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Expand 2 X 3 + 7 X 2 + X − 6 in Powers of (X-2) - Applied Mathematics 1

Sum

Expand `2x^3+7x^2+x-6` in powers of (x-2)

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Solution

Let `f(x)=2x^3+7x 62+x-6` and a = 2

∴ f'(x) = 6x2+14x+1,  f''(x)=12x+14,  f'''(x)=12

∴ f(2) = 45, f’(2) = 53, f’’(2) = 38, f’’’(2) = 12.

Now, f(x) = f(a) + f(x-a)f’(a) +`(x-a)^2/(2!)`f''(a) +................

∴ f(x) = f(2) + (x - 2)f’(2) + `(x-2^2)/(2!)`f''(2) +.................

`2x^3+7x^2+x-6=45+(x-2)53+(x-2)^(2.19)+(x-2)^(3.2)`

Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
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