Please help. I got stuck on this problem all week. And please show steps. Use Taylor Inequality to estimate approximation f(x)Tn(x) when x lies in the given interval. Round your answer to six decimal places. a=3, n=3, 2.5<=x<=3.5?
well, the formula is
|Rk(x)| <= M/(n+1)! |x-a|n+1
Now, 2.5 <= x <= 3.5 means |x-3| <= 1/2
Using your values, that means that
|R3(x)| <= M/4! * 1/16
where M is a bound on the 4th derivative of f(x), which you have not specified, but maybe you can fix that
The function is f(x)=ln(1+2x) but the 4th derivative I don't know. So, how do I plug in now?
To use Taylor's Inequality to estimate the error in approximating f(x) by its Taylor polynomial Tn(x), we need to know the value of the (n+1)-th derivative of f(x) and the upper bound M for the absolute value of the (n+1)-th derivative on the interval of interest.
In this case, you mentioned that n=3, so we need to find the 4th derivative of f(x) and determine the upper bound M on the interval [2.5, 3.5].
First, let's find the 4th derivative of f(x). You haven't provided the function f(x), so we cannot find the exact value of the derivative without that information. However, I can guide you through the process if you provide the function.
Once we have the 4th derivative, we can find the maximum absolute value of the 4th derivative, M, on the interval [2.5, 3.5].
Based on the provided information, we only have the value of a=3, so it's not possible to determine the exact function or its derivatives without additional information.
If you provide the function f(x) or additional details, I can guide you through finding the 4th derivative and estimate the approximation.