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Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
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
YouTube Link: https://youtu.be/UoHu27xoTyc
** Machine Learning Engineer Masters Program: https://www.edureka.co/machine-learning-certification-training **
This Edureka PPT on Least Squares Regression Method will help you understand the math behind Regression Analysis and how it can be implemented using Python.
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Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
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
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
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
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
In this chapter, we shall study the nature of a
function which is governed by the sign of its derivative. If the graph of a function is in upward going direction or in downward coming direction then it is called as monotonic function, and this property of the function is called Monotonicity. If a function is defined in any interval, and if in some part of the interval, graph moves upwards
A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain.
NOTE : If x < x f(x ) < f(x ) x , x D, then
and in the remaining part moves downward then
1 2 1
2 1 2
function is not monotonic in that interval.
f(x) is called strictly increasing in domain D.
These are of two types –
2.1 Monotonic Increasing :
A function f(x) defined in a domain D is said to be monotonic increasing function if the value of f(x) does not decrease (increase) by increasing (decreasing) the value of x or
We can say that the value of f(x) should increase (decrease) or remain equal by increasing
(Decreasing) the value of x.
Similarly if x1 < x2 f(x1) > f(x2), x1, x2 D then it is called strictly decreasing in domain D.
R
If Tor x1 x2 f(x1) f(x2 )
, x1 , x2 D
or SR x1 x2 f(x1) f(x2 ) , x1 , x2 D
2.2 Monotonic Decreasing :
A function f(x) defined in a domain D is said to be monotonic decreasing function if the value of f(x) does not increase (decrease) by increasing (decreasing) the value of x or
We can say that the value of f(x) should decrease (increase) or remain equal by increasing
(Decreasing) the value of x.
For Example
(i) f(x) = ex is a monotonic increasing function where as g(x) = 1/x is monotonic decreasing function.
(ii) f(x) = x2 and g(x) = | x | are monotonic increasing for x > 0 and monotonic decreasing for x < 0. In general they are not monotonic functions.
(iii) Sin x, cos x are not monotonic function whereas tan x, cot x are monotonic.
(i) At a Point : A function f(x) is said to be monotonic increasing (decreasing) at a point x = a of its domain if it is monotonic increasing
R
If Tor x1 x2 f(x1) f(x2 )
, x1 , x2 D
(decreasing) in the interval (a – h, a + h) where h is a small positive number. Hence we may
or SR x1 x2 f(x1) f(x2 ) , x1 , x2 D
observe that if f(x) is monotonic increasing at x = a, then at this point tangent to its graph will make an acute angle with the x–axis where as if the function is monotonic decreasing these tangent will make
an obtuse angle with x–axis. Consequently f' (a) will be positive or negative according as f(x) is monotonic increasing or decreasing at x = a.
Ex. Function f(x) = x2 + 1 is monotonically decreasing in [ –1, 0] because
f' (x) = 2x < 0, x (–1, 0)
Ex. Function f(x) = x2 is not a monotonic function in the interval [–1, 1] because
f' (x) > 0, when x = 1/2
Ex. Function f(x) = sin2x + cos2x is constant
Computing integrals with Riemann sums is like computing derivatives with limits. The calculus of integrals turns out to come from antidifferentiation. This startling fact is the Second Fundamental Theorem of Calculus!
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
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.
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.
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.
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
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• 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
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.
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.
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. 3/20
• A function is a special kind of binary
relation.
• A binary relation f ⊆ A × B is a function if
for each a ∈ A there is a unique b ∈ B
Function Definition
1
2
3
α
β
γ
x
y
6. 6/20
An (n+1)-ary relation f ⊆ A1 × A2 × … × An
× B is a function if for each < a1, a2, …,
an> ∈ A1 × A2 × … × An there is a unique
b ∈ B.
