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


Definition

The 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

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