SlideShare a Scribd company logo
Topic – General Quadrature Formula
Subject Incharge: Dr. Dharm Raj Singh
Name: Mrinal Dev
Basic Topics
 Calculus
 Function
 Differentiation
 Limit
 Range
Main Topic
 Integration
 Quadrature
 Quadratic Equation
 Quadrature Formula
 Derivation
Calculus: Calculus, is the mathematical study of continuous
change, in the same way that geometry is the study of shape and
algebra is the study of generalizations of arithmetic operations.
Function: A function was originally the idealization of how a
varying quantity depends on another quantity.
Differentiation: Differentiation is a process of finding a function
that outputs the rate of change of one variable with respect to
another variable.
Limit: A limit is the value that a function (or sequence)
"approaches" as the input (or index) "approaches"
some value. Limits are essential
to calculus (and mathematical analysis in general) and
are used to define continuity, derivatives, and integrals.
It is of Two Types and they are:
1- Lower Limit:- The lower class limit of a class is the
smallest data value that can go into the class.
2- Upper Limit:- The upper class limit of a class is the
largest data value that can go into the class.
Range: Values can occur between the smallest and
largest values in a set of observed values or data points.
Given a set of values, or data points, the range is
determined by subtracting the smallest value from the
largest value.
General Quadrature Equation
Integration is the reverse of differentiation.
However:
If y = 2x + 3, dy/dx = 2
If y = 2x + 5, dy/dx = 2
If y = 2x, dy/dx = 2
So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc.
For this reason, when we integrate, we have to add a
constant. So the integral of 2 is 2x + c, where c is a
constant.
A "S" shaped symbol is used to mean the integral of,
and dx is written at the end of the terms to be
integrated, meaning "with respect to x". This is the
same "dx" that appears in dy/dx .
To integrate a term, increase its power by 1 and
divide by this figure. In other words:
∫ xn dx = 1/n+1 (xn+1) + c
Examples:
∫ x5 dx = 1/6 (x6) + c
 In mathematics, quadrature is a historical term
which means determining area. Quadrature
problems served as one of the main sources of
problems in the development of calculus, and
introduce important topics
in mathematical analysis.
An equation where the highest exponent of the
variable (usually "x") is a square (2). So it will
have something like x2, but not x3 etc.
A Quadratic Equation is usually written
ax2 + bx + c = 0.
Example: 2x2 + 5x − 3 = 0.
Quadrature Equation is also known as Newton’s
Forward Interpolation Formula.
The Gauss–Kronrod quadrature formula is an adaptive method for
numerical integration. ... It is an example of what is called a
nested quadrature rule: for the same set of function evaluation points, it has
two quadrature rules, one higher order and one lower order (the latter called
an embedded rule).
The Formula is given by:
y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎
𝒖(𝒖−𝟏)
𝟐!
𝟐 𝒚 𝟎 +
𝒖(𝒖−𝟏)(𝒖−𝟐)
𝟑!
𝟑 𝒚 𝟎 + ⋯
Let I = ydx where y = f (x)
Also assume that f (x) be given for certain equidistant values of x, say
x0, x1, x2, x3,….xn. Let the range (b-a) be divided into n equal parts,
each of width h, so that
h =
𝑏 − 𝑎
𝑛
Thus, we have
x0 = a, x1 = a + h, x2 = a + 2h, … xn = a + nh = b
Now , let
yk = f (xk), k = 0,1,2…n
Consider, I = y dx = ydx
b
a

b
a

0
0
x nh
x


We have
y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎 +
𝒖(𝒖−𝟏)
𝟐!
𝟐 𝒚 𝟎 +
𝒖(𝒖−𝟏)(𝒖−𝟐)
𝟑!
𝟑 𝒚 𝟎 + ⋯
where u =
du =
dx = hdu
𝑥 − 𝑥0
ℎ
1
ℎ


