On-line Math 21

2.4  Applications of the derivative.

Example 1 Use differentials to find the relative error (the percentage of error) in the computation of the volume of a cube, if the measurements of the sides might be off by as much as 1%.

Solution

Let's say that the side of the cube is measured as x . The percentage error is of course (expressed as a percent) 100 times the fraction of error

error value ,

so if the error in measurement is dx , the fraction of the measurement error is

.01 = dx x ,

if the error is 1%. Now, since the volume of the cube is V = x³ , then the differential of the volume is dV = 3x²dx . The exact error in the measurement of the volume is DV , which is approximated by dV , The fraction of the volume that is the error is

DV V » dV V = 3x2dx x3 = 3 dx x = .03,

so the percentage of volume error is about 3%, three times the measurement error percentage of the sides.

Copyright (c) 2000 by David L. Johnson.