Taylor Polynomial (n=4) for g(x): Showing 0<E4(x)<80x^5

In summary, the conversation discusses finding a TP (n=4) for the function g(x) = (1+5x)^1/5 and proving that 0<E4(x)<80x^5 when x>0. The summary also includes the calculation for E4(x) as (399/[5(1+5X)^24/5] ) *x^5 and the realization that taking X=0 results in the largest possible error.
  • #1
sony
104
0
Hi

I have found the following TP (n=4) for g(x) = (1+5x)^1/5
P4(x) = 1+x-2x^2+6x^3-21x^4

Then they ask me to show that 0<E4(x)<80x^5 when x>0.

I don't know how to start, or exactly what I am supposed to show...?

I have found E4(x) to be( 399/[5(1+5X)^24/5] ) *x^5...

And 0<X<x ...?
 
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  • #2
To show that the error is LESS than something, you want to think about the X that gives the LARGEST possible error. Since X is in the denominator, the largest value of the fraction will be when X= 0. If you take X= 0 what is that value in the parentheses? (Gosh, 399 is awful close to 400!)
 
  • #3
Oh now I see how I get 0<E4(x)<80x^5

Thanks!
 

1. What is a Taylor Polynomial for g(x)?

A Taylor Polynomial for g(x) is a mathematical expression that approximates the value of a function at a specific point by using a finite number of terms in its power series. It is useful in calculus for calculating values of functions that are difficult to evaluate directly.

2. What does "n=4" in the Taylor Polynomial refer to?

The "n=4" in the Taylor Polynomial indicates that the polynomial is using the first four terms of the function's power series to approximate the value at a specific point. In this case, n=4 means that the polynomial will use the first four derivatives of the function to calculate the approximation.

3. What does "0

This statement means that the error, or the difference between the actual value of the function and the approximation, is between 0 and 80 times x raised to the power of 5. This is a measure of the accuracy of the approximation, and a smaller error indicates a more precise approximation.

4. How is the Taylor Polynomial for g(x) calculated?

The Taylor Polynomial for g(x) is calculated by finding the first four derivatives of the function at the specific point, evaluating those derivatives at the point, and then plugging those values into the Taylor Polynomial formula, which is: f(a) + f'(a)(x-a) + f''(a)(x-a)^2 + f'''(a)(x-a)^3 + ... + f(n)(a)(x-a)^n. The result is the approximation of the function's value at that point.

5. What is the significance of using a Taylor Polynomial with n=4?

Using a Taylor Polynomial with n=4 allows for a more accurate approximation of the function's value at a specific point compared to using a polynomial with a lower value of n. However, using a higher value of n, such as n=10, would result in an even more precise approximation. The choice of n depends on the desired level of accuracy and the complexity of the function.

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