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   102.01100 
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.