206.11.3.27 first three nonzero terms of the Taylor series

In summary, the first three nonzero terms of the Taylor series are $\displaystyle \begin{align*} f^{(0)}(a) \end{align*}$, $\displaystyle \begin{align*} f^{(2)}(a) \end{align*}$ and $\displaystyle \begin{align*} -\frac{\sqrt{2}}{2}\left(x- \frac{3 \pi}{4}\right) \end{align*}$.
  • #1
karush
Gold Member
MHB
3,269
5
$\textsf{a. Find the first three nonzero terms
of the Taylor series $a=\frac{3\pi}{4}$}$
\begin{align}
\displaystyle
f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\
f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\
f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\
\end{align}

$\textsf{so then}$
\begin{align}
\displaystyle
f\left(x\right)&
\approx\frac{\frac{\sqrt{2}}{2}}{0!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{0}+\frac{- \frac{\sqrt{2}}{2}}{1!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{1}+\frac{- \frac{\sqrt{2}}{2}}{2!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{2}\\
\sin{\left (x \right )}&\approx
\frac{\sqrt{2}}{2}
- \frac{\sqrt{2}}{2}\left(x- \frac{3 \pi}{4}\right)
- \frac{\sqrt{2}}{4}\left(x- \frac{3 \pi}{4}\right)^{2}
\end{align}
 
Last edited:
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  • #2
Re: 206.11.3.27 first three nonzero terms of the Taylor serie

karush said:
$\textsf{a. Find the first three nonzero terms
of the Taylor series $a=\frac{3\pi}{4}$}$
\begin{align}
\displaystyle
f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\sin{x} \\
f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\
f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=\frac{\sqrt{2}}{2} \\
\end{align}

$\textsf{so then}$
\begin{align}
\displaystyle
f\left(x\right)&
\approx\frac{\frac{\sqrt{2}}{2}}{0!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{0}+\frac{- \frac{\sqrt{2}}{2}}{1!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{1}+\frac{- \frac{\sqrt{2}}{2}}{2!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{2}\\
\sin{\left (x \right )}&\approx
\frac{\sqrt{2}}{2}
- \frac{\sqrt{2}}{2}\left(x- \frac{3 \pi}{4}\right)
- \frac{\sqrt{2}}{4}\left(x- \frac{3 \pi}{4}\right)^{2}
\end{align}

Well you have calculated $\displaystyle \begin{align*} f^{(0)}(a) \end{align*}$ and $\displaystyle \begin{align*} f^{(2)}(a) \end{align*}$ incorrectly, but somehow written them correctly in the final answer...
 
  • #3
Re: 206.11.3.27 first three nonzero terms of the Taylor serie

$\textsf{a. Find the first three nonzero terms
of the Taylor series $a=\frac{3\pi}{4}$}$
\begin{align}
\displaystyle
f^0(x)&=\sin{x} &\therefore \ \ f^0(a)&=\frac{\sqrt{2}}{2} \\
f^1(x)&=\cos{x} &\therefore \ \ f^1(a)&= -\frac{\sqrt{2}}{2}\\
f^2(x)&=- \sin{x}&\therefore \ \ f^2(a)&=-\frac{\sqrt{2}}{2} \\
\end{align}

$\textsf{so then}$
\begin{align}
\displaystyle
f\left(x\right)&
\approx\frac{\frac{\sqrt{2}}{2}}{0!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{0}+\frac{- \frac{\sqrt{2}}{2}}{1!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{1}+\frac{- \frac{\sqrt{2}}{2}}{2!}\left(x-\left(\frac{3 \pi}{4}\right)\right)^{2}\\
\sin{\left (x \right )}&\approx
\frac{\sqrt{2}}{2}
- \frac{\sqrt{2}}{2}\left(x- \frac{3 \pi}{4}\right)
- \frac{\sqrt{2}}{4}\left(x- \frac{3 \pi}{4}\right)^{2}
\end{align}
 

Related to 206.11.3.27 first three nonzero terms of the Taylor series

1. What is the significance of the first three nonzero terms in the Taylor series for 206.11.3.27?

The first three nonzero terms in the Taylor series for 206.11.3.27 represent the approximation of the function at a specific point. They provide a way to estimate the value of the function near that point by using a polynomial of increasing degree.

2. How is the Taylor series for 206.11.3.27 calculated?

The Taylor series for 206.11.3.27 is calculated using the function's derivatives at a given point. The first term is simply the value of the function at that point, while the subsequent terms involve the derivatives evaluated at that point.

3. What is the formula for finding the first three nonzero terms in the Taylor series for 206.11.3.27?

The formula for finding the first three nonzero terms in the Taylor series for 206.11.3.27 is: f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + (f'''(a)/3!)(x-a)^3, where a is the point at which the series is being evaluated.

4. How accurate is the approximation provided by the first three nonzero terms in the Taylor series for 206.11.3.27?

The accuracy of the approximation provided by the first three nonzero terms in the Taylor series for 206.11.3.27 depends on the function being approximated and the distance from the point of evaluation. In general, the more terms included in the series, the more accurate the approximation will be.

5. Can the Taylor series for 206.11.3.27 be used to approximate the function at any point?

Yes, the Taylor series for 206.11.3.27 can be used to approximate the function at any point. However, the closer the point is to the point of evaluation, the more accurate the approximation will be.

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