Consider the following function f(x) = \ln(1 + 2x),\ a = 3,\ n = 3,\ 2.6 ≤ x

Falak Kinney

Falak Kinney

Answered question

2021-09-07

Consider the following function f(x)=ln(1+2x), a=3, n=3, 2.6x3.4 Approximate f by a Taylor polynomial with degree n at the number a. Use Taylor's Inequality to estimate the accuracy of the approximation f(x)Tn(x) when x lies in the given interval. (Round your answer to six decimal places.) |R3(x)|0.000001

Answer & Explanation

Dora

Dora

Skilled2021-09-08Added 98 answers

The given function is f(x)=ln(1+2x). We have to find Tayulor polynomial of degree 3.
Note that the formula for the Taylor Polynomial, about the point a, of degree n is given by
Tn(x)=k=0nf(k)(a)k!(xa)k
For n=3 and a=3 we have
P(x)=T3(x)=f(3)+f(3)1!(x)+f(3)2!(x)2+f(3)3!(x)3
Now note the following
f(3)=ln(7)
f(x)=(ln(2x+1))=22x+1f(3)27
f(x)=(22x+1)=4(2x+1)2f(3)=449
f(x)=(4(2x+1)2)=16(2x+1)3f(3)=16343
Plugging above values in the formula for Taylor polynomial we get
P(x)=ln(7)0!(x(3))0+271!(x(3))1+4492!(x(3))2+163433!(x(3))3
We know that the error in the approximation f(x)Tn(x) is given by
Rn(x)=M(n+1)!(xa)n+1
where

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