MINIQUIZ 10/2/08  REVISED SOLUTIONS
Name:
Topics covered:
• §
4.1  Maximum and minimum values
• §
4.2  The mean value theorem
TrueFalse: (3 pts. each)
1.
If
f
is differentiable at
x
=
a
, then
f
is continuous at
x
=
a
.
TRUE. The following proof, which you do NOT need to know, is taken from pg. 128 of Stewart 6e.
Assume
f
is differentiable at
x
=
a
. Then
f
(
a
) = lim
x
→
a
f
(
x
)

f
(
a
)
x

a
exists (and is finite). To show that
f
is continuous at
x
=
a
, we need to show that lim
x
→
a
f
(
x
) =
f
(
a
),
or equivalently, lim
x
→
a
(
f
(
x
)

f
(
a
)) = 0. Trivially, we have
f
(
x
)

f
(
a
) =
f
(
x
)

f
(
a
)
x

a
(
x

a
)
and so
lim
x
→
a
[
f
(
x
)

f
(
a
)]
=
lim
x
→
a
f
(
x
)

f
(
a
)
x

a
(
x

a
)
=
lim
x
→
a
f
(
x
)

f
(
a
)
x

a
·
lim
x
→
a
(
x

a
)
=
f
(
a
)
·
0
=
0
.
2.
If
f
is continuous at
x
=
a
, then
f
is differentiable at
x
=
a
.
FALSE. A counterexample is
f
(
x
) =

x

. We claim that this function is continuous at
x
= 0. For this
we need to show that lim
x
→
0
f
(
x
) =
f
(0), or in our case that lim
x
→
0

x

= 0. We compute the left
and righthand limits and show they are equal:
lim
x
→
0+

x

=
lim
x
→
0+
x
= 0
,
lim
x
→
0


x

=
lim
x
→
0

(

x
) = 0
.
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 Fall '10
 EdwardL.Robinson
 Mean Value Theorem, Quadratic equation, lim, Continuous function

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