Notes: `f(x)=sin(3x) ,c=0` Use the definition of Taylor series to find the Taylor series, centered at c for the function.

Taylor series is an example of infinite series derived from the expansion of $\displaystyle{f{{\left({x}\right)}}}$ about a single point. It is represented by infinite sum of $\displaystyle{{f}}^{{n}}{\left({x}\right)}$ centered at $\displaystyle{x}={c}$ . The general formula for Taylor series is:

$\displaystyle{f{{\left({x}\right)}}}={\sum_{{{n}={0}}}^{\infty}}\frac{{{{f}}^{{n}}{\left({c}\right)}}}{{{n}!}}{{\left({x}-{c}\right)}}^{{n}}$

$\displaystyle{f{{\left({x}\right)}}}={f{{\left({c}\right)}}}+{f{'}}{\left({c}\right)}{\left({x}-{c}\right)}+\frac{{{{f}}^{{2}}{\left({c}\right)}}}{{{2}!}}{{\left({x}-{c}\right)}}^{{2}}+\frac{{{{f}}^{{3}}{\left({c}\right)}}}{{{3}!}}{{\left({x}-{c}\right)}}^{{3}}+\frac{{{{f}}^{{4}}{\left({c}\right)}}}{{{4}!}}{{\left({x}-{c}\right)}}^{{4}}+\ldots$

To apply the definition of Taylor series for the given function $\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({3}{x}\right)}}}$ , we list $\displaystyle{{f}}^{{n}}{\left({x}\right)}$ using the derivative formula for trigonometric function: $\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\sin{{\left({u}\right)}}}={\cos{{\left({u}\right)}}}\cdot\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}$ and $\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\cos{{\left({u}\right)}}}=-{\sin{{\left({u}\right)}}}\cdot\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}$ .

Let $\displaystyle{u}={3}{x}$ then $\displaystyle\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}={3}$ .

$\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({3}{x}\right)}}}$

$\displaystyle{f{'}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle={\cos{{\left({3}{x}\right)}}}\cdot{3}$

$\displaystyle={3}{\cos{{\left({3}{x}\right)}}}$

$\displaystyle{{f}}^{{2}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}{3}{\cos{{\left({3}{x}\right)}}}$

$\displaystyle={3}\frac{{d}}{{{\left.{d}{x}\right.}}}{\cos{{\left({3}{x}\right)}}}$

$\displaystyle={3}\cdot{\left(-{\sin{{\left({3}{x}\right)}}}\cdot{3}\right)}$

$\displaystyle=-{9}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle{{f}}^{{3}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}-{9}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle=-{9}\frac{{d}}{{{\left.{d}{x}\right.}}}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle=-{9}\cdot{\cos{{\left({3}{x}\right)}}}\cdot{3}$

$\displaystyle=-{27}{\cos{{\left({3}{x}\right)}}}$

$\displaystyle{{f}}^{{4}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}-{27}{\cos{{\left({3}{x}\right)}}}$

$\displaystyle=-{27}\cdot\frac{{d}}{{{\left.{d}{x}\right.}}}{\cos{{\left({3}{x}\right)}}}$

$\displaystyle=-{27}\cdot{\left(-{\sin{{\left({3}{x}\right)}}}\cdot{3}\right)}$

$\displaystyle={81}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle{{f}}^{{5}}{\left({x}\right)}=\frac{{d}}{{{\left.{d}{x}\right.}}}{81}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle={81}\cdot\frac{{d}}{{{\left.{d}{x}\right.}}}{\sin{{\left({3}{x}\right)}}}$

$\displaystyle={81}\cdot{\left({\cos{{\left({3}{x}\right)}}}\cdot{3}\right)}$

$\displaystyle={243}{\cos{{\left({3}{x}\right)}}}$

Plug-in $\displaystyle{x}={0}$ on each $\displaystyle{{f}}^{{n}}{\left({x}\right)}$ , we get:

$\displaystyle{f{{\left({0}\right)}}}={\sin{{\left({3}\cdot{0}\right)}}}$

$\displaystyle={\sin{{\left({0}\right)}}}$

$\displaystyle={0}$

$\displaystyle{f{'}}{\left({0}\right)}={3}{\cos{{\left({3}\cdot{0}\right)}}}$

$\displaystyle={3}{\cos{{\left({0}\right)}}}$

$\displaystyle={3}\cdot{1}$

$\displaystyle={3}$

$\displaystyle{{f}}^{{2}}{\left({0}\right)}=-{9}{\sin{{\left({3}\cdot{0}\right)}}}$

