1. The document discusses concepts in differential calculus including envelopes, asymptotes, and finding vertical, horizontal, and oblique asymptotes of curves.
2. It provides examples and steps for finding the envelope of a family of curves, and the oblique asymptotes of an algebraic curve using the highest degree terms.
3. Key aspects covered are parametrizing lines, using partial derivatives to eliminate parameters for envelopes, and determining asymptotes by examining the limiting behavior of curves as they tend toward infinity.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
TIU CET Review Math Session 6 - part 2 of 2youngeinstein
College Entrance Test Review
Math Session 6 - part 2 of 2
FUNCTIONS
How to evaluate
Operations on functions
Composite functions
Trigonometric Functions
Pythagorean Theorem
30 60 90 triangle
45 45 90 triangle
Exponential Functions
Logarithmic Functions
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.
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.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
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.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
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.
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
3. Finding the Envelope
• Family of curves given by F(x,y,a) = 0
• For each a the equation defines a curve
• Take the partial derivative with respect to a
• Use the equations of F and Fa to eliminate the parameter
a
• Resulting equation in x and y is the envelope
4. Parametrize Lines
• L is the length of ladder
• Parameter is angle a
• Note x and y intercepts
1sincos aa L
y
L
x
Lyx
aa sincos
7. Example: No intersections
• Start with given ellipse
• At each point construct the osculating circle (radius =
radius of curvature)
• Original ellipse is the envelope of this family of circles
• Neighboring ellipses are disjoint!
15. VERTICAL ASYMPTOTES
The line x = a is a vertical asymptote of the graph of the
function
y = ƒ (x) if at least one of the following statements is true:
Lim f (x) =+∞ or -∞ or Lim f (x)=+∞ or -∞
X → a+ X → a-
17. HORIZONTAL ASYMPTOTES
Horizontal asymptotes are horizontal lines that the graph
of the function approaches as x → ±∞. The horizontal
line y = c is a horizontal asymptote of the function
y = ƒ(x)
if Lim f(x)=c
or
Lim f(x)=c
In the first case, ƒ(x) has y = c as asymptote when x
tends to −∞, and in the second that ƒ(x) has y = c as an
asymptote as x tends to +∞
x→∞
x→ -∞
19. OBLIQUE ASYMPTOTES
When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or
slant asymptote. A function f(x) is asymptotic to the straight line y = mx + n (m ≠ 0)
if Lim [f (x) - (mx+n) ]=0
Or
Lim [f (x) - (mx+n) ]=0
X → -∞
x→∞
21. WORKING RULETO FIND OBLIQUE
ASYMPTOTESOF AN ALGEBRAIC CURVES
1. In the highest degree terms,put x=1,y=m and obtain Φ
(m).Then in the next highest degree term again put
x=1,y=m to obtain Φ (m) and so on.
2. Put Φ (m)=0,then its roots say m1,m2…. are the slopes of
the asymptotes.
n
n-1
n
22. 3.For the non repeated roots of Φ (m)=0,find c from the relation
c=-Φ (m)/Φ (m) for each value
of m.
The required asymptotes are
y=m1x+c1
y=m2x+c2
y=m3x+c3 …………
n
n-1 n
23. 4.If Φ (m)=0 for some value of m but
Φ (m)≠0 ,then there is no asymptotes corresponding to that
value of m.
5.If Φ "'(m)=0 and also Φ (m)=0 which is the case when two of
the asymptotes are parallel, then find c from the equation
(c²/2!)Φ (m)+ cΦ (m)+Φ (m)=0
which gives two values of c.Thus there are two parallel
asymptotes corresponding to this value of m.
n
n-1
n
‘
n-1
n n-1 n-2
‘
′
″
′
24. 6.If Φ (m)=Φ (m)=Φ (m)=0,then the values of c corresponding
to this value of m are determined from the equation
(c³/3!)Φ (m)+(c²/2!)Φ (m)+cΦ’ (m)=0.
n
n-1 n-2
n n-1 n-1
″
″′
″
′
26. Find the asymptotes of the curve
x³+3x²y-4y³-x+y+3=0?
SOLUTION:
The given curve is
x³+3x²y-4y³-x+y+3=0
To find the oblique asymptotes
Putting x=1,y=m in the third, second and first degree
terms one by one, we get
27. Φ (m)=1+3-4m³
Φ (m)=0
Φ (m)=-1+m
Now Φ (m)=0
1+3m-4m³=0
(1-m)(1+4m+4m²)=0
m=1,m=-½,m=-½
Also Φ (m)=3-12m²
and Φ (m)=-24m
3
2
1
3
3
′
3
″
28. therefore, c=-Φ (m)/Φ (m)
=-Φ (m)/Φ (m)
=-(0/(3-12m²))
When m=1, c=-(0/(3-12))=0
When m=-1/2
c=-(0/(3-3))=0
Therefore in this case, c is given by
(c²/2!)Φ (m)+cΦ (m)+Φ(m)=0
(c²/2!)(-24m)+c.0+(m-1)=0
n-1 n
′
2 3
′
3
″
2
′
1
