Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - 12.3 Taylor Polynomials at 0 - 12.3 Exercises - Page 631: 11

Answer

$${P_4}\left( x \right) = - x - \frac{1}{2}{x^2} - \frac{1}{3}{x^3} - \frac{1}{4}{x^4}$$

Work Step by Step

$$\eqalign{ & f\left( x \right) = \ln \left( {1 - x} \right) \cr & {\text{Use the definition of Taylor Polynomial of Degree }}n\,\,\,\left( {{\text{see page 629}}} \right) \cr & {\text{Let }}f{\text{ be a function that can be differentiated }}n{\text{ times at 0}}{\text{. The Taylor }} \cr & {\text{polynomial of degree }}n{\text{ for }}f{\text{ at 0 is }} \cr & {P_n}\left( x \right) = f\left( 0 \right) + \frac{{{f^{\left( 1 \right)}}\left( 0 \right)}}{{1!}}x + \frac{{{f^{\left( 2 \right)}}\left( 0 \right)}}{{2!}}{x^2} + \cdots + \frac{{{f^{\left( n \right)}}\left( 0 \right)}}{{n!}}{x^n} = \sum\limits_{i = 0}^n {\frac{{{f^{\left( n \right)}}\left( 0 \right)}}{{i!}}} {x^i} \cr & {\text{Find the Taylor polynomials of degree 4 at 0}}{\text{. }} \cr & {\text{then }}n = 4. \cr & {\text{The }}n{\text{ - th derivatives are}} \cr & {f^{\left( 1 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ {\ln \left( {1 - x} \right)} \right] = - \frac{1}{{\left( {1 - x} \right)}} = - {\left( {1 - x} \right)^{ - 1}} \cr & {f^{\left( 2 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ { - {{\left( {1 - x} \right)}^{ - 1}}} \right] = {\left( {1 - x} \right)^{ - 2}}\left( { - 1} \right) = - {\left( {1 - x} \right)^{ - 2}} \cr & {f^{\left( 3 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ { - {{\left( {1 - x} \right)}^{ - 2}}} \right] = - \left( { - 2} \right){\left( {1 - x} \right)^{ - 3}}\left( { - 1} \right) = - 2{\left( {1 - x} \right)^{ - 3}} \cr & {f^{\left( 4 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ { - 2{{\left( {1 - x} \right)}^{ - 3}}} \right] = - 2\left( { - 3} \right){\left( {1 - x} \right)^{ - 4}}\left( { - 1} \right) = - 6{\left( {1 - x} \right)^{ - 4}} \cr & {\text{evaluate }}f\left( 0 \right),{f^{\left( 1 \right)}}\left( 0 \right),{f^{\left( 2 \right)}}\left( 0 \right),{f^{\left( 3 \right)}}\left( 0 \right),{f^{\left( 4 \right)}}\left( 0 \right) \cr & f\left( 0 \right) = \ln \left( {1 - 0} \right) = 0 \cr & {f^{\left( 1 \right)}}\left( 0 \right) = - {\left( {1 - 0} \right)^{ - 1}} = - 1 \cr & {f^{\left( 2 \right)}}\left( 0 \right) = - {\left( {1 - 0} \right)^{ - 2}} = - 1 \cr & {f^{\left( 3 \right)}}\left( 0 \right) = - 2{\left( {1 - 0} \right)^{ - 3}} = - 2 \cr & {f^{\left( 4 \right)}}\left( 0 \right) = - 6{\left( {1 - 0} \right)^{ - 4}} = - 6 \cr & {\text{Replace the found values into the definition of Taylor Polynomial of Degree }}n \cr & {\text{for }}n = 4 \cr & {P_4}\left( x \right) = f\left( 0 \right) + \frac{{{f^{\left( 1 \right)}}\left( 0 \right)}}{{1!}}x + \frac{{{f^{\left( 2 \right)}}\left( 0 \right)}}{{2!}}{x^2} + \frac{{{f^{\left( 3 \right)}}\left( 0 \right)}}{{3!}}{x^3} + \frac{{{f^{\left( 4 \right)}}\left( 0 \right)}}{{4!}}{x^4} \cr & {P_4}\left( x \right) = 0 + \frac{{ - 1}}{{1!}}x + \frac{{ - 1}}{{2!}}{x^2} + \frac{{ - 2}}{{3!}}{x^3} + \frac{{ - 6}}{{4!}}{x^4} \cr & {\text{simplify by using a calculator}} \cr & {P_4}\left( x \right) = - x - \frac{1}{2}{x^2} - \frac{1}{3}{x^3} - \frac{1}{4}{x^4} \cr} $$
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