Taylor/Polynomial Approximation Question

In summary, when creating an nth order Taylor Polynomial of a polynomial function and centering it at a specific value, the resulting approximate polynomial will be exact for all values of x if the degree of the approximation matches the original polynomial. Otherwise, it will only be a good approximation near the centered value.
  • #1
Lebombo
144
0
If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a.

So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5

and I want to find an approximate function, g(x), centered at 5.

From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5).

However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)?

Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.
 
Physics news on Phys.org
  • #2
Lebombo said:
If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a.

So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5

and I want to find an approximate function, g(x), centered at 5.

From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5).

However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)?

Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.

Your "approximation" polynomial g(x) will agree exactly with f(x) for all values. All you're really doing is changing from a polynomial that is a sum of powers of x (f) to one that is a sum of powers of x - 5.
 
  • #3
Lebombo said:
If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a.

So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5

and I want to find an approximate function, g(x), centered at 5.

From what I gather, this means that after I create the Taylor Polynomial of 5 degrees, when I plug in 5 to both f(x) and g(x), they will be equal values. That is, f(5) = g(5).

However, does this also mean that f(x) = g(x) for all values of x (not just the centered value) if one polynomial is approximated (Taylor polynomial) from another polynomial? (as opposed to a polynomial being approximated from a non-polynomial)?

Note: I'm not considering infinite series in this question, only nth term Taylor polynomials.

If your expansion around x=5 is degree 5, like the original polynomial, the 'approximate' polynomial will be exact everywhere. It will be the same as the original polynomial if you expand it out, because the terms in the Taylor expansion vanish after degree 5. If it's less than degree 5 then it will only be a good approximation near x=5.
 

What is Taylor/Polynomial Approximation?

Taylor/Polynomial Approximation is a mathematical technique used to approximate a function with a polynomial expression. It involves finding a polynomial that closely matches the behavior of the given function at a particular point or interval.

Why is Taylor/Polynomial Approximation important?

Taylor/Polynomial Approximation is important because it allows us to approximate functions that are difficult to integrate or differentiate. It is also used in fields such as physics, engineering, and economics to model real-world phenomena.

What is the difference between Taylor and Polynomial Approximation?

Taylor Approximation is a specific type of Polynomial Approximation that uses derivatives of a function to determine the coefficients of the polynomial. Polynomial Approximation, on the other hand, is a more general method that can be used with or without derivatives.

What is the Taylor series?

The Taylor series is an infinite sum of terms that represent the coefficients of the Taylor polynomial. It is used to approximate a function at a particular point or interval, with the accuracy of the approximation increasing as more terms are added to the series.

How is Taylor/Polynomial Approximation used in real life?

Taylor/Polynomial Approximation has many practical applications, such as in physics for approximating the motion of objects, in finance for predicting stock prices, and in computer graphics for creating smooth curves and surfaces.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
879
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
960
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
244
  • Calculus and Beyond Homework Help
Replies
9
Views
426
  • Calculus and Beyond Homework Help
Replies
5
Views
937
  • Calculus and Beyond Homework Help
Replies
27
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top