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:
Applications

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 = x^{3}
So that
V ' = 3x^{2} =
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

We can use differentials to approximate
We
let
f(x) = x^{1/2}
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/r^{3
}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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