ngac (xtn62) – NewFinalReview01 – Gilbert – (58365)
1
This printout should have 18 questions.
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be±ore answering.
001
10.0 points
I±
f
is a ±unction defned on (
−
2
,
2) whose
graph is
1
2
−
1
−
2
1
2
−
1
−
2
which o± the ±ollowing is the graph o± its
derivative
f
′
?
1.
1
2
−
1
−
2
1
2
−
1
−
2
2.
1
2
−
1
−
2
1
2
−
1
−
2
3.
1
2
−
1
−
2
1
2
−
1
−
2
4.
1
2
−
1
−
2
1
2
−
1
−
2
correct
5.
1
2
−
1
−
2
1
2
−
1
−
2
6.
1
2
−
1
−
2
1
2
−
1
−
2
Explanation:
Since the graph on (
−
2
,
2) consists o±
straight lines joined at
x
=
−
1 and
x
= 1,
the derivative o±
f
will exist at all points in
(
−
2
,
2) except
x
=
−
1 and
x
= 1, eliminating
the answer whose graph contain flled dots at
x
=
−
1 and
x
= 1. On the other hand, the
graph o±
f
(i) has slope
−
2 on (
−
2
,
−
1),
(ii) has slope 1 on (
−
1
,
1), and
(iii) has slope
−
2 on (1
,
2).
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2
Consequently, the graph of
f
′
consists of
the horizontal lines and ‘holes’ in
1
2
−
1
−
2
1
2
−
1
−
2
002
10.0 points
For which values of
x
is the function
f
(
x
) =
x
−
3
x
2
+ 3
not continuous?
1.
x
=
−
√
3
2.
f
is continuous for all
x
correct
3.
x
= 3
4.
x
= 3
,
−
√
3
5.
x
=
−
√
3
,
√
3
6.
x
=
√
3
Explanation:
Because
f
is a rational function it will fail
to be continuous only at zeros of the denomi
nator. But there are no real solutions to
x
2
=
−
3
,
so the function
f
is continuous for all values of
x
.
003
10.0 points
Below is the graph of a function
f
.
2
4
6
−
2
−
4
−
6
2
4
6
8
−
2
−
4
Use the graph to ±nd all values of
x
at which
f
is continuous but not di²erentiable.
1.
x
= 0
,
3
2.
x
=
−
3
,
0
,
3
3.
No such values
4.
x
=
−
3
,
0
5.
x
= 0
correct
Explanation:
A function is continuous but not di²eren
tiable at a point if and only if the function has
a cusp (wedge) at that point. In the graph
above the only such point is
x
= 0
.
Notice that
f
fails to be continuous at
x
=
−
3
and
x
= 3.
keywords: not di²erentiable, discontinuity,
continuous, graph
004
10.0 points
Determine
f
′
(
x
) when
f
(
x
) = 9
x
6
−
4
x
3
−
3
π.
1.
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 Fall '08
 Gilbert
 Calculus, Derivative, Limit, lim g

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