User:Bchethan

Things I have learned about Wiki:

 * Create a User Page
 * How to create heading
 * Inserting Table and modifying it
 * Checking the History and undoing the changes
 * how to insert links to redirect from one page to other.

Taylor's Theorem
Taylor's theorem asserts that any sufficiently smooth function can locally be approximated by polynomials. A simple example of application of Taylor's theorem is the approximation of the exponential function ex near x = 0:


 * $$ \textrm{e}^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!}.$$

The approximation is called the n-th order Taylor approximation to ex because it approximates the value of the exponential function by a polynomial of degree n. This approximation only holds for x close to zero, and as x moves further away from zero, the approximation becomes worse. The quality of the approximation is controlled by the remainder term:

$$R_n(x) = \textrm{e}^x - \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!}\right).$$

More generally, Taylor's theorem applies to any sufficiently differentiable function &fnof;, giving an approximation, for x near a point a, of the form


 * $$f(x)\approx f(a) + f'(a)(x-a) +\frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n.$$

The remainder term is just the difference of the function and its approximating polynomial


 * $$R_n(x) = f(x) - \left(f(a) + f'(a)(x-a) +\frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n\right).$$

Changes That Can be Included
I read Floating point algorithm and loss of significance and I felt it is good. In Taylor's Theorem limitations were not discussed.I feel below limitation can be added to the wikipidea page.

Limitations of Taylor's theorem:
Taylor's theorem can not be implemented to a function at a point if the function is not continuous at that point.For example step function can not have a Taylor's series at a point where it is moving to next step.

Project Proposal:
I am interested to work on creating exercises,quiz related to Numerical Analysis topics covered in our class.My major focus will be on interpolating polynomials and solutions for Ordinary Differential equations using different methods.I will create multiple quizzes of each topic so that if some one interested to take quiz on only one method he don't ave to go through quiz which contains all the methods and waste his time.