Question: c) ii) Expand $2x^3+7x^2+x-6$ in powers of (x-2)

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

$∴f’(x) = 6x^2 + 14x +1 , f’’(x) = 12x +14, f’’’(x) = 12$

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

Now , $f(x) = f(a) + f(x-a)f’(a) + \frac{(x-a)^2}{2!}f’’(a) +……$………………

$∴ f(x) = f(2) + (x-2)f’(2) + \frac{(x-2)^2}{2!}f’’(2) +………………..$

$2x^3+7x^2+x-6 = 45 + (x-2){.53} + (x-2)^{2.19} + (x-2)^{3.2}$

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