Answer :
To find the value of [tex]F(7)[/tex] using Newton's divided difference formula, let's first understand what Newton's divided difference formula is. This formula is used for constructing an interpolating polynomial that passes through a given set of points.
Given the points:
- [tex](3, 84)[/tex]
- [tex](5, 180)[/tex]
- [tex](8, 504)[/tex]
- [tex](9, 790)[/tex]
- [tex](10, 1716)[/tex]
We're going to find the divided differences and construct the polynomial step-by-step.
- Calculate the first divided differences:
[tex]\begin{align*}
[f(x_0), f(x_1)] &= \frac{f(5) - f(3)}{5 - 3} = \frac{180 - 84}{2} = 48, \\
[f(x_1), f(x_2)] &= \frac{f(8) - f(5)}{8 - 5} = \frac{504 - 180}{3} = 108, \\
[f(x_2), f(x_3)] &= \frac{f(9) - f(8)}{9 - 8} = \frac{790 - 504}{1} = 286, \\
[f(x_3), f(x_4)] &= \frac{f(10) - f(9)}{10 - 9} = \frac{1716 - 790}{1} = 926.
\end{align*}[/tex]
- Calculate the second divided differences:
[tex]\begin{align*}
[f(x_0), f(x_1), f(x_2)] &= \frac{[f(x_1), f(x_2)] - [f(x_0), f(x_1)]}{8 - 3} = \frac{108 - 48}{3} = 20, \\
[f(x_1), f(x_2), f(x_3)] &= \frac{[f(x_2), f(x_3)] - [f(x_1), f(x_2)]}{9 - 5} = \frac{286 - 108}{4} = 44.5, \\
[f(x_2), f(x_3), f(x_4)] &= \frac{[f(x_3), f(x_4)] - [f(x_2), f(x_3)]}{10 - 8} = \frac{926 - 286}{2} = 320.
\end{align*}[/tex]
- Calculate the third divided differences:
[tex]\begin{align*}
[f(x_0), f(x_1), f(x_2), f(x_3)] &= \frac{[f(x_1), f(x_2), f(x_3)] - [f(x_0), f(x_1), f(x_2)]}{9 - 3} = \frac{44.5 - 20}{4} = 6.125, \\
[f(x_1), f(x_2), f(x_3), f(x_4)] &= \frac{[f(x_2), f(x_3), f(x_4)] - [f(x_1), f(x_2), f(x_3)]}{10 - 5} = \frac{320 - 44.5}{5} = 55.1.
\end{align*}[/tex]
- Calculate the fourth divided difference:
[tex][f(x_0), f(x_1), f(x_2), f(x_3), f(x_4)] = \frac{[f(x_1), f(x_2), f(x_3), f(x_4)] - [f(x_0), f(x_1), f(x_2), f(x_3)]}{10 - 3} = \frac{55.1 - 6.125}{5} = 9.795.[/tex]
- Construct the polynomial:
The polynomial [tex]P(x)[/tex] based on Newton's formula is:
[tex]P(x) = f(x_0) + [f(x_0), f(x_1)](x - x_0) + [f(x_0), f(x_1), f(x_2)](x - x_0)(x - x_1) + \cdots[/tex]
Substituting the divided differences, we have:
[tex]P(x) = 84 + 48(x - 3) + 20(x - 3)(x - 5) + 6.125(x - 3)(x - 5)(x - 8) + 9.795(x - 3)(x - 5)(x - 8)(x - 9)[/tex]
- Use this polynomial to find [tex]F(7)[/tex]:
Substitute [tex]x = 7[/tex] into the polynomial to find [tex]F(7)[/tex].
[tex]\begin{align*}
F(7) &= 84 + 48(7 - 3) + 20(7 - 3)(7 - 5) + 6.125(7 - 3)(7 - 5)(7 - 8) + 9.795(7 - 3)(7 - 5)(7 - 8)(7 - 9) \\
&= 84 + 48 imes 4 + 20 imes 4 imes 2 + 6.125 imes 4 imes 2 imes (-1) + 9.795 imes 4 imes 2 imes (-1) imes (-2)
\end{align*}[/tex]
Now, simplify the expression:
[tex]F(7) = 84 + 192 + 160 - 49 - 156.72 = 230.28[/tex]
So, [tex]F(7)[/tex] is approximately [tex]230.28[/tex]. This completes the calculation using Newton's divided difference formula.
