Question

Find the third divided difference with
arguments 2, 4, 9, 10 of the function
f (x) = x - x
3 2 .

Answer 1

Answer:

The Third Dividend Difference with  arguments 2, 4, 9, 10 of the function f(x) = x³-2x is 1.

Step-by-step explanation:

The given function is f(x) = x³-2x

Given arguments - 2,4,9,10

Let x₀ = 2, x₁ = 4, x₂ = 9, x₃ = 10

∴ f(x₀) = f(2) = 2³ - 2×2 = 4

∴ f(x₁) = f(4) = 4³ - 2×4 = 56

∴ f(x₂) = f(9) = 9³ - 2×9 = 711

∴ f(x₃) = f(10) = 10³ - 2×10 = 980

∴ First Dividend Difference -

i) Δₓ₁f(x₀) = f(x₀,x₁) = {f(x₁) - f(x₀)}/(x₁ - x₀) = (56-4)/(4-2) = 26

ii) Δₓ₂f(x₁) = f(x₁,x₂) = {f(x₂) - f(x₁)}/(x₂ - x₁) = (711-56)/(9-4) = 131

iii) Δₓ₃f(x₂) = f(x₂,x₃) = {f(x₃) - f(x₂)}/(x₃ - x₂) = (980-711)/(10-9) = 269

∴ Second Dividend Difference -

i) Δ₍ₓ₁,ₓ₂₎f(x₀) = f(x₀,x₁,x₂) = {f(x₁,x₂) - f(x₀,x₁)}/(x₂ - x₀) = (131-26)/(9-2) = 15

ii) Δ₍ₓ₂,ₓ₃₎f(x₁) = f(x₁,x₂,x₃) = {f(x₂,x₃) - f(x₁,x₂)}/(x₃ - x₁) = (269-131)/(10-4) = 23

∴ Third Dividend Difference -

Δ₍ₓ₁,ₓ₂,ₓ₃₎f(x₀) = f(x₀,x₁,x₂,x₃) = {f(x₁,x₂,x₃) - f(x₀,x₁,x₂)}/(x₃ - x₀)

= (23-15)/(10-2) = 1

∴ The Third Dividend Difference with  arguments 2, 4, 9, 10 of the function f(x) = x³-2x is 1.

Answer 2

Answer:

The Third Dividend Difference with arguments 2, 4, 9, 10 of the function f(x) = x³-2x is 1

Step-by-step explanation:

the given function is f(x)=$x^{3}-2x$

given  arguments,

$x_{0$=2

$x_1=4\\x_{2}= 9\\x_{3} =10$

Let x₀ = 2, x₁ = 4, x₂ = 9, x₃ = 10

substitute value of $x_{0} ,x_{1} ,x_{2} ,x_{3}$ in f(x)=f(x)=$x^{3}-2x$

∴ f(x₀) = f(2) = 2³ - 2×2 = 8-4=4

∴ f(x₁) = f(4) = 4³ - 2×4 = 64-8=56

∴ f(x₂) = f(9) = 9³ - 2×9 = 729-18=711

∴ f(x₃) = f(10) = 10³ - 2×10 = 1000-20=980

First dividend difference

1)Δₓ₁f(x₀) = f(x₀,x₁)=f(x₁) - f(x₀)}/(x₁ - x₀)=56-4)/(4-2) = 26

2) Δₓ₂f(x₁) = f(x₁,x₂) = {f(x₂) - f(x₁)}/(x₂ - x₁) = (711-56)/(9-4) = 131

3) Δₓ₃f(x₂) = f(x₂,x₃) = {f(x₃) - f(x₂)}/(x₃ - x₂) = (980-711)/(10-9) = 269

Second Dividend Difference

1) Δ₍ₓ₁,ₓ₂₎f(x₀) = f(x₀,x₁,x₂) = {f(x₁,x₂) - f(x₀,x₁)}/(x₂ - x₀) = (131-26)/(9-2) = 15

2)Δ₍ₓ₂,ₓ₃₎f(x₁) = f(x₁,x₂,x₃) = {f(x₂,x₃) - f(x₁,x₂)}/(x₃ - x₁) = (269-131)/(10-4) = 23

Third Dividend Difference

Δ₍ₓ₁,ₓ₂,ₓ₃₎f(x₀) = f(x₀,x₁,x₂,x₃) = {f(x₁,x₂,x₃) - f(x₀,x₁,x₂)}/(x₃ - x₀)

= (23-15)/(10-2) = 1

∴ The Third Dividend Difference with  arguments 2, 4, 9, 10 of the function f(x) = x³-2x is 1.

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