Functions
with N-Dimensional Domains
α
β
γ
<1,1>
<1,2>
<1,3>
7. 7/20
• We can use various notation for functions: for
f = {(1, α),(2, β),(3, β)}
Notation for Functions
Notation (x, y) ∈ f f : x→y y = f(x)
Example (2, β) ∈ f f : 2→β β = f(2)
• In the notation, x is the argument or preimage
and y is the image. We can also have the
image of a set of arguments.
• For functions with n-ary domains, use <x0, x1,
…, xn> in place of x.
8. 8/20
Function Domain and Range
• f : A → B
– A is the domain space
• same as the domain (since all elements participate)
• dom f, dom(f), or domain(f)
– B is the range space
• may or may not be the same as the range, which is:
– {y | ∃x(y=f(x))}
– All rhs values in pairs (all that get “hit”)
πBf
• ran f, ran(f), range(f)
• f : D1 × D2 × … × Dn → Z
• f : Dn
→ Z (when all domains are the same)
9. 9/20
Remove the requirement that each a ∈ A must
participate. Retain the uniqueness requirement.
Partial Functions
Partial Function:
α
β
γ
f = {(<1,2>, β),(<1,3>, β),(<1,3>, γ)}
<1,3> not unique
<1,1>
<1,2>
<1,3>
α
β
γ
<1,1>
<1,2>
<1,3>
NOT a Partial Function:
α
β
γ
<1,1>
<1,2>
<1,3>
Partial Function: (A
Total Function is also a
Partial Function.)
10. 10/20
• Identity Function
– IA : A → A
– IA = {(x, x) | x ∈ A}
• Constant Function
– C : A → B
– C = {(x, c) | x ∈ A ∧ c ∈ B }
– Often A and B are the same
• C : A → A
• C= {(x, c) | x ∈ A ∧ c ∈ A}
Special Functions
11. Discussion #29 11/20
Composition of Functions
• Composition is written “°”
• Range space of f =
domain space of g
a
c
1
2
4
f
g
b
α
β
3
f(a) = 2 g(2) = α g(f(a)) = α
g°f(a) = α
f(b) = 2 g(2) = α g(f(b)) = α
g°f(b) = α
f(c) = 4 g(4) = β g(f(c)) = β
g°f(c) = β
12. Discussion #29 12/20
Injection: “one-to-one” or “1-1”
∀x∀y(f(x) = f(y) ⇒ x = y)
– For f : A → B, the elements in B are “hit” at most once
Injection
a
b
d
1
2
3
c
Injective
a
b
d
1
2
3
c
NOT Injective
x
y
x
y
13. 13/20
Surjection: “onto”
∀y∃x(y = f(x))
– For f : A → B, the elements in B are all “hit” at least once
Surjection
1
2
4
a
b
c
3
Surjective NOT Surjective
x
y
x
y
1
2
4
a
b
c
3
{not
“hit”
14. 14/20
Bijection: “one-to-one and onto” or “1-1 correspondence”
∀x∀y(f(x) = f(y) ⇒ x = y) ∧ ∀y∃x(y = f(x))
– For f : A → B, every B element is “hit” once and only once
Bijection
1
2
a
b
c
3
Bijective NOT Bijective
x
y
x
y
1
2
4
a
b
c
3
NOT Surjective
NOT injective
15. 15/20
Notes on Bijection
1. |A| = |B|
– An “extra” B cannot be “hit” (not a surjection)
– An “extra” A requires that at least one B must be
“hit” twice (not an injection)
1. If f is a bijection, swapping the elements of the
ordered pairs is a function
– Called the inverse
– Denoted f-1
– Is also a bijection
– f-1
(f(x)) is the identity function, i.e. f-1
(f(x)) = x.
16. 16/20
Notes on Bijection (continued …)
3. The inverse of an injection is a partial function.
If f : A →B is an injection, then f-1
is a partial function
4. Restricting the range space of an injective
function to the range yields a bijection
Remove b
a
b
d
1
2
3
c
f
a
b
d
1
2
3
c
f-1