Let f be continuous on the interval [0, 1] to R and such that f(0)=f.pdf
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Let f be continuous on the interval [0, 1] to R and such that f(0)=f(1). Prove that there exists a point c in [0, 1/2] such that f(c) = f(c + 1/2). Solution This is actually Rolles theorm which is true for all numbers that satisfy the given condition.
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![Let f be continuous on the interval [0, 1] to R and such that f(0)=f(1). Prove that there exists a
point c in [0, 1/2] such that f(c) = f(c + 1/2).
Solution
This is actually Rolles theorm which is true for all numbers that satisfy the given
condition](https://image.slidesharecdn.com/letfbecontinuousontheinterval01torandsuchthatf0f-230323140537-a9dcb8e8/85/Let-f-be-continuous-on-the-interval-0-1-to-R-and-such-that-f-0-f-pdf-1-320.jpg)
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Let F(x,y) be the statement x can fool y,where the domainconsi.pdf
Let F(x,y) be the statement \"x can fool y,\"where the domainconsists of all people in the world.
Use quantifiers to expresseach of these statements.
1) Everybody can fool fred.
2) Evelyn can fool everybody.
3) Everybody can fool somebody.
4) There is no one who can fool everybody
5) Everyone can be fooled by somebody.
6) No one can fool both fred and jerry.
7) Nancy can fool exactly two people.
8) There is exactly one person whom everybody can fool.
9) No one can fool himself or herself.
10)There is someone who can fool exactly one person besideshimself or herself.
Let F(x,y) be the statement \"x can fool y,\"where the domainconsists of all people in the world.
Use quantifiers to expresseach of these statements.
1) Everybody can fool fred.
2) Evelyn can fool everybody.
3) Everybody can fool somebody.
4) There is no one who can fool everybody
5) Everyone can be fooled by somebody.
6) No one can fool both fred and jerry.
7) Nancy can fool exactly two people.
8) There is exactly one person whom everybody can fool.
9) No one can fool himself or herself.
10)There is someone who can fool exactly one person besideshimself or herself.
Solution
1. for all x, F(x, fred) 2. for all x, F(evelyn, x) 3. for all x, there exists y, F(x, y) 4
not there exists y, for all x, F(y, x) [or,equivalently, for all y, there exists x, not F(y, x)] 5 for all
x, there exists y, F(y, x) 6 not there exists x, F(x, fred) and F(x, jerry) 7 (there exists x, there
exists y, x not equal y andF(nancy, x) and F(nancy, y)) and not (there exists x, there existsy,
there exists z, x not equal y, x not equal z, y not equal z, andF(nancy, x) and F(nancy, y) and
F(nancy, z)) 8 (there exists x, for all y F(y, x)) and not (thereexists x, there exists y, x not equal
y, and for all z F(z, x) andF(z, y)) 9 not there exists x, F(x, x) 10 there exists x, there exists y, x
not equal y and F(x, y)and for all z not equal y, x, not F(x, z).
Let F(x, y) be the statement x can fool y, where the universe of.pdf
Let F(x, y) be the statement \"x can fool y,\" where the universe of discourse consists of all the
human beings on Earth. Translate the statement into plain English:
a. There is someone that can fool everyone.
b. There is one person who can be fooled by everybody.
c. Everyone can fool somebody.
d. Everyone can be fooled by somebody.
Solution
it says for any x, x can fool y => this means that every person can fool atleast some
person. therefore (c).
let f(x) = csc^2xfind f(x) find f(4)SolutionTreat csc.pdf
let f(x) = csc^2x
find f\'(x)
find f\'(/4)
Solution
Treat csc x just like any other variable squared (dy/dx x^2 = 2x)
So dy/dx csc^2(x) is 2csc(x)
but now you have to take the derivative of the \"inside\" function, which is csc (x):
dy/dx csc(x) = -csc(x)cot(x)
multiply the two and you get your answer:
so f\'(x) = dy/dx (csc^2(x)) = -2csc^2(x)cot(x)
so f\'(pi/4)=-2*(2)(1)=-4.
Let f(x) and g(x) be increasing and Riemann integrable. h is defined.pdf
Let f(x) and g(x) be increasing and Riemann integrable. h is defined as h=f-g. Show that h is
Riemann integrable.
Solution
f(x) is increasing and Riemann integrable.
g(x) is also increasing and Riemann integrable.
