Differentials

I.  Quiz

II.  Homework

III.  Differentials (Definitions)

Recall that the derivative is defined by

If we drop the limit and assume that delta x is small we have:

we can rearrange this equation to get:

Application:

Suppose that a die is manufactured so that each side is .5 inches plus or minus .01 inches.  Then its volume is

V = x³ 

So that v' = 3x²  = 3(.5)²  = .75

delta y  approx (.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

Example:  We can use differentials to approximate

sqrt(1.01)

f(x) = x^1/2 

Since 

We have

f(1) = 1, f''(1) = 1/2, delta x = .01 we have

f(1 + delta x)  approx= 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.

Definition:  The relative error is defined as the error/total, while the percent error is defined by error/total x 100%

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 +- .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 = -.0015

V' delta v = (-.0015)(.5) = -.00075

V = .005 +- -.00075

We have a percent error of .00075/.005 = 15%