Question

using finite difference, show that the data f(-3)=13, f(-2)=7,f(-1)=3,f(0)=1,f(1)=1,f(2)=3,f(3)=7. represents a second degree polynomial. obtain this polynomial using interpolation and find f(2.5).

Answer 1

Answer to Question #74946, Math / Quantitative Methods

Using finite difference, show that the data $f(-3) = 13, f(-2) = 7, f(-1) = 3, f(0) = 1, f(1) = 1, f(2) = 3, f(3) = 7$, represents a second degree polynomial. Obtain this polynomial using interpolation and find $f(2,5)$.

Solution.

We have:

$$\Delta x = 1$$ $$\Delta f _ {1} = f (- 2) - f (- 3) = 7 - 13 = -6$$ $$\Delta f _ {2} = f (- 1) - f (- 2) = 3 - 7 = -4$$ $$\Delta f _ {3} = f (0) - f (- 1) = 1 - 3 = -2$$ $$\Delta f _ {4} = f (1) - f (0) = 1 - 1 = 0$$ $$\Delta f _ {5} = f (2) - f (1) = 3 - 1 = 2$$ $$\Delta f _ {6} = f (3) - f (2) = 7 - 3 = 4$$ $$\Delta^ {2} f = \Delta f _ {2} - \Delta f _ {1} = \Delta f _ {3} - \Delta f _ {2} = \Delta f _ {4} - \Delta f _ {3} = \Delta f _ {5} - \Delta f _ {4} = \Delta f _ {6} - \Delta f _ {5} = 2$$

Since $\Delta^3 f = 0$ then we have second degree polynomial of the form:

$$f (x) = a _ {2} x ^ {2} + a _ {1} x + a _ {0}$$

Then:

$$\Delta f = 2 a _ {2} x (\Delta x) + a _ {2} (\Delta x) ^ {2} + a _ {1} (\Delta x)$$ $$\Delta^ {2} f = 2 a _ {2} (\Delta x) ^ {2}$$ $$\Delta^ {2} f = 2 a _ {2} \cdot 1 = 2 \Rightarrow a _ {2} = 1$$ $$\Delta f _ {4} = 1 \cdot 1 + a _ {1} \cdot 1 = 0 \Rightarrow a _ {1} = - 1$$ $$f (0) = a _ {0} = 1$$

Answer:

$$f (x) = x ^ {2} - x + 1$$ $$f (2. 5) = 2. 5 ^ {2} - 2. 5 + 1 = 4. 7 5$$

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