Differentials and Amount of Error Cont.
Calculating the Amount of Error
Any type of measurement is prone to a certain amount of error. In many applications, certain quantities are calculated based on measurements. For example, the area of a circle is calculated by measuring the radius of the circle. An error in the measurement of the radius leads to an error in the computed value of the area. Here we examine this type of error and study how differentials can be used to estimate the error.
Consider a function [latex]f[/latex] with an input that is a measured quantity. Suppose the exact value of the measured quantity is [latex]a[/latex], but the measured value is [latex]a+dx[/latex]. We say the measurement error is [latex]dx[/latex] (or [latex]\Delta x[/latex]). As a result, an error occurs in the calculated quantity [latex]f(x)[/latex]. This type of error is known as a propagated error and is given by
Since all measurements are prone to some degree of error, we do not know the exact value of a measured quantity, so we cannot calculate the propagated error exactly. However, given an estimate of the accuracy of a measurement, we can use differentials to approximate the propagated error [latex]\Delta y[/latex].
Specifically, if [latex]f[/latex] is a differentiable function at [latex]a[/latex], the propagated error is
Unfortunately, we do not know the exact value [latex]a[/latex]. However, we can use the measured value [latex]a+dx[/latex], and estimate
In the next example, we look at how differentials can be used to estimate the error in calculating the volume of a box if we assume the measurement of the side length is made with a certain amount of accuracy.
Suppose the side length of a cube is measured to be [latex]5[/latex] cm with an accuracy of [latex]0.1[/latex] cm.
- Use differentials to estimate the error in the computed volume of the cube.
- Compute the volume of the cube if the side length is (i) [latex]4.9[/latex] cm and (ii) [latex]5.1[/latex] cm to compare the estimated error with the actual potential error.
The measurement error [latex]dx \, (=\Delta x)[/latex] and the propagated error [latex]\Delta y[/latex] are absolute errors. We are typically interested in the size of an error relative to the size of the quantity being measured or calculated.
Given an absolute error [latex]\Delta q[/latex] for a particular quantity, we define the relative error as [latex]\frac{\Delta q}{q}[/latex], where [latex]q[/latex] is the actual value of the quantity. The percentage error is the relative error expressed as a percentage.
relative and percentage error
Relative error ([latex]\frac{\Delta q}{q}[/latex]) and percentage error measure how significant an error is compared to the true value of a quantity. Relative error gives a scale-independent error measure, while percentage error expresses this error as a percentage, providing a clear indication of accuracy in measurements.
For example, if we measure the height of a ladder to be [latex]63[/latex] in. when the actual height is [latex]62[/latex] in., the absolute error is [latex]1[/latex] in. but the relative error is [latex]\frac{1}{62}=0.016[/latex], or [latex]1.6 \%[/latex].
By comparison, if we measure the width of a piece of cardboard to be [latex]8.25[/latex] in. when the actual width is [latex]8[/latex] in., our absolute error is [latex]\frac{1}{4}[/latex] in., whereas the relative error is [latex]\frac{0.25}{8}=\frac{1}{32}[/latex], or [latex]3.1\%[/latex].
Therefore, the percentage error in the measurement of the cardboard is larger, even though [latex]0.25[/latex] in. is less than [latex]1[/latex] in.
An astronaut using a camera measures the radius of Earth as [latex]4000[/latex] mi with an error of [latex]\pm 80[/latex] mi. Let’s use differentials to estimate the relative and percentage error of using this radius measurement to calculate the volume of Earth, assuming the planet is a perfect sphere.
Determine the percentage error if the radius of Earth is measured to be [latex]3950[/latex] mi with an error of [latex]\pm 100[/latex] mi.