38. (a)
Note that
A
5
p
r
2
and
}
d
d
A
r
} 5
2
p
r
, so
dA
5
2
p
r dr
.
When
r
changes from
a
to
a
1
dr
, the change in area is
approximately 2
p
a dr
. Substituting 2 for
a
and 0.02 for
dr
, the change in area is approximately
2
p
(2)(0.02)
5
0.08
p
<
0.2513
(b)
}
d
A
A
} 5 }
0.
4
0
p
8
p
} 5
0.02
5
2%
39.
Let
A
5
cross section area,
C
5
circumference, and
D
5
diameter. Then
D
5 }
C
p
}
, so
}
d
d
D
C
} 5 }
p
1
}
and
dD
5 }
p
1
}
dC
.
Also,
A
5
p
1
}
D
2
}
2
2
5
p
1
}
2
C
p
}
2
2
5 }
4
C
p
2
}
, so
}
d
d
C
A
} 5 }
2
C
p
}
and
dA
5 }
2
C
p
}
dC
. When
C
increases from 10
p
in. to
10
p
1
2 in. the diameter increases by
dD
5 }
p
1
}
(2)
5 }
p
2
}
<
0.6366 in. and the area increases by
approximately
dA
5 }
1
2
0
p
p
}
(2)
5
10 in
2
.
40.
Let
x
5
edge length and
V
5
volume. Then
V
5
x
3
, and so
dV
5
3
x
2
dx
. With
x
5
10 cm and
dx
5
0.01
x
5
0.1 cm, we
have
V
5
10
3
5
1000 cm
3
and
dV
5
3(10)
2
(0.1)
5
30 cm
3
,
so the percentage error in the volume measurement is
approximately
}
d
V
V
} 5 }
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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