This document discusses numerical differentiation and integration using Newton's forward and backward difference formulas. It provides examples of using these formulas to calculate derivatives from tables of ordered data pairs. Specifically, it shows how to calculate derivatives at interior points using central difference formulas, and at endpoints using forward or backward formulas depending on if the point is near the start or end of the data range. Formulas are derived for calculating the first and second derivatives, and examples are worked through to find acceleration and rates of cooling from given temperature-time tables.
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
Gauss Forward And Backward Central Difference Interpolation Formula Deep Dalsania
This PPT contains the topic called Gauss Forward And Backward Central Difference Interpolation Formula of subject called Numerical and Statistical Methods for Computer Engineering.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
How do you calculate the particular integral of linear differential equations?
Learn this and much more by watching this video. Here, we learn how the inverse differential operator is used to find the particular integral of trigonometric, exponential, polynomial and inverse hyperbolic functions. Problems are explained with the relevant formulae.
This is useful for graduate students and engineering students learning Mathematics. For more videos, visit my page
https://www.mathmadeeasy.co/about-4
Subscribe to my channel for more videos.
finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATIONTouhidul Shawan
This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.
Gauss Forward And Backward Central Difference Interpolation Formula Deep Dalsania
This PPT contains the topic called Gauss Forward And Backward Central Difference Interpolation Formula of subject called Numerical and Statistical Methods for Computer Engineering.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
How do you calculate the particular integral of linear differential equations?
Learn this and much more by watching this video. Here, we learn how the inverse differential operator is used to find the particular integral of trigonometric, exponential, polynomial and inverse hyperbolic functions. Problems are explained with the relevant formulae.
This is useful for graduate students and engineering students learning Mathematics. For more videos, visit my page
https://www.mathmadeeasy.co/about-4
Subscribe to my channel for more videos.
finite difference Method, For Numerical analysis. working matlab code. numeric analysis finite difference method. MATLAB provides tools to solve math. Using linear programing techniques we can easily solve system of equations. This file provides a running code of Finite difference matlab code
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
Tips and Tricks to Clear - CSIR-UGC NET- Physical SciencesMeenakshisundaram N
Power point presentation on Tips and Tricks to clear entrance exams at Post-graduate level of Physics Curriculum in India, primarily focusing on CSIR-UGC NET. Its evolved based on the author's experience in guiding hundreds students for more than 18 years in clearing various entrance exams in Physical sciences such as GATE/NET/SET/JEST/TIFR//Polytechnic Colleges Faculty Selection/PG-TRB etc., Entrance Exams.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Study Material Numerical Differentiation and Integration
1. Page | 1
VIVEKANANDA COLLEGE, TIRUVEDAKAM WEST
(Residential & Autonomous – A Gurukula Institute of Life-Training)
(Affiliated to Madurai Kamaraj University)
PART III: PHYSICS MAJOR – FOURTH SEMESTER-CORE SUBJECT PAPER-II
NUMERICAL METHODS – 06CT42
(For those who joined in June 2018 and after)
Reference Text Book: Numerical Methods – P.Kandasamy, K.Thilagavathy & K.Gunavathi,
S.Chand & Company Ltd., New Delhi, 2014.
UNIT – IV Numerical Differentiation and Integration
Numerical Differentiation
Introduction
We found the polynomial curve y = f (x), passing through the (n+1) ordered pairs (xi, yi), i=0, 1,
2…n. Now we are trying to find the derivative value of such curves at a given x = xk (say), whose x0 <
xk < xn.
To get derivative, we first find the curve y = f (x) through the points and then differentiate and get its
value at the required point.
If the values of x are equally spaced. We get the interpolating polynomial due to Newton-Gregory.
• If the derivative is required at a point nearer to the starting value in the table, we use
Newton’s forward interpolation formula.
• If we require the derivative at the end of the table, we use Newton’s backward interpolation
formula.
• If the value of derivative is required near the middle of the table value, we use one of the
central difference interpolation formulae.
