This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
This is a PowerPoint on teaching the subject of Polynomials, Monomials, and Rational Expressions, dealing with how to add, subtract, multiply, and simplify all of these.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
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.
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.
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.
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.
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
Objectives
Write the general solution of a differential equation and
use indefinite integral notation for antiderivatives.
Use basic integration rules to find antiderivatives.
Find a particular solution of a differential equation.
5. 5
Antiderivatives
To find a function F whose derivative is f(x) = 3x2, you
might use your knowledge of derivatives to conclude that
The function F is an antiderivative of f .
6. 6
Antiderivatives
Note that F is called an antiderivative of f, rather than the
antiderivative of f. To see why, observe that
are all derivatives of f (x) = 3x2. In fact, for any constant C,
the function F(x)= x3 + C is an antiderivative of f.
7. Antiderivatives
Using Theorem 4.1, you can represent the entire family of
antiderivatives of a function by adding a constant to a known
antiderivative.
For example, knowing that Dx [x2] = 2x, you can represent the
family of all antiderivatives of f(x) = 2x by
7
G(x) = x2 + C Family of all antiderivatives of f(x) = 2x
where C is a constant. The constant C is called the constant
of integration.
8. 8
Antiderivatives
The family of functions represented by G is the general
antiderivative of f, and G(x) = x2 + C is the general
solution of the differential equation
G'(x) = 2x. Differential equation
A differential equation in x and y is an equation that
involves x, y, and derivatives of y.
For instance, y' = 3x and y' = x2 + 1 are examples of
differential equations.
9. 9
Example 1 – Solving a Differential Equation
Find the general solution of the differential equation y' = 2.
Solution:
To begin, you need to find a function whose derivative is 2.
One such function is
y = 2x. 2x is an antiderivative of 2.
Now, you can use Theorem 4.1 to conclude that the
general solution of the differential equation is
y = 2x + C. General solution
10. 10
Example 1 – Solution
The graphs of several functions of the form y = 2x + C
are shown in Figure 4.1.
Figure 4.1
cont’d
12. 12
Notation for Antiderivatives
When solving a differential equation of the form
it is convenient to write it in the equivalent differential form
The operation of finding all solutions of this equation is
called antidifferentiation (or indefinite integration) and is
denoted by an integral sign ∫.
13. 13
Notation for Antiderivatives
The general solution is denoted by
The expression ∫f(x)dx is read as the antiderivative of f with
respect to x. So, the differential dx serves to identify x as
the variable of integration. The term indefinite integral is a
synonym for antiderivative.
15. 15
Basic Integration Rules
The inverse nature of integration and differentiation can be
verified by substituting F'(x) for f(x) in the indefinite
integration definition to obtain
Moreover, if ∫f(x)dx = F(x) + C, then
16. 16
Basic Integration Rules
These two equations allow you to obtain integration
formulas directly from differentiation formulas, as shown in
the following summary.
18. 18
Example 2 – Applying the Basic Integration Rules
Describe the antiderivatives of 3x.
Solution:
So, the antiderivatives of 3x are of the form where
C is any constant.
19. 19
Basic Integration Rules
In Example 2, note that the general pattern of integration is
similar to that of differentiation.
21. 21
Initial Conditions and Particular Solutions
You have already seen that the equation y = ∫f(x)dx has
many solutions (each differing from the others by a
constant).
This means that the graphs of any two antiderivatives of f
are vertical translations of each other.
22. 22
Initial Conditions and Particular Solutions
For example, Figure 4.2 shows the
graphs of several antiderivatives
of the form
for various integer values of C.
Each of these antiderivatives is a solution
of the differential equation
Figure 4.2
23. 23
Initial Conditions and Particular Solutions
In many applications of integration, you are given enough
information to determine a particular solution. To do this,
you need only know the value of y = F(x) for one value of x.
This information is called an initial condition.
For example, in Figure 4.2, only one curve passes through
the point (2, 4).
To find this curve, you can use the general solution
F(x) = x3 – x + C General solution
and the initial condition
F(2) = 4 Initial condition
24. 24
Initial Conditions and Particular Solutions
By using the initial condition in the general solution, you
can determine that
F(2) = 8 – 2 + C = 4
which implies that C = –2.
So, you obtain
F(x) = x3 – x – 2. Particular solution
25. 25
Example 8 – Finding a Particular Solution
Find the general solution of
and find the particular solution that satisfies the initial
condition F(1) = 0.
Solution:
To find the general solution, integrate to obtain
26. 26
Example 8 – Solution
Using the initial condition F(1) = 0, you
can solve for C as follows.
So, the particular solution, as shown
in Figure 4.3, is
cont’d
Figure 4.3