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WORKSHEET 15  Fall 1995
1.
The IntermediateValue Theorem
Let
f
be continuous throughout the closed interval [
a, b
]. Let
m
be any number between
f
(
a
)and
f
(
b
).
a) Draw several pictures of functions satisfying this hypotheses. Make sure you include both the case
that
f
(
a
)
≥
f
(
b
)andthat
f
(
a
)
≤
f
(
b
).
b) In each of your pictures, pick a value for
m
and draw the line
y
=
m
. How many times does this
line intersect the graph of
f
in each case. Can you draw an example where they
don’t
intersect?
c) Give an argument that given the hypotheses above, there must be at least one
c
in [
a, b
] such that
f
(
c
)=
m
.
2. Use the Intermediate Value Theorem to show that there is a number
c
such that 4
−
c
=2
c
.
3. A set in the plane bounded by a curve is
convex
if for any two points
P
and
Q
in the set, the line
segment joining them also lies in the set.
Q
Q
P
P
nonconvex
convex
a) Let
L
be a line in the plane and let
K
be a convex set. Show that there is a line parallel to
L
that
cuts
K
into two pieces of equal area.
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Spring '08 term at North Texas.
 Spring '08
 Grether

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