers.
A complex number can be viewed as a point or position vector in a two-dimensional
Cartesian coordinate system called the complex plane.
The numbers are conventionally plotted using the real part as the horizontal
component, and imaginary part as vertical .These two values used to identify a given
complex number are therefore called its Cartesian, rectangular, or algebraic form.
Ardatz erreala
Ardatz irudikaria
Z1= 3+2i
Z2= -4-i
9. 1. COMPLEX NUMBERS.
4. Complex number’s forms:
4.3. The trigonometric form;
푧1 = 푟 ∙ 푐표푠휑 + 푠푖푛휑 푖
In this case, take care that: 푧1 = 푟 ∙ 푐표푠휑 + 푠푖푛휑 푖 = r 푐표푠휑 + 푟 푠푖푛휑 푖 = a + bi
4.4. The affix form; 푧1 = 푎, 푏
10. 2. REAL NUMBER’S SEQUENCES
In mathematics, informally speaking, a sequence is an ordered list of objects (or
events).
It contains members (also called elements, or terms); a1, a2, …, an.
The terms of a sequence are commonly denoted by a single variable, say an, where
the index n indicates the nth element of the sequence.
Indexing notation is used to refer to a sequence in the abstract. It is also a natural
notation for sequences whose elements are related to the index n (the element's
position) in a simple way
11. 2. REAL NUMBER’S SEQUENCES
Examples;
• an=1/n is the next sequence: 1, ½, 1/3, ¼, 1/5, 1/6, …)
• If we have a1=3, and an+1= an+2, we obtein the next sequence: 3, 5, 7, 9, … where
the general term is an=2n+1
• A sequence can be constant if all the terms have the same value; for instance:
(-3, -3, -3, …), so in this case an=-3. See that the general term hasn’t any variable
n.
12. 2. REAL NUMBER’S SEQUENCES
Definitions:
A sequence 푎푛 푛∈ℕi s said to be monotonically increasing if each term is greater
than or equal to the one before it. For a sequence 푎푛 푛∈ℕ, this can be written as
푎푛 ≤ 푎푛+1, ∀푛 ∈ 푁.
If each consecutive term is strictly greater than (>) the previous term then the
sequence is called strictly monotonically increasing
A sequence 푎푛 푛∈ℕi s said to be monotonically decreasing if each term is less than
or equal to the previous one. For a sequence 푎푛 푛∈ℕ, this can be written as
푎푛 ≥ 푎푛+1, ∀푛 ∈ 푁.
If each consecutive term is strictly less than the previous
term then the sequence is called strictly monotonically
decreasin
13. 2. REAL NUMBER’S SEQUENCES
Definitions:
If a sequence is either increasing or decreasing it is called a monotone sequence.
This is a special case of the more general notion of a monotonic function.
Examples:
푎푛 = 2푛 + 1, (3, 5, 7, 9 … ) is a monotonically increasing sequence.
푎푛 =
1
푛
, (1, 1
2 , 1
3 , 1
4 … ) is a monotonically decreasing sequence.
14. 2. REAL NUMBER’S SEQUENCES
Definitions:
If the sequence of real numbers 푎푛, is such that all the terms, after a certain one, are
less than some real number M, then the sequence is said to be bounded from above.
In less words, this means 푎푛 ≤ 푀, ∀푀 ∈ 푁. Any such k is called an upper bound.
Likewise, if, for some real m, 푎푛 ≥ 푚, ∀푚 ∈ 푁, then the sequence is bounded from
below and any such m is called a lower bound.
If a sequence is both bounded from above and bounded from below then the
sequence is said to be bounded.
The sequence 푎푛 =
1
푛
is bounded from above, because all the
elements are less tan 1.
15. 3. Limit of a SEQUENCE.
One of the most important properties of a sequence is convergence.
Informally, a sequence converges if it has a limit.
Continuing informally, a (singly-infinite) sequence has a limit if it approaches some
value L, called the limit, as n becomes very large.
lim
푛→∞
푎푛 = 퐿
If a sequence converges to some limit, then it is convergent; otherwise it is
divergent.
16. 3. Limit of a SEQUENCE.
• If an gets arbitrarily large as n → ∞ we write
lim
푛→∞
푎푛 = ∞
In this case the sequence (an) diverges, or that it converges to infinity.
• If an becomes arbitrarily "small" negative numbers (large in magnitude) as n → ∞
we write
lim
푛→∞
푎푛 = −∞
and say that the sequence diverges or converges to minus infinity.
18. 3. Limit of a SEQUENCE.
• Usual cases:
lim
푛→∞
푘 = 퐾 ∀푘 ∈ ℜ
lim
푛→∞
푘
푛푝 = 0
∀푘 ∈ ℜ
∀푝 ∈ ℕ
lim
푛→∞
푝(푛)
푞(푛)
=
Where p(n) and q(n) are polinomies, the limit is the limit
of the division of the main grade of both polinomies.
If p(n)’s grade is greater, then the limit is infinity.
If 1(n)’s grade is greater, then the limit is 0.
If both have the same grade, then the limit is the
division of de coeficient of both polinomies.
19. 3. Limit of a SEQUENCE.
• Usual cases:
lim
푛→∞
1
푛
= 0
lim
푛→∞
1
푛 + 푎
= 0
lim
푛→∞
1
푛 + 푛 + 푎
= 0
lim
푛→∞
푛 − 푛 + 푎 = If we have the rest of tow square root, we will
multipicate and divide with the conjugate.
20. 4. The “e” number.
The number e is an important mathematical constant that is the base of the natural
logarithm.
It is approximately equal to 2.718281828, and is the limit of 1 +
1
푛
푛
as n approaches
infinity.
푒 = lim
푛→∞
1 +
1
푛
푛
It is a convergent sequence, and it is bounded from above.
21. 4. The “e” number.
We can find some sequence’s limits knowing the e number;
lim
푛→∞
1 +
1
푛
푘푛
= 푒푘 ∀푘 ∈ ℤ
lim
푛→∞
1 +
1
푛 + 푘
푛
= 푒 ∀푘 ∈ ℝ
lim
푛→∞
1 +
1
푛
푘+푛
= 푒 ∀푘 ∈ ℤ