- The document discusses differentiation and integration of algebraic functions.
- It provides rules for finding the derivatives of functions such as y = xn, y = axn, and the sum or difference of functions.
- It also discusses that the derivative of a constant function is equal to 0.
- The document concludes by discussing integration as the reverse process of differentiation and provides rules for indefinite and definite integrals of simple algebraic functions.
Newton Cotes Integration Method, Open Newton Cotes, Closed Newton Cotes Gauss...HadiaZahid2
the description about weddle's rule and newton cotes method and Gaussian quadrature method of numerical computing course.It contains introduction , Rules and Examples
Linear equations in two variables. Please download the powerpoint file to enable animation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
Newton Cotes Integration Method, Open Newton Cotes, Closed Newton Cotes Gauss...HadiaZahid2
the description about weddle's rule and newton cotes method and Gaussian quadrature method of numerical computing course.It contains introduction , Rules and Examples
Linear equations in two variables. Please download the powerpoint file to enable animation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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.
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.
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.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
13. 1. Given that 𝑦 = 3𝑥2 − 𝑥3, find
𝑑𝑦
𝑑𝑥
.
A. 6𝑥 − 3𝑥4
B. 6𝑥 − 3
C. 3𝑥 − 3𝑥2
D. 6𝑥 − 3𝑥2
E. 6 − 3𝑥2
The correct answer is 6𝑥 − 3𝑥2
Solution
𝑦 = 3𝑥2
− 𝑥3
𝑑𝑦
𝑑𝑥
= 6𝑥 − 3𝑥2
14. 2. Evaluate 0
1
𝑥 + 1 1 − 𝑥 𝑑𝑥
A.
1
3
B. −
1
3
C.
2
3
D. −
2
3
E. 1
The correct answer is
2
3
Solution
0
1
𝑥 + 1 1 − 𝑥 𝑑𝑥
=
0
1
1 − 𝑥2
𝑑𝑥
= 𝑥 −
𝑥3
3
1
0
= 1 −
13
3
− 0 =
2
3
15. 3. Given that 𝑓 𝑥 = 2 −
1
𝑥
2
, find 𝑓′(1).
A. 2
B. 8
C. 4
D. 6
E. 5
The correct answer is 2
Solution
𝑓 𝑥 = 2 −
1
𝑥
2
→ 𝑓 𝑥 = 4 −
4
𝑥
+
1
𝑥2
𝑓 𝑥 = 4 − 4𝑥−1 + 𝑥−2
𝑓′ 𝑥 = 4𝑥−2
−2𝑥−3
𝑓′ 𝑥 =
4
𝑥2
−
2
𝑥3
𝑓′(1) =
4
12
−
2
13
= 2
16. 4. If 𝑓′(𝑥) = 2𝑥 − 1 and f 1 = 2, find 𝑓(𝑥)?
A. 𝑓 𝑥 = 𝑥2 − 𝑥 − 2
B. 𝑓 𝑥 = 𝑥2 + 𝑥 + 2
C. 𝑓 𝑥 = 𝑥2 − 𝑥 + 2
D. 𝑓 𝑥 = 2𝑥2
− 𝑥 + 2
E. 𝑓 𝑥 = 3𝑥2 − 𝑥 + 2
The correct answer is 𝑓 𝑥 = 𝑥2 − 𝑥 + 2
Solution
𝑓′
(𝑥) = 2𝑥 − 1
Integrating both sides with respect to
𝑥
𝑓 𝑥 =
2𝑥2
2
− 𝑥 + 𝑐
𝑓 𝑥 = 𝑥2 − 𝑥 + 𝑐
We were given that 𝑓 1 = 2
2 = 12
− 1 + 𝑐
𝑐 = 2
𝑓 𝑥 = 𝑥2
− 𝑥 + 2
17. 5. Find the gradient of 𝑦 = 2𝑥3 − 𝑥2 − 2𝑥 at 𝑥 = −1
A. 6
B. 8
C. 4
D. 6
E. 5
The correct answer is 6
Solution
𝑦 = 2𝑥3 − 𝑥2 − 2𝑥
𝑑𝑦
𝑑𝑥
= 6𝑥2 − 2𝑥 − 2
𝑑𝑦
𝑑𝑥
= 6 −1 2
− 2 −1 − 2
𝑑𝑦
𝑑𝑥
= 6
⟹ 𝑇ℎ𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑦 = 2𝑥3
− 𝑥2
− 2𝑥
is 6
18. 6. Find the derivative of 𝑦 =
𝑥4−3𝑥2
𝑥
.
