Question

If y = xⁿ  then the ratio of relative errors in y and x is

Options

• 1 : 1

• 2 : 1

• 1 : n

• n : 1

Answer 1

 n : 1

\[\text { Let } \frac{∆ x}{x} \text { be the relative error in x and} \frac{∆ y}{y} \text { be the error in y } . \]

\[\text { Now,} y = x^n \]

\[ \Rightarrow \frac{dy}{dx} = n x^{n - 1} \]

\[ \Rightarrow \frac{∆ y}{y} = \frac{n x^{n - 1}}{y}dx\]

\[ \Rightarrow \frac{∆ y}{y} = \frac{n x^{n - 1}}{x^n}dx = n\frac{∆ x}{x}\]

\[ \Rightarrow \frac{∆ y}{y}: \frac{∆ x}{x} = n: 1\]