As f(x) and g(x) are both Riemann integrable, we can say the limit of the sum of the partitioned
area exists.
This again implies the difference of the function also is Riemann integrable.
As f(x) - g(x) also will have limit as L1-L2 where L1 is Riemann limit of f(x) and L2 is Riemann
limit of g(x)
Thus h(x) = f(x) - g(x) also will have a limit L1-L2
Hence h(x) is Riemann integrable..
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Solution
Is it always true that f(x) less and equal to 1 for all x in the range of X? No,
f(x) is not a probability (like the cdf). f(x) can be bigger than 1. For example, for uniform
distribution on [0,1/2], f(x)=2 Does it have to be true for some x in the range of X?? Yes,
otheriwse if f(x)>1 always then 1=integral f(x) >1 contradiction.
Let F be a field. Prove that the only ideals of F are F and {0}S.pdf
Let F be a field. Prove that the only ideals of F are F and {0}
Solution
It should be clear that {0} is an ideal of F. Now, suppose I is an ideal of F containing some
nonzero element x. Then x is a unit, so it has an inverse, call it x^-1. Then, since I is an ideal, it
contains x(x^-1) = 1. Now, let y ? F. Since I is an ideal, we have y = 1(y) ? I. Thus, F = I..
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Let c represent \"I was my care,\" let r represent \"it rains\" and let p represent \"the play is
postponed\" Write the compound statement in symbols.
If it does not rain, then I was my car.
The statement is written symbolically as ~____---->_____
Solution
p and -p.
Let B= { 1+x, x+x^2, 1+x^2} be an ordered basis of P2. And let B b.pdf
Let B= { 1+x, x+x^2, 1+x^2} be an ordered basis of P2. And let B\' be the standard basis of P2.
Find the coordinate vector of p(x) = 1+2x-x^2 with respect to B. Please show steps.
Solution
2+2x i +0 j+0 k
2+2[1+x]i
[4+2x ]i.
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Let F(x,y) be the statement x can fool y,where the domainconsi.pdf
Let F(x,y) be the statement \"x can fool y,\"where the domainconsists of all people in the world.
Use quantifiers to expresseach of these statements.
1) Everybody can fool fred.
2) Evelyn can fool everybody.
3) Everybody can fool somebody.
4) There is no one who can fool everybody
5) Everyone can be fooled by somebody.
6) No one can fool both fred and jerry.
7) Nancy can fool exactly two people.
8) There is exactly one person whom everybody can fool.
9) No one can fool himself or herself.
10)There is someone who can fool exactly one person besideshimself or herself.
Let F(x,y) be the statement \"x can fool y,\"where the domainconsists of all people in the world.
Use quantifiers to expresseach of these statements.
1) Everybody can fool fred.
2) Evelyn can fool everybody.
3) Everybody can fool somebody.
4) There is no one who can fool everybody
5) Everyone can be fooled by somebody.
6) No one can fool both fred and jerry.
7) Nancy can fool exactly two people.
8) There is exactly one person whom everybody can fool.
9) No one can fool himself or herself.
10)There is someone who can fool exactly one person besideshimself or herself.
Solution
1. for all x, F(x, fred) 2. for all x, F(evelyn, x) 3. for all x, there exists y, F(x, y) 4
not there exists y, for all x, F(y, x) [or,equivalently, for all y, there exists x, not F(y, x)] 5 for all
x, there exists y, F(y, x) 6 not there exists x, F(x, fred) and F(x, jerry) 7 (there exists x, there
exists y, x not equal y andF(nancy, x) and F(nancy, y)) and not (there exists x, there existsy,
there exists z, x not equal y, x not equal z, y not equal z, andF(nancy, x) and F(nancy, y) and
F(nancy, z)) 8 (there exists x, for all y F(y, x)) and not (thereexists x, there exists y, x not equal
y, and for all z F(z, x) andF(z, y)) 9 not there exists x, F(x, x) 10 there exists x, there exists y, x
not equal y and F(x, y)and for all z not equal y, x, not F(x, z).
Let F(x, y) be the statement x can fool y, where the universe of.pdf
Let F(x, y) be the statement \"x can fool y,\" where the universe of discourse consists of all the
human beings on Earth. Translate the statement into plain English:
a. There is someone that can fool everyone.
b. There is one person who can be fooled by everybody.
c. Everyone can fool somebody.
d. Everyone can be fooled by somebody.
