Find the Taylor polynomials of orders 0,1,2, and 3 generated by f(x)=\ln(9+x) at x=4

Rui Baldwin

Rui Baldwin

Answered question

2021-09-03

Find the Taylor polynomials of orders 0,1,2, and 3 generated by f(x)=ln(9+x) at x=4

Answer & Explanation

Clelioo

Clelioo

Skilled2021-09-04Added 88 answers

We have to find the Taylor polynomial of order 0,1,2, and 3 generated by f(x)=ln(9+x) at x=4
We begin with the formula for a Taylor polynomial generated at x=a:
k=0fk(a)k!(xa)k=f(a)+f(a)1!(xa)1+f(a)2!(xa)2++fn(a)n!(xa)n+
The Taylor polynomial of order n refers to the first n+1 (to the nth derivative) terms of this Taylor polynomial.
We are given f(x)=ln(9+x) and we are to construct the first three Taylor polynomials centered at a=4. First, let's find the first three derivatives as follows:
f(x)=ln(9+x)f(4)=ln(9+4)=ln(13)
f(x)=19+xf(4)=19+4=113
f(x)=1(9+x)2f(4)=1(9+4)2=1169
f(x)=2(9+x)3f(4)=2(9+4)3=2133
We can now construct the first Taylor polynomials of orders 0,1,2, and 3 as follows:
P0(x)=f(4)=ln(13)
P1(x)=f(4)+f(4)1!(x4)1=ln(13)+113(x4)

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