Estimating Change Section 45b Recall that we sometimes use the notation dy dx to represent the derivative of y with respect to x this notation is not truly a ratio ID: 264715
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Slide1
Differentials,Estimating Change
Section 4.5bSlide2
Recall that we sometimes use the notation
dy
/dx to represent the derivative of y with respect to x this notation is not truly a ratio!!!
This leads us to the definition of new variables:Differentials
Let be a differentiable function. Thedifferential dx is an independent variable. Thedifferential dy is
(
dy
is always a dependent variable thatdepends on both x and dx)Slide3
Guided Practice
Find if
Find
ifSlide4
Guided Practice
Find
and evaluate for the given values of and .
With the given data:Slide5
Differentials can be used to
estimate change:
Let
be differentiable at . Theapproximate change in the value of when changes from to isSlide6
Guided Practice
The given function
changes value when x changesfrom a to a + dx.
the absolute change
Find:
the estimated changeSlide7
Guided Practice
The given function
changes value when x changesfrom a to a
+ dx.
Find:
the approximation errorSlide8
Guided Practice
The radius r of a circle increases from
a
= 10 m to 10.1 m.Use dA to estimate the increase in the circle’s area A.Compare this estimate with the true change in A
.
Estimated increase is
dA
:
m
2
True change:
m
2
dA
errorSlide9
Guided Practice
Write a differential formula that estimates the given
change
in area.
The change in the surface area of a spherewhen the radius changes from a to a + dr.
When
r
changes from
a
to
a +
dr
…
The change in surface area is approximatelySlide10
Guided Practice
Write a differential formula that estimates the given
change
in area.
The change in the surface area of a cube whenthe edge lengths change from a to a + dx.
When
x
changes from
a
to
a + dx
…
The change in surface area is approximatelySlide11
Guided Practice
The differential equation
tells
us how sensitive the output of is toa change in input at different values of x
.The larger the value of at x, the greaterthe effect of a given change dx.Slide12
Guided Practice
You want to calculate the depth of a well from the
given
equation by timing how long it takes a heavy stone youdrop to splash into the water below. How sensitive will
your calculations be to a 0.1 second error in measuringthe time?
The size of
ds
in the equation
depends on how big
t
is. If
t
= 2 sec, the error caused
by
dt
=
0.1 is
only
ft
Three seconds later at
t
= 5 sec, the error caused by
the
same
dt
:
ftSlide13
Guided Practice
The height and radius of a right circular cylinder are
equal,
so the cylinder’s volume is . The volume is to becalculated with an error of no more than 1% of the truevalue. Find
approx. the greatest error that can be toleratedin the measurement of h, expressed as a percentage of h.
We want
, which gives
The height should be measured with
an error of no more than .Slide14
Guided Practice
A manufacturer contracts to mint coins for the
federal
government. How much variation dr in the radius of thecoins can be tolerated if the coins are to weigh
within1/1000 of their ideal weight? Assume the thickness doesnot vary.
We want
, which gives
The variation of the radius should not
exceed 1/2000 of the ideal radius,
that is, 0.05% of the ideal radius.