roberts (eer474) – Quest HW 4 – seckin – (56425)
1
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printout
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have
17
questions.
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001
10.0 points
What is the significance of the expression
f
(1 +
h
)
−
f
(1)
h
in the following graph of
f
when
h
=
7
2
?
1
2
3
4
5
1
2
3
4
5
P
Q
R
S
T
U
1.
slope of line through
P
and
U
2.
slope of line through
P
and
R
3.
equation of line through
P
and
T
4.
length of line segment
PR
5.
equation of line through
P
and
R
6.
slope of tangent line at
P
7.
length of line segment
PT
8.
slope of line through
P
and
T
correct
9.
equation of line through
P
and
U
10.
length of line segment
PU
Explanation:
When
h
=
7
2
the expression
f
(1 +
h
)
−
f
(1)
h
is the ratio of the rise and the run between the
points
P
and
T
. Thus the expression is the
slope of line through
P
and
T
.
002
10.0 points
If
f
is a differentiable function, then
f
′
(
a
)
is given by which of the following?
I. lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
II. lim
x
→
a
f
(
x
)
−
f
(
a
)
x
−
a
III. lim
x
→
a
f
(
x
+
h
)
−
f
(
x
)
h
1.
I, II, and III
2.
I only
3.
II only
4.
I and III only
5.
I and II only
correct
Explanation:
Both of
f
′
(
a
) =
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
and
f
′
(
a
) =
lim
x
→
a
f
(
x
)
−
f
(
a
)
x
−
a
are valid definitions of
f
′
(
a
). By contrast,
lim
x
→
a
f
(
x
+
h
)
−
f
(
x
)
h
=
f
(
a
+
h
)
−
f
(
a
)
h
because
f
is continuous. Consequently,
f
′
(
a
)
is given only by
I and II
.
003
10.0 points
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roberts (eer474) – Quest HW 4 – seckin – (56425)
2
Let
f
be a function such that
lim
h
→
0
f
(1 +
h
) = 2
,
and
lim
h
→
0
f
(1 +
h
)
−
f
(1)
h
= 3
.
Which of the following statements are true?
A.
f
(1) = 2
,
f
′
(1) = 3 ,
B.
f
is continuous at
x
= 1 ,
C.
f
is differentiable at
x
= 1 .
1.
all are true
correct
2.
none are true
3.
A and B only
4.
A and C only
5.
C only
6.
A only
7.
B and C only
8.
B only
Explanation:
A. True: by definition,
f
is differentiable at
x
= 1, hence also continuous at
x
= 1.
B. True:
f
is differentiable at
x
= 1, so also
continuous at
x
= 1.
C. True: by definition.
004
10.0 points
Let
f
be the function defined by
f
(
x
) = 7
x
−
(
x
−
3 +

x
−
3

)
2
.
Determine if
lim
h
→
0
f
(1 +
h
)
−
f
(1)
h
exists, and if it does, find its value.
1.
limit doesn’t exist
2.
limit = 9
3.
limit = 8
4.
limit = 7
correct
5.
limit = 10
6.
limit = 11
Explanation:
Since
f
(
x
) =
braceleftbigg
7
x,
x <
3,
7
x
−
4(
x
−
3)
2
,
x
≥
3,
we see that
lim
h
→
0
f
(1 +
h
)
−
f
(1)
h
=
f
′
(1)
because 1
,
1 +
h <
3 for all small
h
. Conse
quently,
limit = 7
.
005
10.0 points
If
f
is a function having
2
4
6
−
2
−
4
2
4
6
8
10
−
2
as its graph, which of the following could be
the graph of
f
′
?
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 Spring '09
 Gualdani
 Derivative, Differential Calculus, Quest HW

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