Question 1209694: If f is a polynomial of degree 4 such that
f(0) = 1, f(1) = 2, f(2) = -7, f(3) = 0, f(4) = 3,
then determine f(5).
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to determine f(5) using the method of finite differences:
1. **Set up a difference table:**
| x | f(x) | Δf(x) | Δ²f(x) | Δ³f(x) | Δ⁴f(x) |
|---|---|---|---|---|---|
| 0 | 1 | | | | |
| 1 | 2 | 1 | | | |
| 2 | -7 | -9 | -10 | | |
| 3 | 0 | 7 | 16 | 26 | |
| 4 | 3 | 3 | -4 | -20 | -46 |
Where Δf(x) represents the first difference, Δ²f(x) the second difference, and so on. Each difference is calculated by subtracting the previous value from the current value. For example, Δf(1) = f(1) - f(0) = 2 - 1 = 1, and Δ²f(2) = Δf(2) - Δf(1) = -9 - 1 = -10.
2. **Since f(x) is a polynomial of degree 4, the fourth differences are constant.** We can use this fact to find the next values in the table. The last entry in the Δ⁴f(x) column is -46.
3. **Extend the table:**
We can extend the table by working backwards.
* Δ³f(4) = Δ³f(3) + Δ⁴f(3) = -20 + (-46) = -66
* Δ²f(5) = Δ²f(4) + Δ³f(4) = -4 + (-66) = -70
* Δf(5) = Δf(4) + Δ²f(5) = 3 + (-70) = -67
* f(5) = f(4) + Δf(5) = 3 + (-67) = -64
Therefore, f(5) = -64.
Final Answer: The final answer is $\boxed{-64}$
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