A. 3𝑥2
− 6
B. 𝑥2 − 6𝑥
C. 𝑥2
− 3𝑥
D. 3𝑥2
− 6𝑥
E. 3𝑥2 + 6𝑥
The correct answer is3𝑥2 − 6𝑥
Solution
𝑦 =
𝑥4
− 3𝑥2
𝑥
This implies that
𝑦 =
𝑥4
𝑥
−
3𝑥2
𝑥
𝑦 = 𝑥3
− 3𝑥2
𝑑𝑦
𝑑𝑥
= 3𝑥2
− 6𝑥
19. 7. The curve 3𝑦 + 2𝑥2 = 6, passes through the point (1,
4
3
). Find the
gradient of the tangent to the curve at this point.
A. −
4
3
B.
4
3
C.
3
4
D. −
3
4
E. 4
Solution
3𝑦 + 2𝑥2
= 6
Making 𝑦 the subject we have;
𝑦 = 2 −
2
3
𝑥2
𝑑𝑦
𝑑𝑥
= −
4
3
𝑥
When 𝑥 = 1
𝑑𝑦
𝑑𝑥
= −
4
3
= 𝑚
20. 8. The curve 3𝑦 + 2𝑥2 = 6, passes through the point (1,
4
3
). Find the
equation of the tangent to the curve at this point.
A. 3𝑦 − 4𝑥 − 8 = 0
B. 3𝑦 + 4𝑥 + 8 = 0
C. 3𝑦 + 4𝑥 − 8 = 0
D. 4𝑦 + 3𝑥 − 8 = 0
E. 4𝑦 − 3𝑥 − 8 = 0
The correct answer is3𝑦 + 4𝑥 − 8 = 0
Solution
3𝑦 + 2𝑥2 = 6
Making 𝑦 the subject we have;
𝑦 = 2 −
2
3
𝑥2
𝑑𝑦
𝑑𝑥
= −
4
3
𝑥
When 𝑥 = 1
𝑑𝑦
𝑑𝑥
= −
4
3
= 𝑚
The equation of the tangent is given by;
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)
𝑦 −
4
3
= −
4
3
(𝑥 − 1)
3𝑦 − 4 = −4(𝑥 − 1)
3𝑦 − 4 = −4𝑥 + 4
The equation is;
3𝑦 + 4𝑥 − 8 = 0
21. 9. Evaluate the indefinite integral 3𝑥2 − 4𝑥3 𝑑𝑥
A. 𝑥3
− 𝑥4
B.
𝑥3
3
−
𝑥4
4
+ 𝐶
C. 𝑥3
+ 𝑥4
+ 𝐶
D. 3𝑥3
− 4𝑥4
+ 𝐶
E. 𝑥3
− 𝑥4
+ 𝐶
The correct answer is𝑥3 − 𝑥4 + 𝐶
Solution
3𝑥2
− 4𝑥3
𝑑𝑥
=
3𝑥3
3
−
4𝑥4
4
+ 𝐶
= 𝑥3 − 𝑥4 + 𝐶
22. 10. Evaluate 0
1
3𝑥2 + 4𝑥3 𝑑𝑥
A. −2
B. 2
C. 3
D. 4
E. -3
The correct answer is 2
Solution
0
1
3𝑥2 + 4𝑥3 𝑑𝑥
=
3𝑥3
3
+
4𝑥4
4
1
0
= 𝑥3
+ 𝑥4 1
0
= 13 + 14 − 0 = 2
23. SUMMARY
If we have 𝑦 = 𝑥 𝑛, then
𝑑𝑦
𝑑𝑥
= 𝑛𝑥 𝑛−1
If we have 𝑦 = 𝑥 𝑛, then
𝑥 𝑛 𝑑𝑥 =
1
𝑛 + 1
𝑥 𝑛+1 + 𝐶, 𝑛 ≠ −1
Integration is the reverse of differentiation