Approximation by polynomails

• wajed
In summary, when approximating a function with a polynomial, we use the Taylor series, which involves using derivatives of the function at a point to create a polynomial that closely matches the function in a neighborhood around that point. Higher derivatives give more information about the behavior of the function and can lead to more accurate approximations.

wajed

Im studying Series/Sequences/Approximation by polynomials..

-We approximate a function f(x) by getting a polynomial (I dont know how we get it, and I dont know what characteristics it should have, and Id like to know please)

-when we need more accuracy we add a higher derivatives, but why is adding a higher derivative gives more accuracy? I tried to imagine what a second derivative represents on a graph, and I came out with a result that may be true, If a second derivative is the rate of change of a first derivative then I should draw a new graph where the independent variable is the function that we defferentiated for once, is that true? I think understanding what a second, third, etc... derivatives represent on a graph can make it clear for me to understand how a higher derivative gives more accuracy.

So, to sum up the questions:
1- how do we find the polynomial when we approximate a function?
2- what does a second, third, etc... derivative represent on a graph?
3- why adding higher derivatives add more accuracy to the "function"? -in other words: precisely, what's the relationship between taking higher derivatives and the accuracy of the result of the polynomial?-

Thank you,

Try http://en.wikipedia.org/wiki/Taylor_series" [Broken].

It's in any introductory calculus book. If you want a proof, try an introductory complex analysis book, Taylor's theorem follows easily from the Laurent series.

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A very basic version to get you started:

1. Pick a point in the neighborhood of where you want to approximate the function.
2. The approximation is given by the Taylor series, and the polynomial is:

[f(a) (x-a)^0 / 0!] + [f '(a) (x-a)^1 / 1!] + [f ''(a) (x-a)^2 / 2!] + ... + [f^n(a) (x-a)^n / n!]

Example:

f(x) = e^x. Let's approximate around x=0. So a=0.

f(0) = 1.
f '(0) = 1.
f ''(0) = 1.

So

p(x) = 1 + (x) + [(x)^2] / 2 is the best approximation to e^x around x=0 using a 2nd order polynomial.

Higher derivatives represent slopes of slopes of slopes ... of slopes. In terms of the original function, higher order derivatives start losing meaning after the first few (first derivative is slope, 2nd derivative is concavity, third derivative is... what? the 78th derivative...?) However, each provides a little more information on exactly how the function behaves.

The thing about Taylor polynomials is that by matching the derivatives of a function at a point, it becomes harder and harder to distinguish the function and your polynomial in a neighborhood around that point. Imagine using a magnifying glass around a point. If the first, second, third, etc. derivatives match, it starts looking like the same function - exactly. Of course, not all functions work like this, but many do.

1. What is the purpose of using polynomial approximations?

Polynomial approximations are used to simplify complex mathematical functions into a series of simpler polynomial functions. This allows for easier computation and analysis of the original function.

2. How accurate are polynomial approximations?

The accuracy of polynomial approximations depends on the degree of the polynomial used. Generally, the higher the degree, the more accurate the approximation will be. However, using a higher degree polynomial may also result in a more complex function.

3. Can polynomial approximations be used for any type of function?

No, polynomial approximations are most effective for functions that are continuous and smooth. They may not accurately represent functions with sharp discontinuities or oscillations.

4. How do I choose the degree of polynomial for approximation?

The choice of degree for polynomial approximation depends on the desired level of accuracy and the complexity of the original function. It may require some trial and error to find the optimal degree for a given function.

5. Are there any limitations to using polynomial approximations?

Yes, polynomial approximations are limited by the degree of the polynomial used. As the degree increases, so does the complexity of the function, making it more difficult to analyze and interpret. Additionally, polynomial approximations may not accurately represent functions with sharp discontinuities or oscillations.