Approximating y by Newton’s forward formula taking limit of
integration becomes 0 to n.
I = h y0 + u y0+
𝑢(𝑢−1)
2!
2 𝑦0 +
𝑢(𝑢−1)(𝑢−2)
3!
3 𝑦0 + ⋯ du
n
= h y0u +
𝒖 𝟐
2
y0 +
𝒖 𝟑
𝟑
−
𝒖 𝟐
𝟐
2!
2 y0 +
𝑢4
4
−𝑢3
+𝑢2
3!
3y0 + …
0
=h ny0 +
𝒏 𝟐
2
y0 +
𝒏 𝟑
𝟑
−
𝒏 𝟐
𝟐
2!
2y0 +
𝑛4
4
−𝑛3
+𝑛2
3!
3y0 + …
0
n

I = nh y0 +
𝒏
2
y0 +
𝒏(𝟐𝒏−𝟑)
12
2y0 +
𝒏 𝒏−𝟐 𝟐
24
3y0+…
This is called General Quadrature formula.
General Quadrature Equation

More Related Content

What's hot

First order linear differential equation
First order linear differential equationFirst order linear differential equation
First order linear differential equation
Nofal Umair
 
Ode powerpoint presentation1
Ode powerpoint presentation1Ode powerpoint presentation1
Ode powerpoint presentation1
Pokkarn Narkhede
 
Higher order ODE with applications
Higher order ODE with applicationsHigher order ODE with applications
Higher order ODE with applications
Pratik Gadhiya
 
Linear differential equation
Linear differential equationLinear differential equation
Linear differential equation
Pratik Sudra
 
Chapter 1: First-Order Ordinary Differential Equations/Slides
Chapter 1: First-Order Ordinary Differential Equations/Slides Chapter 1: First-Order Ordinary Differential Equations/Slides
Chapter 1: First-Order Ordinary Differential Equations/Slides
Chaimae Baroudi
 
Differential equations
Differential equationsDifferential equations
Differential equations
Seyid Kadher
 
graph theory
graph theory graph theory
graph theory
ganith2k13
 
partial diffrentialequations
partial diffrentialequationspartial diffrentialequations
partial diffrentialequations
8laddu8
 
Introduction to Functions of Several Variables
Introduction to Functions of Several VariablesIntroduction to Functions of Several Variables
Introduction to Functions of Several Variables
Nhan Nguyen
 
Differentiation
DifferentiationDifferentiation
Differentiation
timschmitz
 
Numerical differentiation
Numerical differentiationNumerical differentiation
Numerical differentiation
andrushow
 
Ordinary differential equation
Ordinary differential equationOrdinary differential equation
Ordinary differential equation
DnyaneshwarPardeshi1
 
first order ode with its application
 first order ode with its application first order ode with its application
first order ode with its application
Krishna Peshivadiya
 
Beta gamma functions
Beta gamma functionsBeta gamma functions
Beta gamma functions
Dr. Nirav Vyas
 
Ordinary differential equation
Ordinary differential equationOrdinary differential equation
Ordinary differential equation
JUGAL BORAH
 
Partial differential equations
Partial differential equationsPartial differential equations
Partial differential equations
aman1894
 
Matrix algebra
Matrix algebraMatrix algebra
Matrix algebra
Farzad Javidanrad
 
Curve fitting
Curve fittingCurve fitting
Curve fitting
Mayank Bhatt
 
LinearAlgebra.ppt
LinearAlgebra.pptLinearAlgebra.ppt
LinearAlgebra.ppt
vijaykumar838577
 
Gaussian Elimination Method
Gaussian Elimination MethodGaussian Elimination Method
Gaussian Elimination Method
Andi Firdaus
 

What's hot (20)

First order linear differential equation
First order linear differential equationFirst order linear differential equation
First order linear differential equation
 
Ode powerpoint presentation1
Ode powerpoint presentation1Ode powerpoint presentation1
Ode powerpoint presentation1
 
Higher order ODE with applications
Higher order ODE with applicationsHigher order ODE with applications
Higher order ODE with applications
 
