Taylor polynomial approximation (HELP ME)

In summary, the conversation discusses determining the degree of the Taylor polynomial about c=1 that should be used to approximate ln(1.2) with an error less than 0.001. The book provides steps and an expression, |Rn(1.2)| = (.02)^(n+1)/(z^(n+1)*(n+1)), but then introduces a second expression, (.02)^(n+1)/(n+1) < 0.001, with an error pointing to 1000 < (n+1)(5^(n+1)) and the question of why the z term was left out. The conversation ends with a request for an explanation of how the process works in general.
  • #1
frasifrasi
276
0
Ok, we are asked to determined the degree of the the taylor polynomial about c =1 that should be used to approximate ln (1.2) so the error is less than .001


the book goes throught the steps and arrives at:

|Rn(1.2)| = (.02)^(n+1)/(z^(n+1)*(n+1)


but then, it states that

(.02)^(n+1)/(n+1) < .001

and there is an error pointing this expression to

1000 < (n+1)(5^(n+1))

--> I have no idea how the book arrived at this second expression, could anyone please explain?

- Also, if anyone knows, why was the z term left out in the second step?


Thank you.
 
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  • #2
why was the z term dropped?

does anyone know?
 
  • #3
I just don't believe your book has what you wrote. For one thing, |Rn(1.2)| = (.02)^(n+1)/(z^(n+1)*(n+1) is missing a ")". For another, the remainder formula has z in the numerator not the denominator- You can't make the error smaller by choosing z larger!
 
  • #4
OK, in general how does this process work? I am having a little trouble following the steps...

*Yeah, I forgot the last parenthesis
 

FAQ: Taylor polynomial approximation (HELP ME)

What is Taylor polynomial approximation?

Taylor polynomial approximation is a mathematical technique used to approximate a function using a polynomial that is centered at a specific point. This method is based on the Taylor series, which is an infinite sum of terms that represent the derivatives of the function evaluated at the center point.

Why is Taylor polynomial approximation useful?

Taylor polynomial approximation is useful because it allows us to approximate complex functions with simpler polynomials. This can make it easier to analyze and work with these functions, especially in cases where the function cannot be easily integrated or differentiated.

How do you find the Taylor polynomial approximation of a function?

To find the Taylor polynomial approximation of a function, you need to determine the center point and the number of terms in the polynomial. Then, you can use the Taylor series formula to find the coefficients of the polynomial. Finally, substitute these coefficients into the polynomial to get the approximation.

What are the limitations of Taylor polynomial approximation?

One limitation of Taylor polynomial approximation is that it is only accurate within a certain interval around the center point. Outside of this interval, the approximation may not be accurate. Additionally, as the number of terms in the polynomial increases, the accuracy of the approximation may decrease.

How is Taylor polynomial approximation different from linear approximation?

Taylor polynomial approximation is a generalization of linear approximation, which only uses the first two terms of the Taylor series. Taylor polynomial approximation can use any number of terms, making it more accurate than linear approximation. Additionally, Taylor polynomial approximation can be used for any type of function, while linear approximation is limited to linear functions only.

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