Solution
it says for any x, x can fool y => this means that every person can fool atleast some
person. therefore (c).
let f(x) = csc^2xfind f(x) find f(4)SolutionTreat csc.pdf
let f(x) = csc^2x
find f\'(x)
find f\'(/4)
Solution
Treat csc x just like any other variable squared (dy/dx x^2 = 2x)
So dy/dx csc^2(x) is 2csc(x)
but now you have to take the derivative of the \"inside\" function, which is csc (x):
dy/dx csc(x) = -csc(x)cot(x)
multiply the two and you get your answer:
so f\'(x) = dy/dx (csc^2(x)) = -2csc^2(x)cot(x)
so f\'(pi/4)=-2*(2)(1)=-4.
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Let f(x) and g(x) be increasing and Riemann integrable. h is defined as h=f-g. Show that h is
Riemann integrable.
Solution
f(x) is increasing and Riemann integrable.
g(x) is also increasing and Riemann integrable.
As f(x) and g(x) are both Riemann integrable, we can say the limit of the sum of the partitioned
area exists.
This again implies the difference of the function also is Riemann integrable.
As f(x) - g(x) also will have limit as L1-L2 where L1 is Riemann limit of f(x) and L2 is Riemann
limit of g(x)
Thus h(x) = f(x) - g(x) also will have a limit L1-L2
Hence h(x) is Riemann integrable..
Let f be the p.d.f of a continuous random variables. Is it always tr.pdf
Let f be the p.d.f of a continuous random variables. Is it always true that f(x) less and equal to 1
for all x in the range of X? Does it have to be true for some x in the range of X??
Solution
Is it always true that f(x) less and equal to 1 for all x in the range of X? No,
f(x) is not a probability (like the cdf). f(x) can be bigger than 1. For example, for uniform
distribution on [0,1/2], f(x)=2 Does it have to be true for some x in the range of X?? Yes,
otheriwse if f(x)>1 always then 1=integral f(x) >1 contradiction.
Let F be a field. Prove that the only ideals of F are F and {0}S.pdf
Let F be a field. Prove that the only ideals of F are F and {0}
Solution
It should be clear that {0} is an ideal of F. Now, suppose I is an ideal of F containing some
nonzero element x. Then x is a unit, so it has an inverse, call it x^-1. Then, since I is an ideal, it
contains x(x^-1) = 1. Now, let y ? F. Since I is an ideal, we have y = 1(y) ? I. Thus, F = I..
Let c represent I was my care, let r represent it rains and .pdf
Let c represent \"I was my care,\" let r represent \"it rains\" and let p represent \"the play is
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If it does not rain, then I was my car.
The statement is written symbolically as ~____---->_____
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p and -p.
Let B= { 1+x, x+x^2, 1+x^2} be an ordered basis of P2. And let B b.pdf
Let B= { 1+x, x+x^2, 1+x^2} be an ordered basis of P2. And let B\' be the standard basis of P2.
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2+2x i +0 j+0 k
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Let E and F be two events for which one knows that the probability that atleast one of them
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Solution
P(A or B) = P(A) + P(B) - P(A & B) = .21+.34-.05 = .5 P(neither A nor B) = 1 -
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Let A= {-1, 0, 1}. Which of the following field properties does not hold for this set?
Identity for addition
Commutative for multiplication
Inverse for multiplication
Distributive
Identity for addition
Commutative for multiplication
Inverse for multiplication
Distributive
Solution
Inverse for multiplication
remaining all other option the set holds
you can verify it bi dividing the A in to 2 or more subsets
thank you.
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Let A be an n
Solution
As there are two eigen values lemda 1 and lemda 2 we must get n linearly independent eigen
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As for lemda 1 we have n-1 dimensional null space we have n-1 eigen vectors.
For lemda 2 atleast one eigen vector we have
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Let A be a set of n distinct elements. then the total number ofdistinct functions from A to A is
and out of these are onto functions
Solution
Each input has n possible outputs, and there are a total of ninputs so, the number of distinct
functions from A to A isnn.
The first input has n possible outputs, the next input has n-1possible outputs (all outputs
exculding the first selected), thenext has n-2 (all outputs excluding the first two
selected),continuing on in this vein, it can be concluded, the number ofdistinct onto functions is
n!.