Linear differential equation
Linear differential equationLinear differential equation
Linear differential equation
 
Chapter 1: First-Order Ordinary Differential Equations/Slides
Chapter 1: First-Order Ordinary Differential Equations/Slides Chapter 1: First-Order Ordinary Differential Equations/Slides
Chapter 1: First-Order Ordinary Differential Equations/Slides
 
Differential equations
Differential equationsDifferential equations
Differential equations
 
graph theory
graph theory graph theory
graph theory
 
partial diffrentialequations
partial diffrentialequationspartial diffrentialequations
partial diffrentialequations
 
Introduction to Functions of Several Variables
Introduction to Functions of Several VariablesIntroduction to Functions of Several Variables
Introduction to Functions of Several Variables
 
Differentiation
DifferentiationDifferentiation
Differentiation
 
Numerical differentiation
Numerical differentiationNumerical differentiation
Numerical differentiation
 
Ordinary differential equation
Ordinary differential equationOrdinary differential equation
Ordinary differential equation
 
first order ode with its application
 first order ode with its application first order ode with its application
first order ode with its application
 
Beta gamma functions
Beta gamma functionsBeta gamma functions
Beta gamma functions
 
Ordinary differential equation
Ordinary differential equationOrdinary differential equation
Ordinary differential equation
 
Partial differential equations
Partial differential equationsPartial differential equations
Partial differential equations
 
Matrix algebra
Matrix algebraMatrix algebra
Matrix algebra
 
Curve fitting
Curve fittingCurve fitting
Curve fitting
 
LinearAlgebra.ppt
LinearAlgebra.pptLinearAlgebra.ppt
LinearAlgebra.ppt
 
Gaussian Elimination Method
Gaussian Elimination MethodGaussian Elimination Method
Gaussian Elimination Method
 

Similar to General Quadrature Equation

Math major 14 differential calculus pw
Math major 14 differential calculus pwMath major 14 differential calculus pw
Math major 14 differential calculus pw
Reymart Bargamento
 
Project in Calcu
Project in CalcuProject in Calcu
Project in Calcu
patrickpaz
 
Cs jog
Cs jogCs jog
integration in maths pdf mathematics integration
integration in maths pdf mathematics integrationintegration in maths pdf mathematics integration
integration in maths pdf mathematics integration
Dr. Karrar Alwash
 
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdfIVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf
42Rnu
 
Basic Cal - Quarter 1 Week 1-2.pptx
Basic Cal - Quarter 1 Week 1-2.pptxBasic Cal - Quarter 1 Week 1-2.pptx
Basic Cal - Quarter 1 Week 1-2.pptx
jamesvalenzuela6
 
.
..
Calculus - Functions Review
Calculus - Functions ReviewCalculus - Functions Review
Calculus - Functions Review
hassaanciit
 
Derivatie class 12
Derivatie class 12Derivatie class 12
Derivatie class 12
Sadiq Hussain
 
mc-ty-polynomial-2009-1.pdf
mc-ty-polynomial-2009-1.pdfmc-ty-polynomial-2009-1.pdf
mc-ty-polynomial-2009-1.pdf
Hazel Mier Timagos Basit
 
Introduction to R
Introduction to RIntroduction to R
Introduction to R
University of Salerno
 
Developing Expert Voices
Developing Expert VoicesDeveloping Expert Voices
Developing Expert Voices
suzanne
 
Numarical values
Numarical valuesNumarical values
Numarical values
AmanSaeed11
 
Numarical values highlighted
Numarical values highlightedNumarical values highlighted
Numarical values highlighted
AmanSaeed11
 
.
..
A Fast Numerical Method For Solving Calculus Of Variation Problems
A Fast Numerical Method For Solving Calculus Of Variation ProblemsA Fast Numerical Method For Solving Calculus Of Variation Problems
A Fast Numerical Method For Solving Calculus Of Variation Problems
Sara Alvarez
 
