Find the Taylor polynomial of degree 4 at c=1 for the equation
f(x)=ln x

and determine the accuracy of this polynomial at x=2.

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I am going to explain two methods of solving the first part. I like the second approach much better of course:

Part a) Method I-

We know that if the derivatives of all orders at x=c exist, then:
f(x)= f(c)+ f'(c)(x-c)+ f''(c)((x-c)^2/2!)+ f'''(c)((x-c)^3/3!)+...
Now let's go back to the problem. Here f(x)=ln(x) and we want the taylor series about x=1. Therefore, for example:
f(x)= ln(x) --> f(1)= ln(1) =0,
f'(x)= 1/x --> f'(1)= 1/1= 1,
....
we use the above results to write f(x)= ln(x) in the form of:
ln(x)= 0+ 1*(x-1)+...
and we stop and truncate once we find the 4th degree of x. (i.e. here it is the term including (x-1)^4 and will not go as far as (x-1)^5).

part a:) Method II-

Well, the better way in my point of view is to enjoy the taylor series of the famous functions and extend them to what we want to find. You can find the Taylor series of these functions in almost all the calculus textbooks. Trig. functions, exponentials ...

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