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प्रश्न
Find f(2.8) from the following table:
| x | 0 | 1 | 2 | 3 |
| f(x) | 1 | 2 | 11 | 34 |
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उत्तर
Since the required value is at the end of the table
Apply backward interpolation formula.
| x | `(f(x))/y` | `Deltay` | `Delta^2y` | `Delta^3y` |
| 0 | 1 | |||
| 1 | ||||
| 1 | 2 | 8 | ||
| 9 | 6 | |||
| 2 | 11 | 14 | ||
| 23 | ||||
| 3 | 34 |
`y_((x = x_0 + "nh")) = y_"n" + "n"/(1!) ∇y_"n" + ("n"("n" + 1))/(2!) ∇^2y_"n" + ("n"("n" + 1)("n" + 2))/(3!) Delta^3y_"n" + .........`
To find y at x = 2.8
∴ x0 + nh = 2.8
∴ 3 + n(1) = 2.8
n = 2.8 – 3
n = – 0.2
`y_((x = 2.8)) = 34 + ((-0.2))/(1!) (23) + ((-0.2)(-0.2 + 1))/(2!) (14) + ((-0.2)(-0.2 + 1)(-0.2 + 2))/(3!) (6) +`
= `34 - 4.6 + ((-0.2)(0.8)(14))/2 + ((-0.2)(0.8)(4))/2 + ((-0.2)(-0.8)(1.8))/6 (6)`
= 34 – 4.6 – 1.12 – 0.288
= 34 – 6.008
= 27.992
∴ f(2.8) = 27.992
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