Algebra part 2
Algebra part 2Algebra part 2
Algebra part 2
Nadrah Afiati
 
Algebra
AlgebraAlgebra
Algebra
Nadrah Afiati
 
Statistics lab 1
Statistics lab 1Statistics lab 1
Statistics lab 1
University of Salerno
 
CALCULUS 2.pptx
CALCULUS 2.pptxCALCULUS 2.pptx
CALCULUS 2.pptx
ShienaMaeIndac
 

Similar to General Quadrature Equation (20)

Math major 14 differential calculus pw
Math major 14 differential calculus pwMath major 14 differential calculus pw
Math major 14 differential calculus pw
 
Project in Calcu
Project in CalcuProject in Calcu
Project in Calcu
 
Cs jog
Cs jogCs jog
Cs jog
 
integration in maths pdf mathematics integration
integration in maths pdf mathematics integrationintegration in maths pdf mathematics integration
integration in maths pdf mathematics integration
 
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdfIVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf
IVS-B UNIT-1_merged. Semester 2 fundamental of sciencepdf
 
Basic Cal - Quarter 1 Week 1-2.pptx
Basic Cal - Quarter 1 Week 1-2.pptxBasic Cal - Quarter 1 Week 1-2.pptx
Basic Cal - Quarter 1 Week 1-2.pptx
 
.
..
.
 
Calculus - Functions Review
Calculus - Functions ReviewCalculus - Functions Review
Calculus - Functions Review
 
Derivatie class 12
Derivatie class 12Derivatie class 12
Derivatie class 12
 
mc-ty-polynomial-2009-1.pdf
mc-ty-polynomial-2009-1.pdfmc-ty-polynomial-2009-1.pdf
mc-ty-polynomial-2009-1.pdf
 
Introduction to R
Introduction to RIntroduction to R
Introduction to R
 
Developing Expert Voices
Developing Expert VoicesDeveloping Expert Voices
Developing Expert Voices
 
Numarical values
Numarical valuesNumarical values
Numarical values
 
Numarical values highlighted
Numarical values highlightedNumarical values highlighted
Numarical values highlighted
 
.
..
.
 
A Fast Numerical Method For Solving Calculus Of Variation Problems
A Fast Numerical Method For Solving Calculus Of Variation ProblemsA Fast Numerical Method For Solving Calculus Of Variation Problems
A Fast Numerical Method For Solving Calculus Of Variation Problems
 
Algebra part 2
Algebra part 2Algebra part 2
Algebra part 2
 
Algebra
AlgebraAlgebra
Algebra
 
Statistics lab 1
Statistics lab 1Statistics lab 1
Statistics lab 1
 
CALCULUS 2.pptx
CALCULUS 2.pptxCALCULUS 2.pptx
CALCULUS 2.pptx
 

Recently uploaded

Imagination in Computer Science Research
Imagination in Computer Science ResearchImagination in Computer Science Research
Imagination in Computer Science Research
Abhik Roychoudhury
 
How to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POSHow to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POS
Celine George
 
E-learning Odoo 17 New features - Odoo 17 Slides
E-learning Odoo 17  New features - Odoo 17 SlidesE-learning Odoo 17  New features - Odoo 17 Slides
E-learning Odoo 17 New features - Odoo 17 Slides
Celine George
 
modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025
NurFitriah45
 
modul ajar kelas x bahasa inggris 24/254
modul ajar kelas x bahasa inggris 24/254modul ajar kelas x bahasa inggris 24/254
modul ajar kelas x bahasa inggris 24/254
NurFitriah45
 
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
Celine George
 
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
Nguyen Thanh Tu Collection
 
How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17
Celine George
 
Allopathic M1 Srudent Orientation Powerpoint
Allopathic M1 Srudent Orientation PowerpointAllopathic M1 Srudent Orientation Powerpoint
Allopathic M1 Srudent Orientation Powerpoint
Julie Sarpy
 
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
thanhluan21
 
Odoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List ScanningOdoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List Scanning
Celine George
 
