18.01
Calculus
Jason
Starr
Fall
2005
Lecture
9.
September
29,
2005
Homework.
Problem
Set
2
all
of
Part
I
and
Part
II.
Practice
Problems.
Course
Reader:
2B1,
2B2,
2B4,
2B5.
1.
Application
of
the
Mean
Value
Theorem.
A
realworld
application
of
the
Mean
Value
Theorem
is
error
analysis
.
A
device
accepts
an
input
signal
x
and
returns
an
output
signal
y
.
If
the
input
signal
is
always
in
the
range
−
1
/
2
≤
x
≤
1
/
2
and
if
the
output
signal
is,
1
y
=
f
(
x
)
=
1
+
x
+
x
2
+
x
3
,
what
precision
of
the
input
signal
x
is
required
to
get
a
precision
of
±
10
−
3
for
the
output
signal?
If
the
ideal
input
signal
is
x
=
a
,
and
if
the
precision
is
±
h
,
then
the
actual
input
signal
is
in
the
range
a
−
h
≤
x
≤
a
+
h
.
The
precision
of
the
output
signal
is
f
(
x
)
−
f
(
a
) .
By
the
Mean
Value


Theorem,
f
(
x
)
−
f
(
a
)
=
f
(
c
)
,
x
−
a
for
some
c
between
a
and
x
.
The
derivative
f
(
x
)
is,
f
(
x
)
=
−
(3
x
2
+
2
x
+
1)
.
(1
+
x
+
x
2
+
x
3
)
2
For
−
1
/
2
≤
x
≤
1
/
2,
this
is
bounded
by,
3(1
/
2)
2
+
2(1
/
2)
+
1

f
(
x
)
=
7
.
04
.
 ≤
[1
+
(
−
1
/
2)
+
(
−
1
/
2)
2
+
(
−
1
/
2)
3
]
2
Thus
the
Mean
Value
Theorem
gives,
f
(
x
)
−
f
(
a
) =
f
(
c
≤
7
.
04
x
−
a
≤
7
.
04
h.



)

x
−
a



Therefore
a
precision
for
the
input
signal
of,
h
=
10
−
3
/
7
.
04
≈
10
−
4
guarantees
a
precision
of
10
−
3
for
the
output
signal.
2.
First
derivative
test.
A
function
f
(
x
)
is
increasing
,
respectively
decreasing
,
if
f
(
a
)
is
less
than
f
(
b
),
resp.
greater
than
f
(
b
),
whenever
a
is
less
than
b
.
In
symbols,
f
is
increasing,
respectively
decreasing,
if
f
(
a
)
< f
(
b
)
whenever
a < b,
resp.
f
(
a
)
> f
(
b
)
whenever
a < b.