To determine the degree of the polynomial function that fits the given data using finite differences, follow these steps:
### Step-by-Step Solution:
1. List the Given Data:
The data points provided are:
- [tex]\( x = [1, 2, 3, 4, 5] \)[/tex]
- [tex]\( f(x) = [20, 4, 0, 4, 16] \)[/tex]
2. Understand Finite Differences:
Finite differences are used to estimate the derivatives of functions at discrete points. Specifically, they help to identify the degree of the polynomial that fits the data.
3. Compute First Differences:
Calculate the first differences [tex]\(\Delta f(x)\)[/tex] as follows:
[tex]\[
\Delta f(x) = f(x + 1) - f(x)
\][/tex]
[tex]\[
\begin{align*}
\Delta f(x) [1] &= f(2) - f(1) = 4 - 20 = -16\\
\Delta f(x) [2] &= f(3) - f(2) = 0 - 4 = -4\\
\Delta f(x) [3] &= f(4) - f(3) = 4 - 0 = 4\\
\Delta f(x) [4] &= f(5) - f(4) = 16 - 4 = 12
\end{align*}
\][/tex]
So the first differences are:
[tex]\[
\Delta f(x) = [-16, -4, 4, 12]
\][/tex]
4. Compute Second Differences:
Calculate the second differences [tex]\(\Delta^2 f(x)\)[/tex] as follows:
[tex]\[
\Delta^2 f(x) = \Delta f(x+1) - \Delta f(x)
\][/tex]
[tex]\[
\begin{align*}
\Delta^2 f(x) [1] &= (-4) - (-16) = 12\\
\Delta^2 f(x) [2] &= 4 - (-4) = 8\\
\Delta^2 f(x) [3] &= 12 - 4 = 8
\end{align*}
\][/tex]
So the second differences are:
[tex]\[
\Delta^2 f(x) = [12, 8, 8]
\][/tex]
5. Compute Third Differences:
Calculate the third differences [tex]\(\Delta^3 f(x)\)[/tex] as follows:
[tex]\[
\Delta^3 f(x) = \Delta^2 f(x+1) - \Delta^2 f(x)
\][/tex]
[tex]\[
\begin{align*}
\Delta^3 f(x) [1] &= 8 - 12 = -4\\
\Delta^3 f(x) [2] &= 8 - 8 = 0
\end{align*}
\][/tex]
So the third differences are:
[tex]\[
\Delta^3 f(x) = [-4, 0]
\][/tex]
6. Compute Fourth Differences:
Calculate the fourth differences [tex]\(\Delta^4 f(x)\)[/tex] as follows:
[tex]\[
\Delta^4 f(x) = \Delta^3 f(x+1) - \Delta^3 f(x)
\][/tex]
[tex]\[
\Delta^4 f(x) [1] = 0 - (-4) = 4
\][/tex]
So the fourth differences are:
[tex]\[
\Delta^4 f(x) = [4]
\][/tex]
7. Identify the Degree:
The finite difference method shows that the differences do not become zero until we compute the fourth differences. Since the fourth differences are the first to become constant but non-zero, the polynomial that fits the data is of degree 4.
Thus, the degree of the polynomial function that fits the given data is:
[tex]\[ \boxed{4} \][/tex]
