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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, $y=x^{n}$

$\Rightarrow \frac{d y}{d x}=n x^{n-1}$

$\Rightarrow \frac{\Delta y}{y}=\frac{n x^{n-1}}{y} d x$

$\Rightarrow \frac{\Delta y}{y}=\frac{n x^{n-1}}{x^{n}} d x=n \frac{\Delta x}{x}$

$\Rightarrow \frac{\Delta y}{y}: \frac{\Delta x}{x}=n: 1$