Taylor Polynomials: Exploring Different Derivations

In summary, the conversation discusses the process of deriving Taylor polynomials for various functions, specifically for cos(t^2) and ln(x^2). The use of chain rule is mentioned and it is noted that the result given in the conversation is incorrect. The conversation ends with a suggestion to calculate the derivatives more carefully, as the chain rule should give a correct result.
  • #1
georg gill
153
6
http://bildr.no/view/1030479

The link above, it is my own and it is a bit disorderly, I think should explain taylor polynomials. In one assignent one had an assignment to derive taylor polynomials for

[tex]cost^2[/tex]

If one use the derivation rules with chain one get 2t for first derivative and [tex]4t^4[/tex]
for second and so on. If t=0 and we are looking at a maclaurin series then every term except the first cos(0) becomes zero. I can understand that one could say [tex]u=t^2[/tex] and derieve taylor series for cosx and just use afterwards [tex]u=t^2[/tex].

But what is even more confusing is that in one other assignment one were too find taylor polynomial for [tex]y=ln(x^2)[/tex]. Here one uses chain rule and it does work:

[tex]y'=\frac{2}{x}[/tex] [tex]y''=\frac{-2}{x^2}[/tex] [tex]y''=\frac{4}{x^4}[/tex]


What is the difference?
 
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  • #2
georg gill said:
...
taylor polynomials for

[tex]cost^2[/tex]

If one use the derivation rules with chain one get 2t for first derivative and [tex]4t^4[/tex]
for second and so on. If t=0 and we are looking at a maclaurin series then every term except the first cos(0) becomes zero.

Try to calculate the derivatives more carefully, because what you stated is false. The first term is not the only non-zero term (there are infinitely many non-zero terms). The chain rules gives a correct result. (perhaps you are forgetting about the product rule)
 

1. What are Taylor polynomials?

Taylor polynomials are a type of mathematical function used to approximate more complex functions. They are composed of a series of terms that involve the function's derivatives evaluated at a specific point.

2. How are Taylor polynomials derived?

Taylor polynomials can be derived using several different methods, including the "brute force" method of calculating the derivatives and using the Taylor series formula. Other methods include using the Lagrange form of the remainder term and the binomial theorem.

3. Why are Taylor polynomials useful?

Taylor polynomials are useful because they allow us to approximate more complex functions with simpler, polynomial functions. This makes it easier to perform calculations and analyze the behavior of these functions.

4. Can Taylor polynomials be used for any function?

Technically, Taylor polynomials can be used for any function, but the accuracy of the approximation will depend on the function and the point at which it is evaluated. Some functions may require a higher degree Taylor polynomial to achieve a more accurate approximation.

5. How can I verify the accuracy of a Taylor polynomial approximation?

The accuracy of a Taylor polynomial approximation can be verified by comparing it to the original function at different points. As the degree of the polynomial increases, the approximation should become more accurate. Additionally, the error can be calculated using the remainder term formula to determine the maximum possible error for a given degree polynomial.

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