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:
Suppose that a die is manufactured so that each side is .5 inches .01 inches. Then its volume is
V = x3
V' = 3x2 = 3(.5)2 = .75
Dy @ (.75)(.01) = .0075 cu inches
So that the volume of the die is in the range
(.5)3 .0075 = .125 .0075
or between .1175 and .1375 cubic inches.
We can use differentials to approximate
f(x) = x1/2
f(1 + Dx) - f(1) @ f '(1) Dx
f(1 + Dx) @ f'(1)Dx + f(1)
f(1) = 1 f'(1) = 1/2 Dx = .01
f(1 + Dx) @ 1/2 (.01) + 1 = 1.005
(The true value is 1.00499)
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.
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.
V' = -15r-4 = -15/10,000 = -0.0015
V' D v = (-0.0015)(.5) = -0.00075
V = 0.005 -0.00075
We have a percent error of 0.00075 / 0.005 = 15%
We define the marginal revenue as the additional revenue from selling an additional unit of a product. If D x = 1 then the marginal revenue follows
DR @ R'(x)D x.
Suppose that the demand equation for a bicycle is
p = 1000 - 2x
Use differentials to approximate the change in revenue as sales increase from 100 to 101 units.
R = px = 1000x - 2x2
R' = 1000 - 4x
R'(100) = 600
DR @ 600(1) = 600
Note that the true marginal revenue is
R(101) - R(100) = 598