Newton’s forward difference formula to get the derivative
Newton’s forward difference interpolation formula is
𝑦 (𝑥0 + 𝑢ℎ) = 𝑦𝑢 = 𝑦0 + 𝑢∆𝑦0 +
𝑢(𝑢 − 1)
2!
∆2
𝑦0 +
𝑢(𝑢 − 1)(𝑢 − 2)
3!
∆3
𝑦0 + ⋯
where 𝑦 (𝑥) is a polynomial of degree 𝑛 𝑖𝑛 𝑥 𝑎𝑛𝑑 𝑢 =
𝑥 − 𝑥0
ℎ
where ℎ 𝑖s interval of differencing
The values of first and second derivative at the starting value 𝑥 = 𝑥0 for given by
(
𝑑𝑦
𝑑𝑥
)
𝑥=𝑥0
=
1
ℎ
[∆ 𝑦0 −
1
2
∆2
𝑦0 +
1
3
∆3
𝑦0 − ⋯ ]
(
𝑑2
𝑦
𝑑𝑥2
)
𝑥=𝑥0
=
1
ℎ2
[∆2
𝑦0 − ∆3
𝑦0 +
11
12
∆4
𝑦0 − ⋯ ]
2. Page | 2
Problems:
1. The table given below revels the velocity v of a body during the time ‘t’ specified. find its
acceleration at t = 1.1
t : 1.0 1.1 1.2 1.3 1.4
v : 43.1 47.7 52.1 56.4 60.8
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧. 𝑣 is dependent on time 𝑡 𝑖. 𝑒. , 𝑣 = 𝑣(𝑡). we require acceleration =
𝑑𝑣
𝑑𝑡
.
therefore, we have to find 𝑣′(1.1).
𝐴𝑠
𝑑𝑣
𝑑𝑡
𝑎𝑡 𝑡 = 1.1 is require, (nearer to beginning value), we use 𝑁𝑒𝑤𝑡𝑜𝑛 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑓𝑜𝑟𝑚𝑢𝑙𝑎.
𝑣(𝑡) = 𝑣 (𝑥0 + 𝑢ℎ) = 𝑣0 + 𝑢∆𝑣0 +
𝑢(𝑢 − 1)
2!
∆2
𝑣0 +
𝑢(𝑢 − 1)(𝑢 − 2)
3!
∆3
𝑣0 + ⋯
𝑑𝑣
𝑑𝑡
=
1
ℎ
.
𝑑𝑣
𝑑𝑡
=
1
ℎ
[∆𝑣0 +
2𝑢 − 1
2
∆2
𝑣0 +
3𝑢2
− 6𝑢 + 2
6
∆3
𝑣0 + ⋯ ]
where 𝑢 =
𝑡 − 𝑡0
ℎ
=
1.1 − 1.0
0.1
= 1
(
𝑑𝑣
𝑑𝑡
)
𝑡=1.1
= (
𝑑𝑣
𝑑𝑡
)
𝑛=1
=
1
0.1
[4.6 +
1
2
(−0.2) +
1
6
(0.1) +
1
12
(0.1)]
= 10[4.6 − 0.1 − 0.0166 + 0.0083]
= 𝟒𝟒. 𝟗𝟏𝟕
t
1.0
1.1
1.2
1.3
1.4
v
43.1
47.7
52.1
56.4
60.8
∆𝑣
4.6
4.4
4.3
4.4
∆2
𝑣
-0.2
-0.1
0.1
∆3
𝑣
0.1
0.2
∆4
𝑣
0.1
3. Page | 3
2. Derive the Newton’s forward difference formula to get the derivative.
We are given (𝑛 + 1)ordered pairs (𝑥𝑖, 𝑦𝑖)𝑖 = 0, 1, … 𝑛. we want to find the derivative of
𝑦 = 𝑓(𝑥) passing through the (𝑛 + 1) points, at a point near to the startinng value 𝑥 = 𝑥0
Newton’s forward difference interpolation formula is
𝑦 (𝑥0 + 𝑢ℎ) = 𝑦𝑢 = 𝑦0 + 𝑢∆𝑦0 +
𝑢(𝑢 − 1)
2!