What is Packaging of Products in Odoo 17
What is Packaging of Products in Odoo 17What is Packaging of Products in Odoo 17
What is Packaging of Products in Odoo 17
Celine George
 
JavaScript Interview Questions PDF By ScholarHat
JavaScript Interview  Questions PDF By ScholarHatJavaScript Interview  Questions PDF By ScholarHat
JavaScript Interview Questions PDF By ScholarHat
Scholarhat
 
The Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdfThe Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdf
luzmilaglez334
 
View Inheritance in Odoo 17 - Odoo 17 Slides
View Inheritance in Odoo 17 - Odoo 17  SlidesView Inheritance in Odoo 17 - Odoo 17  Slides
View Inheritance in Odoo 17 - Odoo 17 Slides
Celine George
 
formative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.Vformative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.V
DrRavindrakshirsagar1
 
Parent PD Design for Professional Development .docx
Parent PD Design for Professional Development .docxParent PD Design for Professional Development .docx
Parent PD Design for Professional Development .docx
AntonioJarligoCompra
 
Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi
Power of Ignored Skills: Change the Way You Think and Decide by Manoj TripathiPower of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi
Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi
Pankaj523992
 
How to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POSHow to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POS
Celine George
 
How to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POSHow to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POS
Celine George
 

Recently uploaded (20)

Imagination in Computer Science Research
Imagination in Computer Science ResearchImagination in Computer Science Research
Imagination in Computer Science Research
 
How to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POSHow to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POS
 
E-learning Odoo 17 New features - Odoo 17 Slides
E-learning Odoo 17  New features - Odoo 17 SlidesE-learning Odoo 17  New features - Odoo 17 Slides
E-learning Odoo 17 New features - Odoo 17 Slides
 
modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025
 
modul ajar kelas x bahasa inggris 24/254
modul ajar kelas x bahasa inggris 24/254modul ajar kelas x bahasa inggris 24/254
modul ajar kelas x bahasa inggris 24/254
 
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
 
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
BÀI TẬP BỔ TRỢ 4 KỸ NĂNG TIẾNG ANH LỚP 12 - GLOBAL SUCCESS - FORM MỚI 2025 - ...
 
How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17
 
Allopathic M1 Srudent Orientation Powerpoint
Allopathic M1 Srudent Orientation PowerpointAllopathic M1 Srudent Orientation Powerpoint
Allopathic M1 Srudent Orientation Powerpoint
 
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
 
Odoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List ScanningOdoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List Scanning
 
What is Packaging of Products in Odoo 17
What is Packaging of Products in Odoo 17What is Packaging of Products in Odoo 17
What is Packaging of Products in Odoo 17
 
JavaScript Interview Questions PDF By ScholarHat
JavaScript Interview  Questions PDF By ScholarHatJavaScript Interview  Questions PDF By ScholarHat
JavaScript Interview Questions PDF By ScholarHat
 
The Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdfThe Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdf
 
View Inheritance in Odoo 17 - Odoo 17 Slides
View Inheritance in Odoo 17 - Odoo 17  SlidesView Inheritance in Odoo 17 - Odoo 17  Slides
View Inheritance in Odoo 17 - Odoo 17 Slides
 
formative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.Vformative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.V
 
Parent PD Design for Professional Development .docx
Parent PD Design for Professional Development .docxParent PD Design for Professional Development .docx
Parent PD Design for Professional Development .docx
 
Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi
Power of Ignored Skills: Change the Way You Think and Decide by Manoj TripathiPower of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi
Power of Ignored Skills: Change the Way You Think and Decide by Manoj Tripathi
 
How to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POSHow to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POS
 
How to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POSHow to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POS
 

