Math Help - Percentage Errors using the Maclaurin Series

1. Percentage Errors using the Maclaurin Series

Hi, I'm having trouble trying to find the percentage errors of P3(x) and P4(x).

Here's the question;
When x=0.7891.

f(x)=x/(sin(x)+2)
f(x)=0.2912

P3(x)=1/2*x-1/4*x^2
P3(x) =0.2389

P4(x)=1/2*x-1/4*x^2+1/8*x^3
P4(x) =0.3003

Find the percentage errors in using P3(x), and P4(x) to approximate f(x).

Thanks, DLL.

2. The formula for relative error is simply: $\eta=\left | \frac{\Delta f(x)}{f(x)}\right |$ where $\Delta f(x)=f(x)-f(x)_{approximation}$ for some value of $x$.

3. I'm not sure how to use this formula as I have never come across it before. I was thinking of the Maclarin Remaider Therom. I'm not sure how I use these errors to approximate f(x) however.

4. I don't need to use the relative error formula. I need to find the percentage errors of in usig P3(x) and P4(x) to approximate f(x).

f(x)=x/(sins(x)=2) and the Maclurin polynomials are P3(x)=1/2*x-1/4*x^2 and P4(x)=1/2*x-1/4*x^2+1/8*x^3, when x=0.7891.

I'm not sure how I use the percentage errors or use them to approximate f(x).

5. If you express the error as percentages, then it's relative to the true value. You can find the error of using a Maclaurin series to approximate a function, by using the Lagrange remainder theorem, but the answer you get is going to be the absolute error.

The remainder is $r(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}x^{n+1}$ for some $\xi \in [0,x]$

If you want to the find the absolute error by using the Lagrange remainder theorem, then you first need to find $f^{4}(\xi)$ for the first polynomial ( $P_3$) and $f^{5}(\xi)$ for the second ( $P_4$).

6. Okay, thanks Spec.

I used the relative error as you said and I got reasonable anwers. I think what I did was right so thanks for the insight on that.