Approximate the function f(x) = \ln(1+2x) by a Taylor polynomial with degree n = 3 at the...

Question:

Approximate the function {eq}f(x) = \ln(1+2x) {/eq} by a Taylor polynomial with degree n = 3 at the number a = 3.

Taylor Polynomial:

The Taylor polynomial {eq}T^{3}(x) {/eq} for the function {eq}f {/eq} centered at the number {eq}a=3 {/eq} is:

{eq}\displaystyle f(3) + \dfrac {\dfrac {d}{dx}\left( f(3) \right) }{1!}(x-3) + \dfrac {\dfrac {d^2}{dx^2}\left( f(3) \right) }{2!} (x-3)^2 + \dfrac {\dfrac {d^3}{dx^3}\left( f(3) \right) }{3!}(x-3)^3 =\sum ^{3 }_{n=0}\dfrac {x^{n}f^{\left( n\right) }\left( 3\right) }{n!} {/eq}

Answer and Explanation:

The Taylor polynomial {eq}T_3(x) {/eq} for the given function {eq}f(x) = \ln(1+2x) {/eq} about {eq}a=3 {/eq} is:

{eq}= \ln(7)+\dfrac...

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Taylor Series: Definition, Formula & Examples

from AP Calculus AB & BC: Help and Review

Chapter 8 / Lesson 10
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