Use-the-binomial-theorem-to-write-the-first-four-terms-of-the-expansion-of-2-3x-x-2-

Question Number 161442 by nadovic last updated on 18/Dec/21

Use the binomial theorem to write

the first four terms of the expansion

of \sqrt{2 + 3 x - x^{2}}

Answered by mathmax by abdo last updated on 18/Dec/21

f (x) = f (0) + {xf}^{^{'}} (o) + \frac{x^{2}}{2} f^{(2)} (0) + \frac{x^{3}}{3!} f^{(3)} + \dots .

f (0) = \sqrt{2} and f^{^{'}} (x) = \frac{- 2 x + 3}{2 \sqrt{2 + 3 x - x^{2}}} \Rightarrow f^{^{'}} (0) = \frac{3}{2 \sqrt{2}}

f = \sqrt{- x^{2} + 3 x + 2} \Rightarrow f^{2} = - x^{2} + 3 x + 2 \Rightarrow {2 ff}^{^{'}} = - 2 x + 3 \Rightarrow

2 (f^{^{' 2}} + {ff}^{(2)}) = - 2 \Rightarrow {ff}^{(2)} = - 1 - f^{^{' 2}} \Rightarrow

f^{(2)} = - \frac{1}{f} - \frac{f^{^{' 2}}}{f} = - \frac{1}{\sqrt{2 + 3 x - x^{2}}} - \frac{{(- 2 x + 3)}^{2}}{4 (2 + 3 x - x^{2}) \sqrt{2 + 3 x - x^{2}}} \Rightarrow

f^{(2)} (0) = - \frac{1}{\sqrt{2}} - \frac{9}{8 \sqrt{2}} \dots . .

Commented by nadovic last updated on 18/Dec/21

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