Question:
If $y=x^{n}$, then the ratio of relative errors in $y$ and $x$ is
(a) $1: 1$
(b) $2: 1$
(c) $1: n$
(d) $n: 1$
Solution:
(d) $n: 1$
Let $\frac{\Delta x}{x}$ be the relative error in $x$ and $\frac{\Delta y}{y}$ be the error in $y$.
Now, $y=x^{n}$
$\Rightarrow \frac{d y}{d x}=n x^{n-1}$
$\Rightarrow \frac{\Delta y}{y}=\frac{n x^{n-1}}{y} d x$
$\Rightarrow \frac{\Delta y}{y}=\frac{n x^{n-1}}{x^{n}} d x=n \frac{\Delta x}{x}$
$\Rightarrow \frac{\Delta y}{y}: \frac{\Delta x}{x}=n: 1$
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