General Quadrature Equation

  • 1. Topic – General Quadrature Formula Subject Incharge: Dr. Dharm Raj Singh Name: Mrinal Dev
  • 2. Basic Topics  Calculus  Function  Differentiation  Limit  Range Main Topic  Integration  Quadrature  Quadratic Equation  Quadrature Formula  Derivation
  • 3. Calculus: Calculus, is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. Function: A function was originally the idealization of how a varying quantity depends on another quantity. Differentiation: Differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable.
  • 4. Limit: A limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals. It is of Two Types and they are: 1- Lower Limit:- The lower class limit of a class is the smallest data value that can go into the class. 2- Upper Limit:- The upper class limit of a class is the largest data value that can go into the class. Range: Values can occur between the smallest and largest values in a set of observed values or data points. Given a set of values, or data points, the range is determined by subtracting the smallest value from the largest value.
  • 6. Integration is the reverse of differentiation. However: If y = 2x + 3, dy/dx = 2 If y = 2x + 5, dy/dx = 2 If y = 2x, dy/dx = 2 So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc. For this reason, when we integrate, we have to add a constant. So the integral of 2 is 2x + c, where c is a constant. A "S" shaped symbol is used to mean the integral of, and dx is written at the end of the terms to be integrated, meaning "with respect to x". This is the same "dx" that appears in dy/dx .
  • 7. To integrate a term, increase its power by 1 and divide by this figure. In other words: ∫ xn dx = 1/n+1 (xn+1) + c Examples: ∫ x5 dx = 1/6 (x6) + c
  • 8.  In mathematics, quadrature is a historical term which means determining area. Quadrature problems served as one of the main sources of problems in the development of calculus, and introduce important topics in mathematical analysis.
  • 9. An equation where the highest exponent of the variable (usually "x") is a square (2). So it will have something like x2, but not x3 etc. A Quadratic Equation is usually written ax2 + bx + c = 0. Example: 2x2 + 5x − 3 = 0. Quadrature Equation is also known as Newton’s Forward Interpolation Formula.
  • 10. The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. ... It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order (the latter called an embedded rule). The Formula is given by: y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎 𝒖(𝒖−𝟏) 𝟐! 𝟐 𝒚 𝟎 + 𝒖(𝒖−𝟏)(𝒖−𝟐) 𝟑! 𝟑 𝒚 𝟎 + ⋯
  • 11. Let I = ydx where y = f (x) Also assume that f (x) be given for certain equidistant values of x, say x0, x1, x2, x3,….xn. Let the range (b-a) be divided into n equal parts, each of width h, so that h = 𝑏 − 𝑎 𝑛 Thus, we have x0 = a, x1 = a + h, x2 = a + 2h, … xn = a + nh = b Now , let yk = f (xk), k = 0,1,2…n Consider, I = y dx = ydx b a  b a  0 0 x nh x  
  • 12. We have y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎 + 𝒖(𝒖−𝟏) 𝟐! 𝟐 𝒚 𝟎 + 𝒖(𝒖−𝟏)(𝒖−𝟐) 𝟑! 𝟑 𝒚 𝟎 + ⋯ where u = du = dx = hdu 𝑥 − 𝑥0 ℎ 1 ℎ  
  • 13. Approximating y by Newton’s forward formula taking limit of integration becomes 0 to n. I = h y0 + u y0+ 𝑢(𝑢−1) 2! 2 𝑦0 + 𝑢(𝑢−1)(𝑢−2) 3! 3 𝑦0 + ⋯ du n = h y0u + 𝒖 𝟐 2 y0 + 𝒖 𝟑 𝟑 − 𝒖 𝟐 𝟐 2! 2 y0 + 𝑢4 4 −𝑢3 +𝑢2 3! 3y0 + … 0 =h ny0 + 𝒏 𝟐 2 y0 + 𝒏 𝟑 𝟑 − 𝒏 𝟐 𝟐 2! 2y0 + 𝑛4 4 −𝑛3 +𝑛2 3! 3y0 + … 0 n 
  • 14. I = nh y0 + 𝒏 2 y0 + 𝒏(𝟐𝒏−𝟑) 12 2y0 + 𝒏 𝒏−𝟐 𝟐 24 3y0+… This is called General Quadrature formula.