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प्रश्न
Using interpolation, find the value of f(x) when x = 15
| x | 3 | 7 | 11 | 19 |
| f(x) | 42 | 43 | 47 | 60 |
उत्तर
Here the intervals are unequal
By Lagrange’s in-terpolation formula we have,
x0 = 3
x1 = 7
x2 = 11
x3 = 19
y0 = 42
y1 = 43
y2 = 47
y3 = 60 and x = 15.
y = `"f"(x) = ((x - x_1)(x - x_2)(x - x_3))/((x_0 - x_1)(x_0 - x_2)(x_0 - x_3)) xx y_0 + ((x_0 - x_1)(x - x_2)(x - x_3))/((x_1 - x_0)(x_1 - x_2)(x_1 - x_3)) xx y_1 + ((x - x_0)(x - x_1)(x - x_3))/((x_2 - x_0)(x_2 - x_1)(x_2 - x_3))xx y_2 + ((x - x_0)(x - x_1)(x - x_2))/((x_3- x_0)(x_3 - x_1)(x_3 - x_2)) xx y_3`
y(15) = `"f"(15) = ((15 - 7)(15 - 11)(15 - 19))/((3 - 7)(3 - 11)(3 - 19)) xx 42 + ((15 - 3)(15 - 11)(15 - 19))/((7 - 3)(7 - 11)(7 - 19)) xx 43 + ((15 - 3)(15 - 7)(15 - 19))/((11 - 13)(11 - 7)(11 - 19)) xx 47 + ((15 - 3)(15 - 7)(15 - 11))/((19 - 3)(19 - 7)(19 - 11)) xx 60`
= `((8)(4)(-4))/((-4)(-8)(-4)) xx 42 + ((12)(4)(-4))/((4)(-4)(-12)) xx 43 + ((12) xx (8) xx (-4))/((8) xx (4) xx (-8)) xx 47 + ((12) xx (8) xx (4))/((16) xx (12) xx (8)) xx 60`
= 10.5 – 43 + 70.5 + 15
= 53
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