2. Using Newton's Divided Difference formula to find solution
x = 2
Solution:
The value of table for `x` and `y`
Numerical divided differences method to find solution
Newton's divided difference table is
| x | y | `1^(st)` order | `2^(nd)` order | `3^(rd)` order |
| 0 | 4 | | | |
| | -1 | | |
| 1 | 3 | | 2 | |
| | 7 | | 0 |
| 4 | 24 | | 2 | |
| | 15 | | |
| 5 | 39 | | | |
Newton's divided difference interpolation formula is
`f(x) = y_0 + (x - x_0) f[x_0, x_1] + (x - x_0)(x - x_1) f[x_0, x_1, x_2] + (x - x_0)(x - x_1)(x - x_2) f[x_0, x_1, x_2, x_3]`
`f(x) = 4 + (x -0) xx -1 + (x -0)(x -1) xx 2 + (x -0)(x -1)(x -4) xx 0`
`f(x) = 4 + (x) xx -1 + (x^2-x) xx 2 + (x^3-5x^2+4x) xx 0`
`f(x) = 4 + (-x) + (2x^2-2x) + (0) `
`f(x) = 2x^2-3x+4 `
Now, differentiate with x
`f'(x)=4x-3`
`f''(x)=4`
Now, substitute `x=2`
`f'(2)=4 xx 2-3=5`
`f''(2)=4=4`
This material is intended as a summary. Use your textbook for detail explanation.
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