Find the third degree Taylor polynomial for f(x)=\ln x, centered

hvacwk

hvacwk

Answered question

2021-12-14

Find the third degree Taylor polynomial for f(x)=lnx, centered at a=2?

Answer & Explanation

trisanualb6

trisanualb6

Beginner2021-12-15Added 32 answers

The general form of a Taylor expansion centered at a of an analytical function f:
f(x)=n=0fn(a)n!(xa)n. fn is the n-th derivative of f
The third degree Taylor polynomial consists of the first four (n ranging from 0 to 3) terms of the full Taylor expansion.
f(a)+f(a)(xa)+f(a)2(xa)2+f(a)6(xa)3
f(x)=ln(x), f(x)=1x, f(x)=1x2, f(x)=2x3. Thus, the thirs Taylors polynomial is:
ln(a)+1a(xa)12a2(xa)2+13a3(xa)3
If we have a=2, then
ln(2)+12(x2)18(x2)2+124(x2)3

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