Taylor's Theorem: f(x) & Error Estimation

[snowing_tsw]

Homework Statement

f(x) = (1+x)^(1/3)
Use taylor's theorem to write down an expression for the error R_3(x) = f(x) - P_3(x). In what interval does the unknown constant c lie?

I can't find what value is x where c lies between x and a.

Homework Equations

in previous question, we are required to find 6^(1/3) by doing 2(1- 1/4)^(1/3)

The Attempt at a Solution

i got a= 0 and c must lie between a and x.
and R_3(x) = 10x^4/[243(1-x)^(11/3)]

i know the first boundary of c must be zero but i can't find the other boundary

 

[mighty2000]

of c.
Hello,

Thank you for your question. First, let's review Taylor's theorem. It states that any function f(x) that is infinitely differentiable at a point x=a can be approximated by a polynomial P_n(x) of degree n, where the error R_n(x) = f(x) - P_n(x) is given by:

R_n(x) = (1/n!) * f^(n+1)(c) * (x-a)^(n+1)

where c is some unknown constant between x and a.

In this case, we are given the function f(x) = (1+x)^(1/3) and we want to approximate it using a third-degree polynomial P_3(x). This means that n = 3, and the error R_3(x) is given by:

R_3(x) = (1/3!) * f^(4)(c) * (x-a)^4

To find the unknown constant c, we need to find the value of x where it lies between x and a. In this case, a = 0, so we need to find the value of c between 0 and x. Since we are trying to approximate the value of f(x) = (1+x)^(1/3) at x = 6^(1/3), we can set x = 6^(1/3) in the error equation to get:

R_3(6^(1/3)) = (1/3!) * f^(4)(c) * (6^(1/3) - 0)^4

So the unknown constant c lies between 0 and 6^(1/3). However, we can further narrow down this interval by finding the value of c that makes the error R_3(x) as small as possible. To do this, we need to find the maximum value of the fourth derivative of f(x) between 0 and 6^(1/3). This can be done by taking the derivative of f(x) four times and setting it equal to 0, then solving for x. The result is x = 2^(1/3).

So, the unknown constant c lies between 0 and 2^(1/3) in order to minimize the error R_3(x). I hope this helps! Let me know if you have any further questions.