Taylor Polynomials for $e^{-4x}$ at $x=0$

In summary, we were asked to find the nth-order Taylor polynomials for $n=0,1,2$ centered at $a=0$ for the function $f(x)=e^{-4x}$ using the formula $P_n(x)\approx \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}x^k$. We found $P_0(x)\approx 1$ for $n=0$, $P_1(x)\approx 1-4x$ for $n=1$, and $P_2(x)\approx 1-4x+8x^2$ for $n=2$. These approximations only apply to $P
  • #1
karush
Gold Member
MHB
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5
$\tiny{206.11.1.16-T}$
$\textsf{Find the nth-order Taylor polynomials
centered at 0, for $n=0, 1, 2.$}\\$
$$\displaystyle f(x)=e^{-4x}$$
$\textsf{using}\\$
$$P_n\left(x\right)
\approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$
$\textsf{n=0}\\$
\begin{align}
f^0(x)&\approx e^{-4x}\therefore f^0(0)\approx1 \\
P_0\left(x\right)&\approx\frac{1}{0!}x^{0}\approx 1
\end{align}
$\textsf{n=1}\\$
\begin{align}
f^1(x)&\approx-4e^{-4x}\therefore f^1(0)\approx -4 \\
P_1 f(x)&\approx \frac{1}{0!}x^{0}
+\frac{-4}{1!}x^{1}
\approx 1-4x

\end{align}
$\textsf{n=2}\\$
\begin{align}
f^2 (x)&= 16e^{-4x}\therefore f^2 (0)=16\\
P_2 f(x)&\approx \frac{1}{0!} x^{0}
+\frac{-4}{1!}x^{1}+\frac{16}{2!}x^{2}
\approx 1- 4x+8x^{2}
\end{align}

hopefully
not too hard kinda subtle tho
☕
 
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  • #2
Hi karush! ;)

It looks fine to me, except that every $\approx$ should be a $=$.
The approximation only applies to $P_i(x) \approx f(x)$ for $i=1, 2, ...$, but that's not in your list.
 
  • #3
was wondering about that... some examples were confusing..

😎
 

Related to Taylor Polynomials for $e^{-4x}$ at $x=0$

1. What is a Taylor polynomial?

A Taylor polynomial is a mathematical expression that approximates a function by using a series of terms that are calculated from the function's derivatives at a specific point.

2. How do you find the Taylor polynomial for $e^{-4x}$ at $x=0$?

To find the Taylor polynomial for $e^{-4x}$ at $x=0$, you would need to calculate the function's derivatives at $x=0$ and plug them into the formula for a Taylor polynomial, which is given by:
$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ... $
In this case, $a=0$ and the derivatives of $e^{-4x}$ at $x=0$ are all equal to 1, so the Taylor polynomial for $e^{-4x}$ at $x=0$ is simply $1-x+\frac{x^2}{2}-\frac{x^3}{6}+...$

3. Why is the Taylor polynomial for $e^{-4x}$ at $x=0$ useful?

The Taylor polynomial for $e^{-4x}$ at $x=0$ is useful because it allows us to approximate the value of $e^{-4x}$ at any point near $x=0$ without having to use a calculator or a complex mathematical expression. This is especially helpful in situations where we need to quickly estimate a value without having access to advanced tools or technology.

4. How accurate is the Taylor polynomial for $e^{-4x}$ at $x=0$?

The accuracy of the Taylor polynomial for $e^{-4x}$ at $x=0$ depends on the number of terms used in the polynomial. The more terms we include, the closer the approximation will be to the actual value of $e^{-4x}$ at a specific point. However, it is important to note that the Taylor polynomial is only an approximation and not an exact representation of the function, so there will always be some degree of error.

5. Can the Taylor polynomial for $e^{-4x}$ at $x=0$ be used to find the value of $e^{-4x}$ at points other than $x=0$?

Yes, the Taylor polynomial for $e^{-4x}$ at $x=0$ can be used to approximate the value of $e^{-4x}$ at points other than $x=0$. However, the accuracy of the approximation will decrease as we move further away from the point $x=0$, so it is best to use the polynomial for values of $x$ that are close to $0$.

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