$\displaystyle=-{9}{\sin{{\left({0}\right)}}}$

$\displaystyle=-{9}\cdot{0}$

$\displaystyle={0}$

$\displaystyle{{f}}^{{3}}{\left({0}\right)}=-{27}{\cos{{\left({3}\cdot{0}\right)}}}$

$\displaystyle=-{27}{\cos{{\left({0}\right)}}}$

$\displaystyle=-{27}\cdot{1}$

$\displaystyle=-{27}$

$\displaystyle{{f}}^{{4}}{\left({0}\right)}={81}{\sin{{\left({3}\cdot{0}\right)}}}$

$\displaystyle={81}{\sin{{\left({0}\right)}}}$

$\displaystyle={81}\cdot{0}$

$\displaystyle={0}$

$\displaystyle{{f}}^{{5}}{\left({0}\right)}={243}{\cos{{\left({3}\cdot{0}\right)}}}$

$\displaystyle={243}{\cos{{\left({0}\right)}}}$

$\displaystyle={243}\cdot{1}$

$\displaystyle={243}$

Plug-in the values on the formula for Taylor series, we get:

$\displaystyle{\sin{{\left({3}{x}\right)}}}={\sum_{{{n}={0}}}^{\infty}}\frac{{{{f}}^{{n}}{\left({0}\right)}}}{{{n}!}}{{\left({x}-{0}\right)}}^{{n}}$

$\displaystyle={\sum_{{{n}={0}}}^{\infty}}\frac{{{{f}}^{{n}}{\left({0}\right)}}}{{{n}!}}{{x}}^{{n}}$

$\displaystyle={f{{\left({0}\right)}}}+{f{'}}{\left({0}\right)}{x}+\frac{{{{f{'}}}^{{2}}{\left({0}\right)}}}{{{2}!}}{{x}}^{{2}}+\frac{{{{f}}^{{3}}{\left({0}\right)}}}{{{3}!}}{{x}}^{{3}}+\frac{{{{f}}^{{4}}{\left({0}\right)}}}{{{4}!}}{{x}}^{{4}}+\frac{{{{f}}^{{4}}{\left({0}\right)}}}{{{4}!}}{{x}}^{{4}}+\ldots$

$\displaystyle={0}+{3}{x}+\frac{{0}}{{{2}!}}{{x}}^{{2}}+\frac{{-{27}}}{{{3}!}}{{x}}^{{3}}+\frac{{0}}{{{4}!}}{{x}}^{{4}}+\frac{{243}}{{{5}!}}{{x}}^{{5}}+\ldots$

$\displaystyle={0}+{3}{x}+\frac{{0}}{{2}}{{x}}^{{2}}+\frac{{-{27}}}{{6}}{{x}}^{{3}}+\frac{{0}}{{24}}{{x}}^{{4}}+\frac{{243}}{{120}}{{x}}^{{5}}+\ldots$

$\displaystyle={0}+{3}{x}+{0}-\frac{{9}}{{2}}{{x}}^{{3}}+{0}+\frac{{81}}{{40}}{{x}}^{{5}}+\ldots$

$\displaystyle={3}{x}-\frac{{9}}{{2}}{{x}}^{{3}}+\frac{{81}}{{40}}{{x}}^{{5}}+\ldots$

The Taylor series for the given function $\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({3}{x}\right)}}}$ centered at $\displaystyle{c}={0}$ will be:

$\displaystyle{\sin{{\left({3}{x}\right)}}}={3}{x}-\frac{{9}}{{2}}{{x}}^{{3}}+\frac{{81}}{{40}}{{x}}^{{5}}+\ldots$

Notes

Monday, 26 August 2013

$\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({3}{x}\right)}}},{c}={0}$ Use the definition of Taylor series to find the Taylor series, centered at c for the function.

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Monday, 26 August 2013

Use the definition of Taylor series to find the Taylor series, centered at c for the function.

No comments:

Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

$\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({3}{x}\right)}}},{c}={0}$ Use the definition of Taylor series to find the Taylor series, centered at c for the function.

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?