Differentials

Differentials (Definitions)

Recall that the derivative is defined by If we drop the limit and assume that Dx is small we have: we can rearrange this equation to get:

 Dy @ f '(x)Dx

Applications

1. Suppose that a die is manufactured so that each side is 0.5 inches plus or minus 0.01 inches.  Then its volume is

V = x3

So that

V ' = 3x2  = 3(0.5)2  = 0.75

Dy  @ (0.75)(0.01) = .0075 cu inches.

So that the volume of the die is approximately in the range

(0.5)3 +- 0.0075 = 0.125 0.0075

or between 0.1175 and 0.1325 cubic inches

2. We can use differentials to approximate We let

f(x) = x1/2

Since

f(1 + Dx) - f(1)  @  f '(1) Dx

We have

f(1 + Dx)  @  f '(1) Dx + f(1)

f(1) = 1,     f '(1) = 1/2,     Dx = .01

we have

f(1 + Dx)  @  1/2 (.01) + 1  = 1.005

(The true value is 1.00499)

Exercise:

A spherical bowl is full of jellybeans.  You count that there are 25 1 beans that line up from the center to the edge.  Give an approximate error of the number of jelly beans in the jar for this estimate.

Relative Error and Percent Error

 DefinitionThe relative error is defined as                                         Error           Relative Error  =                                                          Total               while the percent error is defined by                                        Error           Percent Error  =                x 100%                                         Total

Example

The level of sound in decibels is equal to

V = 5/r3

Where r is the distance from the source to the ear.  If a listener stands 10 feet 0.5 feet for optimal listening, how much variation will there be in the sound? What is the relative and percent error?

Solution

15
V' =  -15r-4  =                        =  -0.0015
10,000

DV @ (-0.0015)(.5) = -0.00075

V @ 0.005 0.00075

We have a percent error of

0.00075
Percent Error
@                          =  15%
0.005