Differentials, Estimating Change
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Transcript Differentials, Estimating Change
Differentials,
Estimating Change
Section 4.5b
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 y f x be a differentiable function. The
differential dx is an independent variable. The
differential dy is
dy f x dx
(dy is always a dependent variable that
depends on both x and dx)
Guided Practice
Find
dy
if
y x 37 x
5
dy f x dx 5 x 37 dx
4
Find
dy if y sin 3x
dy f x dx 3cos3x dx
Guided Practice
Find dy and evaluate
x and dx .
2x
y
1 x
dy
2
for the given values of
x 2
dx 0.1
1 x 2 2 2 x 2 x
2
2 2x
dx
dy
dx
2
2
2
2
1 x
1
x
With the given data:
dy
2 2 2
2
1 2
2
6
0.1
0.1 0.024
2
25
Differentials can be used to
estimate change:
Let f x be differentiable at x a . The
approximate change in the value of f when
x changes from a to a dx is
df f a dx
Guided Practice
The given function
from a to a + dx.
f
changes value when x changes
f x x x a 1 dx 0.1
Find: the absolute change f f a dx f a
3
f f 1.1 f 1 0.231 0 0.231
the estimated change
2
f x 3x 1
df f a dx
f 1 2
df 2dx 2 0.1 0.2
Guided Practice
The given function
from a to a + dx.
f
changes value when x changes
f x x x a 1 dx 0.1
3
Find: the approximation error
f df
f df 0.231 0.2 0.031
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.
A r Estimated increase is dA:
dA A a dr 2 a dr 2 10 0.1
2
2
m2
True change:
10.1 10 102.01100
2
m
2 0.01
2
2
dA
error
Guided Practice
Write a differential formula that estimates the given change
in area.
The change in the surface area S 4 r of a sphere
when the radius changes from a to a + dr.
2
dS
8 r dS 8 rdr
dr
When r changes from a to a + dr…
The change in surface area is approximately
dS 8 adr
Guided Practice
Write a differential formula that estimates the given change
in area.
The change in the surface area S 6 x of a cube when
the edge lengths change from a to a + dx.
2
dS
12 x dS 12 xdx
dx
When x changes from a to a + dx…
The change in surface area is approximately
dS 12adx
Guided Practice
The differential equation df f x dx
tells us how sensitive the output of f is to
a change in input at different values of x.
The larger the value of f at x, the greater
the effect of a given change dx.
Guided Practice
You want to calculate the depth of a well from the given
equation by timing how long it takes a heavy stone you
drop to splash into the water below. How sensitive will
your calculations be to a 0.1 second error in measuring
the time?
2
s 16t
The size of ds in the equation
ds 32tdt
depends on how big t is. If t = 2 sec, the error caused by
dt = 0.1 is only
ds 32 2 0.1 6.4 ft
Three seconds later at t = 5 sec, the error caused by the
same dt:
ds 32 5 0.1 16 ft
Guided Practice
The height and radius of a right circular cylinder are equal,
3
so the cylinder’s volume is V h . The volume is to be
calculated with an error of no more than 1% of the true
value. Find approx. the greatest error that can be tolerated
in the measurement of h, expressed as a percentage of h.
dV
2
3 h dV 3 h2 dh
dh
We want
dV 0.01V, which gives
0.01h
dh
3
3 h 2 dh 0.01 h3
The height should be measured with
an error of no more than
1
%.
3
Guided Practice
A manufacturer contracts to mint coins for the federal
government. How much variation dr in the radius of the
coins can be tolerated if the coins are to weigh within
1/1000 of their ideal weight? Assume the thickness does
not vary.
dV
2 rh dV 2 rhdr
V r h
dr
2
We want
dV 0.001V, which gives 2 rhdr 0.001 r h
2
dr 0.0005r
The variation of the radius should not
exceed 1/2000 of the ideal radius,
that is, 0.05% of the ideal radius.