∆2
𝑦0 +
𝑢(𝑢 − 1)(𝑢 − 2)
3!
∆3
𝑦0 + ⋯ … (1)
where 𝑦 (𝑥) is a polynomial of degree 𝑛 𝑖𝑛 𝑥 𝑎𝑛𝑑 𝑢 =
𝑥 − 𝑥0
ℎ
Differentiating 𝑦(𝑥) w. r. t. 𝑥,
𝑑𝑦
𝑑𝑥
=
𝑑𝑦
𝑑𝑢
.
𝑑𝑢
𝑑𝑥
=
1
ℎ
.
𝑑𝑦
𝑑𝑢
𝑑𝑦
𝑑𝑥
=
1
ℎ
[∆𝑦0 +
2𝑢 − 1
2
∆2
𝑦0 +
3𝑢2
− 6𝑢 + 2
6
∆3
𝑦0 +
(4𝑢3
− 18𝑢2
+ 22𝑢 − 6)
24
∆4
𝑦0] … . (2)
Equation (2) gives the value of
𝑑𝑦
𝑑𝑥
at general 𝑥 which may be anywhere in the interval.
In special case like 𝑥 = 𝑥0, 𝑖. 𝑒. , 𝑢 = 0 𝑖𝑛 (2)
(
𝑑𝑦
𝑑𝑥
)
𝑥=𝑥0
= (
𝑑𝑦
𝑑𝑥
)
𝑢=0
=
1
ℎ
[∆𝑦0 +
1
2
∆2
𝑦0 +
1
3
∆3
𝑦0 −
1
4
∆4
𝑦0 + ⋯ ] … (3)
Differentiating (2) again w. r. t. 𝑥,
𝑑2
𝑦
𝑑𝑥2
=
𝑑
𝑑𝑢
(
𝑑𝑦
𝑑𝑥
) .
𝑑𝑢
𝑑𝑥
=
𝑑
𝑑𝑢
(
𝑑𝑦
𝑑𝑥
) .
1
ℎ
𝑑2
𝑦
𝑑𝑥2
=
1
ℎ2
[∆2
𝑦0 + (𝑢 − 1)∆3
𝑦0 +
(6𝑢2
− 18𝑢 + 11)
12
∆4
𝑦0 + ⋯ ] … (4)
Equation (4) give the second derivative value at 𝑥 = 𝑥.
setting 𝑥 = 𝑥0 𝑖. 𝑒. , 𝑢 = 0 𝑖𝑛 (4)
(
𝑑2
𝑦
𝑑𝑥2
)
𝑥=𝑥0
=
1
ℎ2
[∆2
𝑦0 − ∆3
𝑦0 +
11
12
∆4
𝑦0 + ⋯ ] … (5)
This equation (5) give the value of second derivative at the starting value 𝑥 = 𝑥0
3. 𝐓𝐡𝐞 𝐭𝐚𝐛𝐥𝐞 𝐛𝐞𝐥𝐨𝐰 𝐠𝐢𝐯𝐞𝐬 𝐭𝐡𝐞 𝐫𝐞𝐬𝐮𝐥𝐭𝐬 𝐨𝐟 𝐚𝐧 𝐨𝐛𝐬𝐞𝐫𝐯𝐚𝐭𝐢𝐨𝐧: 𝛉 𝐢𝐬 𝐭𝐡𝐞 𝐨𝐛𝐬𝐞𝐫𝐯𝐞𝐝 𝐭𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞 𝐢𝐧
𝐝𝐞𝐠𝐫𝐞𝐞𝐬 𝐜𝐞𝐧𝐭𝐫𝐢𝐠𝐫𝐚𝐝𝐞 𝐨𝐟𝐚 𝐯𝐞𝐬𝐬𝐞𝐥 𝐨𝐟 𝐜𝐨𝐨𝐥𝐢𝐧𝐠 𝐰𝐚𝐭𝐞𝐫; 𝐭 𝐢𝐬 𝐭𝐡𝐞 𝐭𝐢𝐦𝐞 𝐢𝐧 𝐦𝐢𝐧𝐮𝐭𝐞𝐬 𝐟𝐫𝐨𝐦 𝐭𝐡𝐞
𝐛𝐞𝐠𝐢𝐧𝐧𝐢𝐧𝐠 𝐨𝐟 𝐨𝐛𝐬𝐞𝐫𝐯𝐚𝐭𝐢𝐨𝐧.
𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐚𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐞 𝐫𝐚𝐭𝐞 𝐨𝐟 𝐜𝐨𝐨𝐥𝐢𝐧𝐠 𝐚𝐭 𝒕 = 𝟑 𝒂𝒏𝒅 𝟑. 𝟓
t :
𝜽 :
1
85.3
3
74.5
5
67.0
7
60.5
9
54.3
12. Page | 12
ii) Since n = 6, we can use Simpson′
s rule
𝐵𝑦 𝑆𝑖𝑚𝑝𝑠𝑜𝑛′
𝑠 𝑜𝑛𝑒 − 𝑡ℎ𝑖𝑟𝑑 𝑟𝑢𝑙𝑒
𝐼 =
0.2
3
[(1.3862944 + 1.6486586 + 2(1.4816045 + 1.5686159)
+ 4(1.4350845 + 1.5260563)]
= 𝟏. 𝟖𝟐𝟕𝟖𝟒𝟕𝟐𝟒
𝑖𝑖𝑖) 𝐵𝑦 𝑆𝑖𝑚𝑝𝑜𝑛𝑠𝑜𝑛′
𝑠 𝑡ℎ𝑖𝑟𝑑 − 𝑒𝑖𝑔ℎ𝑡ℎ𝑠 𝑟𝑢𝑙𝑒,
𝐼 =
3(0.2)
8
[(1.3862944 + 1.6486586)
+ 3(1.4350845 + 1.4816045 + 1.5686159 + 1.6094379 + 2(1.5260563)]
= 𝟏. 𝟖𝟐𝟕𝟖𝟒𝟕𝟐𝟒
Short Answer:
1. Write down the Newton-Cote’s quadrature formula.
∫ 𝑓(𝑥) 𝑑𝑥 ≈ ℎ [𝑛𝑦0 +
𝑛2
2
∆𝑦0 +
1
2
(
𝑛3
3
−
𝑛2
2
) ∆2
𝑦0 +
1
6
(
𝑛4
4
− 𝑛3
+ 𝑛2
) ∆3
𝑦0 + ⋯ ]
𝑥 𝑛
𝑥0
This equation is called 𝑁𝑒𝑤𝑡𝑜𝑛 − 𝐶𝑜𝑡𝑒′
𝑠 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑢𝑟𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎.
2. What is the nature of y (x) in the case of trapezoidal rule?
In trapezoidal rule, y (x) is a linear function of x.
3. State the nature of y (x) and number of intervals in the case of Simpson’s one-third rule?
In Simpson’s one-third rule, y (x) is a polynomial of degree two. To apply this rule n, the number of
intervals must be even.
4. What is the nature of y (x) in the case of Simpson’s three-eighths rule and when it is
applicable?
In Simpson’s third-eighths rule, y (x) is a polynomial of degree three. This rule is applicable if n, the
number of intervals is a multiple of 3.
5. Differentiate between Simpson’s one-third rule and Simpson’s three-eighths rule.
S.No Simpson’s one-third rule Simpson’s three-eighths rule
1 y (x) is a polynomial of degree two y (x) is a polynomial of degree three
2 The number of intervals must be even. The number of intervals is a multiple of 3.