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HOMEWORK V
Due on January 5,2009 for sections 01,04,05
Due on January 6,2009 for sections 03,06
Due on January 7,2009 for sections 02,07
Bring the homework to the first lecture on the due date.
1)
Suppose
is twice differentiable on an interval
(i.e.,
f(
f
I
x
)
′′
exists on
).Suppose that the
points 0 and 2 belong to
and that f(
I
I
0)
f(1)
0
=
=
and f(2)
1
=
.Prove that:
a)
1
1
f(c)
2
′
=
for some point
in
.
1
c
I
b)
2
1
c)
2
>
.
2
c
I
c)
3
1
111
′
=
.
3
c
I
2)
Do the fcns.
a)
1
f(x)
sin x
=
on
and
b)
(0,
),
∞
2
2x
g(x)
x e
−
=
have absolute maximim or minumum values?Justify your answer.
3)
If
and
for
1x
,how small can f(
possibly be?
10
=
2
′
≥
4
≤≤
4)
4)
Find the length of the longest beam that can be carried horizontally around the corners
from hallway of width a m to hallway of width b m.(See the figure,and assume that the beam
has no width.)
a
m
b
m
5)
Sketch the graph of a function which has the following properties:
xx
f(0)
1,f( 1)
0,f(2)
1,limf(x)
2, lim f(x)
1,f (x)
0
→∞
→−∞
′
=±
=
=
=
=
−
>
on
( )
,0
−∞
and on
()
1,
,
∞
on
on
and on
0
′
<
0,1 ,f (x)
0
>
(
−∞
)
( )
0, 2
,and f (x)
0
<
on
.
2,
∞
Assume that
is continuous and its derivatives exist everywhere unless the contrary is
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 Spring '10
 AhmetGuloglu
 Math, Calculus

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