Taylor Polynomial Approximations.

In summary, the conversation discusses the concept of Taylor polynomial approximation and its use in approximating a function at a given point. The speaker expresses confusion about why all the approximations need to be summed up rather than just using the final one. The responder shares a link explaining the concept and provides an example of Taylor polynomials for the function e^x. The conversation ends with the speaker asking for a better explanation of the concept.
  • #1
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Hello,

I'm new here, nice to meet you guys i was in class today and just didn't understand the taylor polynomial approximation, the professor started out approximating a function by polynomials of degree N, he first showed us how a linear polynomial was a crude approximation of the function at a given point but thing i don't get is when he moved on to the next approximation by using the first derivative, than the second...i just didnt get why the final formula added up all the approximations together?...

I just didnt get why like we had to sum up all the approximations rather than just making a formula for the last approximation of the function?...if I am not making any sense I am sorry but if you guys can explain the taylor polynomial a bit better i would greatly appreciate it.



Thanks,

Moe.
 
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  • #2
I wrote a little bit about this some time ago... does this help you get started? https://www.physicsforums.com/blog.php?b=1758
 
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  • #3
Not sure what you mean but if you take the Taylor polynomials for f(x)=e^x about x=0 (also called the Maclaurin polynomials for e^x) they are:

P_0(x) = 1
P_1(x) = 1 + x
P_2(x) = 1 + x + (1/2)x^2
P_3(x) = 1 + x + (1/2)x^2 + (1/6)x^3
etc.

So the first k+1 terms of a Taylor polynomial of degree n is the Taylor polynomial for the same function about the same point of degree k.

If you mean that we should just take the highest degree terms of the Taylor polynomials, in this example 1, x, (1/2)x^2, (1/6)x^3 etc. these terms alone would NOT be good approximations of the original function.
 

1. What are Taylor Polynomial Approximations?

Taylor Polynomial Approximations are mathematical tools used to approximate a function by using a polynomial of a certain degree. These approximations are based on the Taylor series, which is an infinite series that represents a function as a sum of its derivatives at a single point.

2. How are Taylor Polynomial Approximations different from other types of approximations?

Taylor Polynomial Approximations are considered to be more accurate than other types of approximations because they are based on the derivatives of a function at a specific point. This allows for a more precise estimation of the function's behavior near that point.

3. How do you calculate Taylor Polynomial Approximations?

To calculate a Taylor Polynomial Approximation, you need to know the derivatives of the function at a given point. These derivatives are used to construct the polynomial, and the more derivatives you include, the more accurate the approximation will be. The formula for a Taylor Polynomial is: f(x) ≈ f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + ... + (f(n)(a)/n!)(x-a)^n, where n is the degree of the polynomial and a is the point of approximation.

4. What are some real-life applications of Taylor Polynomial Approximations?

Taylor Polynomial Approximations are commonly used in engineering, physics, and other scientific fields to approximate complex functions. They are also used in computer graphics to create smooth curves and surfaces. In finance, they can be used to estimate the future value of investments based on past performance. Additionally, they are used in numerical analysis to solve differential equations and other problems.

5. Can Taylor Polynomial Approximations be used for any function?

No, Taylor Polynomial Approximations can only be used for functions that are infinitely differentiable, meaning that they have derivatives of all orders at a given point. Additionally, the function must be defined and continuous at that point. If these conditions are not met, then the Taylor series may not converge and the approximation will not be accurate.

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