The taylor polynomial for the function (f(x)=ln(x)) based at b=1 is (f(x) = x-1) and this can be determined by using the Taylor first degree approximation formula.
Given :
f(x) = ln(x)
The following steps can be used n order to determine the Taylor polynomial:
Step 1 - The Taylor approximation is given by:
\(\rm f(x) = f(b) + \dfrac{f'(b)}{1!}(x-b) + \dfrac{f''(b)}{2!}(x-b)^2+ \dfrac{f'''(b)}{3!}(x-b)^3+....\)
Step 2 - Now, differentiate the given function f(x) = ln(x).
\(\rm f'(x) = \dfrac{1}{x}\)
Step 3 - Now, substitute the value of (x = 1) in the above expression.
\(\rm f'(1) = \dfrac{1}{1}=1\)
Step 3 - Now, the first degree Taylor approximation is given by:
\(\rm f(x) = f(b) + \dfrac{f'(b)}{1!}(x-b)\)
Step 4 - Substitute the values of the known terms in the above expression.
\(\rm f(x) = f(1) + \dfrac{f'(1)}{1!}(x-1)\)
Step 5 - Simplify the above expression.
f(x) = ln(1) + (x - 1)
f(x) = x - 1
So, the taylor polynomial for the function (f(x)=ln(x)) based at b=1 is (f(x) = x-1).
For more information, refer to the link given below:
https://brainly.com/question/2096984
