Chapter 12 - Sequences and Series - 12.3 Taylor Polynomials at 0 - 12.3 Exercises - Page 631: 2

$${P_4}\left( x \right) = 1 + 3x + \frac{9}{2}{x^2} + \frac{9}{2}{x^3} + \frac{{27}}{8}{x^4}$$

$$\eqalign{ & f\left( x \right) = {e^{3x}} \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[ {{e^{3x}}} \right] = 3{e^{3x}} \cr & {f^{\left( 2 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ {3{e^{3x}}} \right] = 9{e^{3x}} \cr & {f^{\left( 3 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ {9{e^{3x}}} \right] = 27{e^{3x}} \cr & {f^{\left( 4 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ {27{e^{3x}}} \right] = 81{e^{3x}} \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) = {e^{3\left( 0 \right)}} = 1 \cr & {f^{\left( 1 \right)}}\left( 0 \right) = 3{e^{3\left( 0 \right)}} = 3 \cr & {f^{\left( 2 \right)}}\left( 0 \right) = 9{e^{3\left( 0 \right)}} = 9 \cr & {f^{\left( 3 \right)}}\left( 0 \right) = 27{e^{3\left( 0 \right)}} = 27 \cr & {f^{\left( 4 \right)}}\left( 0 \right) = 81{e^{3\left( 0 \right)}} = 81 \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) = 1 + \frac{3}{{1!}}x + \frac{9}{{2!}}{x^2} + \frac{{27}}{{3!}}{x^3} + \frac{{81}}{{4!}}{x^4} \cr & {\text{simplifying}} \cr & {P_4}\left( x \right) = 1 + 3x + \frac{9}{2}{x^2} + \frac{9}{2}{x^3} + \frac{{27}}{8}{x^4} \cr} $$