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Let A = [ 2 1 0 3]
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Let A and B be events defined on a sample space S. Suppose that P(A) = 0.4, P(B) = 0.5, and the
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occur?
Solution
P(A) = 0.4 P(B) = 0.5 P(A and B) = 0.1 P(A or B) = P(A) + P(B) - P(A and B) =
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Let A and B be events defined on a sample space S. If the probability that at least one of them
occurs is 0.3 and the probability that A occurs but B does not is 0.1, what is P(B)?
Solution
P(AUB)=0.3
P(AUB\')=0.1
or P(B\')-P(A\'B\')=0.1
P(AUB)=P(A)+P(B)-P(AB)
P(A\'B\')=1-P(AUB)
=0.7
or 1-P(B)=0.8
or P(B)=0.2.
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Let A = IN, the set of all positive integers. In each case, give an
example of a function f : A -> A with the indicated properties, or
explain why no such function exists.
(a) f is not the identity function on A, but f is bijective.
(b) f is one-to-one, but not onto
Solution
as it form one to many relation ship . so it is not a function..
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Let A ? ? and define f: A ? ? by f(x) = ceil(x). Define the set A to ensure that f is injective on A.
Solution
since range of ceil function is all the integers.
so for f to be injective, f should be one-one.
and f to be one-one there should not exist two elements in domain that is in A such that when we
apply f on that elements output is same.
and it will be possible only when
A = (set of all integers).
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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.
Acetabularia Information For Class 9 .docx
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.
Synthetic Fiber Construction in lab .pptx
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.
TESDA TM1 REVIEWER FOR NATIONAL ASSESSMENT WRITTEN AND ORAL QUESTIONS WITH A...
TESDA TM1 REVIEWER FOR NATIONAL ASSESSMENT WRITTEN AND ORAL QUESTIONS WITH ANSWERS.
Honest Reviews of Tim Han LMA Course Program.pptx
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.
Embracing GenAI - A Strategic Imperative
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.
Overview on Edible Vaccine: Pros & Cons with Mechanism
This ppt include the description of the edible vaccine i.e. a new concept over the traditional vaccine administered by injection.
1.4 modern child centered education - mahatma gandhi-2.pptx
Child centred education is an educational approach that priorities the interest, needs and abilities of the child in the learning process.
Home assignment II on Spectroscopy 2024 Answers.pdf
Answers to Home assignment on UV-Visible spectroscopy: Calculation of wavelength of UV-Visible absorption
A Strategic Approach: GenAI in Education
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.
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...Nguyen Thanh Tu Collection
https://app.box.com/s/hqnndn05v4q5a4k4jd597rkdbda0fniiUnit 8 - Information and Communication Technology (Paper I).pdf
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.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdf
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.
The Roman Empire A Historical Colossus.pdf
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.
Francesca Gottschalk - How can education support child empowerment.pptx
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?
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Digital Tools and AI for Teaching Learning and Research
Digital Tools and AI for Teaching Learning and Research
TESDA TM1 REVIEWER FOR NATIONAL ASSESSMENT WRITTEN AND ORAL QUESTIONS WITH A...
TESDA TM1 REVIEWER FOR NATIONAL ASSESSMENT WRITTEN AND ORAL QUESTIONS WITH A...
Overview on Edible Vaccine: Pros & Cons with Mechanism
Overview on Edible Vaccine: Pros & Cons with Mechanism
1.4 modern child centered education - mahatma gandhi-2.pptx
1.4 modern child centered education - mahatma gandhi-2.pptx
Home assignment II on Spectroscopy 2024 Answers.pdf
Home assignment II on Spectroscopy 2024 Answers.pdf
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BÀI TẬP BỔ TRỢ TIẾNG ANH GLOBAL SUCCESS LỚP 3 - CẢ NĂM (CÓ FILE NGHE VÀ ĐÁP Á...
Unit 8 - Information and Communication Technology (Paper I).pdf
Unit 8 - Information and Communication Technology (Paper I).pdf
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Francesca Gottschalk - How can education support child empowerment.pptx
Francesca Gottschalk - How can education support child empowerment.pptx
Let f be continuous on the interval [0, 1] to R and such that f(0)=f.pdf
- 1. Let f be continuous on the interval [0, 1] to R and such that f(0)=f(1). Prove that there exists a point c in [0, 1/2] such that f(c) = f(c + 1/2). Solution This is actually Rolles theorm which is true for all numbers that satisfy the given condition