Question

Find f(0.5) if f(– 1) = 202, f(0) = 175, f(1) = 82 and f(2) = 55

Answer 1

From the given data

x	– 1	0	1	2
f(x)	202	175	82	55

Here we have to apply Newton’s forward interpolation formula,

Since the value of f(x) is required near the beginning of the table.

`y_((x = x_0 + "nh")) = "f"(x_0) + "n"/(1!) Delta"f"(x_0) + ("n"("n" - 1))/(2!) Delta^2"f"(x_0) + ("n"("n" - 1)("n" - 2))/(3!) Delta^3"f"(x_0) + .......`

Given:

x = 0.5 and h = 1

x₀ + nh = x

– 1 + n(1) = 0.5

n = 1 + 0.5

∴ n = 1.5

x	f(x)	`Delta"f"(x)`	`Delta^2"f"(x)`	`Delta^3"f"(x)`
– 1	202
		– 27
0	175		– 66
		– 93		132
1	82		66
		– 27
2	55

`"y"_((0.5)) = 202 + 1.5/(1!) (-27) + ((1.5)(1.5 - 1))/(2!) (- 66) + ((1.5)(1.5 - 1)(1.5 - 2))/(3!) (132)`

= `202 + 1.5(- 27) + ((1.5)(0.5)(- 66))/2 + ((1.5)(0.5)(- 0.5)(132))/6`

= 202 – 40.5 – 24.75 – 8.25

= 202 – 73.5

f(0.5) = 128.5