User:La740411ohio

What is x3+2x if x=1? Solution: 3

Simple Quiz
{If 2x + 8x =50, then (x-5)+5= - 10 - -5 + 5 - -10
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{f(x)=C is a -linear function +constant function - greatest integer function -absolute value function
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Homework 7 Question 1
I edited the wikiversity page on the theorem on exercise 3 at []

I corrected and also added some more terms and definition:


 * 1) I first discuss the Legendre Polynomial and it's properties.
 * 2) I mentioned that since $$x_i$$ is a roots of $$P_n$$ for each $$i= 1, 2, ..., n$$
 * 3) I replace * by \times
 * 4) I also mentioned that since $$R(x)$$ is a polynomial of degree less than $$n$$
 * 5) Also I stated that by Legendre property 2 started above

Lastly I also stated that the two polynomials $$Q(x)$$ and $$R(x)$$ are of degree less than $$n$$ and $$P_n(x)$$ is of degree $$n$$.

What I plan to do for Final Project
For my final project, I intend to do the following on the Wikiversity's Numerical Analysis page:


 * 1) To properly define zeroth divided difference, first divided difference, second divided difference and the $$kth$$ divided difference on the Wikipedia.  What is wrong with the current definition (at the top of the page)? Don't get into  Finite differences.
 * 2) To show that the divided differences are symmetry, For example for first divided difference I will show that $$ f[x_0,x_1] = f[x_1,x_0] $$. This would be good. You can add this to the list of properties on the Wikipedia page and link to your Wikiversity work.
 * 3) Explain how to compute the first divided, second divided, third divided and the fourth divided difference in table form. Okay.
 * 4) Write an Algorithm to compute Newton's divided difference and state the Advantages of Newton Divided Differences. Advantages versus what? Don't confuse the divided differences with the Newton form of the interpolating polynomial.
 * 5) Give examples to show how to compute divided differences and several exercises to give readers the opportunity to practice using this method. Okay. Try to make each example illustrate something different so it is not just the same ideas with different numbers. Avoid overlapping with Topic:Numerical analysis/Newton form example or Topic:Numerical analysis/Newton form exercise.
 * 6) State and prove the Generalized Mean Value Theorem for divided differences. It exists already at  Mean value theorem (divided differences).

Project Report for User:la740411ohio
For Introduction to Numerical Analysis, Fall 2012.

Introduction
My final project is about Divided differences and Divided differences are useful for interpolating functions when the values are given for unequally spaced values of the argument. The divided differences have a number of special properties that can simplify work with them and one of the property is called the Symmetry property which stated that the Divided differences remain unaffected by the changing the permutations (rearrangement) of their variables. This symmetry property is very important in divided differences and as an attempt to prove this symmetry property I need to use the Expanded form of divided differences, which lead me to prove this theorem. Also on my project I explain the relationship between Generalization of the Mean Value Theorem and the Derivatives and I explain divided difference in tabular for because difference table is a convenient device for displacing differences.

As an attempt to improve what I found to be lacking on the divided differences page on Wikipedia and to add to the learning materials on Wikiversity, I created a divided differences page consisting of a Symmetry property of divided differences,the divided differences in tabular form, numerical example applying the method, the relationship between Generalization of the Mean Value Theorem and the Derivatives and several exercises to provide practice and a short quiz.

Contribution
I created http://en.wikiversity.org/wiki/Topic:Numerical_Analysis/Divided_differences on wikiversity which contain the following, the symmetry property of divided differences,the divided differences in tabular form, Algorithm to compute the divided differences,the Relationship between Generalization of the Mean Value Theorem and the Derivatives and numerical example applying the method, and some numerical examples to provide practice and also short quiz. I first give another way to define the divided difference which is the Expanded form and then I decided to prove the general form of this Expanded form of divided difference because this form is very important in the prove of the symmetry property of divided differences.

Secondly, I state the symmetry property of divided difference and I prove this property by showing that for $$n=1$$ we have $$f[x_1,x_0]=f[x_0,x_1]$$ which is called symmetry of the first divided difference, also for $$n=2$$, then I show that $$f[x_2,x_1,x_0]=f[x_0,x_1,x_2]=f[x_1,x_0,x_2] etc.$$ which is called symmetry of the second divided difference and finally I also show that for $$n=3$$ we have that $$f[x_3,x_2,x_1,x_0]=f[x_0,x_1,x_2,x_3]=f[x_1,x_0,x_3,x_2] etc $$ which is called symmetry of the third divided difference and I finally generalize the proof by using the Expanded form of divided difference that $$f[x_0,x_1\dots,x_n]=f[x_1,x_0\dots,x_n] etc.$$ which is called symmetry of the $$n_{th}$$ divided difference.

Thirdly, I show how to compute the divided differences in tabular form which is the convenient way for displacing differences and I also provide the Algorithm to compute the divided differences.

Lastly, I explain the relationship between Generalization of the Mean Value Theorem and the Derivatives and I show the simplest case where $$n=1$$ to be $$f[x_0,x_1]= \frac{f[x_1]-f[x_0]}{x_1-x_0}=f'(\xi)$$ which is exactly Mean Value Theorem, and I finally generalized what the theorem is telling us by saying that Generalization of the Mean Value Theorem of Newton's $$n^{th}$$ divided difference is in some sense approximation to the $$n^{th}$$ derivatives of $$f$$.

Future Work
In the future, It would be beneficial if I extent my project to cover the Newton Forward & Backward-Difference formula and provide some numerical examples explaining this formula, It may also be beneficial to provide the prove of the Newton's divided difference interpolation formula and how this method is been use to solve a numerical problem.

Conclusions
In this project I contributed to the Divided differences page in wikipedia. I think this is a valuable contribution because I proved the proof for the Expanded form of divided differences, I stated and prove the symmetry property of divided difference which is an important property in divided differences and I also give full details with a numerical example the relationship between Generalization of the Mean Value Theorem and the Derivatives.I hope this and the numerical examples and some quiz I gave can help people have a better understanding of the method.