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dreamspace
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Homework Statement
[itex]Let f(x) = x^{n} - ax^{n-1},\: and \: set\: g(x) = x^{n} \\
(a) \: Use\, the\, Sensitivity\, Formula\, to\, give\, a\, prediction\, for\, the\, nonzero\, \\
root \, of\,\; f_{\epsilon }(x) = x^{n} -ax^{n-1} + \epsilon x^{n} \, for\,small\,\epsilon.\\
\\ (b) \: Find \, the \, nonzero \, root \, and \, compare \, with \, the \, prediction. [/itex]
Homework Equations
Sensitivity Formula for Roots:
[itex]\Delta r \approx -\frac{\epsilon g(r)}{f'(r)}[/itex]
[itex]f'(x) = x^{n-2} (a + xn - an)[/itex]
The Attempt at a Solution
Some thoughts, and tries...
One of the roots for f(x) (without the g(x) and Epsilon) is obviously the constant A.
[itex]a = \frac{x^{n}}{x^{n-1}}[/itex]
So if I now plugg terms into the sensitivity root
[itex]
\Delta r \approx -\frac{\epsilon a^{n}}{x^{n-2} (a + an - an)}
\\
\Delta r \approx - \epsilon \frac{a^{n}}{a^{n-1}}
[/itex]
So a prediction of the nonzero root would simply then be
[itex]r + \Delta r
\\
nonzero root \approx a - \epsilon \frac{a^{n}}{a^{n-1}}
[/itex]
Would that be correct?