Question

If \[y = {x^n}\], then the ratio of relative error in \[y\] and \[x\] is
A) \[1:1\]
B) \[2:1\]
C) \[1:n\]
D) \[n:1\]

Answer 1

Hint: Before finding the relative error at first, we have to find out the approximate error in terms of \[x\] and \[y\].
By using the approximate error, we can find the relative error.
The relative error gives an indication of how good measurement is relative to the size of the object being measured.
We know that; in Mathematics, differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.

Complete step-by-step answer:
It is given that; \[y = {x^n}\]
We have to find the ratio of relative error in \[y\] and \[x\].
We have,
\[y = {x^n}\]
Differentiate with respect to \[x\] we get,
$\Rightarrow$\[\dfrac{{dy}}{{dx}} = n{x^{n - 1}}\]… (1)
Approximate error in \[y\] is \[dy = \left( {\dfrac{{dy}}{{dx}}} \right)\Delta x\]
Substitute the values from (1) we get,
$\Rightarrow$\[dy = n{x^{n - 1}}\Delta x\]
So, relative error in \[y\] is \[\dfrac{{dy}}{y} = \dfrac{n}{x}\Delta x\]
Approximate error in \[x\] is \[dx = \left( {\dfrac{{dx}}{{dy}}} \right)\Delta y\]
Simplifying we get,
$\Rightarrow$\[dx = \left( {\dfrac{{\dfrac{1}{1}}}{{\dfrac{{dy}}{{dx}}}}} \right)\Delta y\]
Substitute the values from (1) we get,
$\Rightarrow$\[dx = \dfrac{1}{{n{x^{n - 1}}}}\Delta y\]
So, relative error in \[x\] is \[\dfrac{{dx}}{x} = \dfrac{1}{{n{x^n}}}\Delta y\]
Required ratio \[ = \dfrac{{\dfrac{n}{x}\Delta x}}{{\dfrac{1}{{n{x^n}}}\Delta y}}\]
Simplifying we get,
Required ratio \[ = {n^2}{x^{n - 1}}\dfrac{{\Delta x}}{{\Delta y}}\]
Simplifying again we get,
Required ratio \[ = \dfrac{n}{1}\]
So, the ratio of relative error in \[y\] and \[x\] is \[n:1\].

Hence, the correct option is D) \[n:1\].

Note: The relative error is defined as the ratio of the absolute error of the measurement to the actual measurement. Using this method, we can determine the magnitude of the absolute error in terms of the actual size of the measurement.
Absolute error is the difference between measured or inferred value and the actual value of a quantity. The absolute error is inadequate due to the fact that it does not give any details regarding the importance of the error.