86
Section 3.1
54. (a)
fx
a
x
()
2
22
=−−
=−
42
2
a
a
(b)
242
2
=−=
−
2
48 4
2
(c)
For
x
≠
2,
f
is continuous. For
x
=
2, we have
lim
( )
( )
( )
xx
f
→→
−+
==
=
−
24
as long as
a
=±
2.
55. (a)
gx
x
x
x
3
2
(b)
x
x
=
+
=
32
2
21
3
1
=
+
3
3
x
x
3
3
1
=
Chapter 3
Derivatives
Section 3.1
Derivative of a Function
(pp. 99–108)
Exploration 1
Reading the Graphs
1.
The graph in Figure 3.3b represents the rate of change of
the depth of the water in the ditch with respect to time.
Since
y
is measured in inches and
x
is measured in days, the
derivative
dy
dx
would be measured in inches per day. Those
are the units that should be used along the
y
-axis in Figure
3.3b.
2.
The water in the ditch is 1 inch deep at the start of the first
day and rising rapidly. It continues to rise, at a gradually
decreasing rate, until the end of the second day, when it
achieves a maximum depth of 5 inches. During days 3, 4, 5,
and 6, the water level goes down, until it reaches a depth of
1 inch at the end of day 6. During the seventh day it rises
again, almost to a depth of 2 inches.
3.
The weather appears to have been wettest at the beginning
of day 1 (when the water level was rising fastest) and driest
at the end of day 4 (when the water level was declining the
fastest).
4.
The highest point on the graph of the derivative shows
where the water is rising the fastest, while the lowest point
(most negative) on the graph of the derivative shows where
the water is declining the fastest.
5.
The
y
-coordinate of point
C
gives the maximum depth of
the water level in the ditch over the 7-day period, while the
x
-coordinate of
C
gives the time during the 7-day period
that the maximum depth occurred. The derivative of the
function changes sign from positive to negative at
C
′
,
indicating that this is when the water level stops rising
and begins falling.
6.
Water continues to run down sides of hills and through
underground streams long after the rain has stopped falling.
Depending on how much high ground is located near the
ditch, water from the first day’s rain could still be flowing
into the ditch several days later. Engineers responsible for
flood control of major rivers must take this into consideration
when they predict when floodwaters will “crest,” and at
what levels.
Quick Review 3.1
1.
hh
h
h
+−
=
++ −
0
2
0
2
4
44
4
(
)
=+
=+=
→
h
h
0
4
404
2.
x
x
→
+
+
=
+
=
2
3
2
23
2
5
2
3.
Since
y
y
1 for
y
y
y
y
<=
−
→
−
01
0
, lim
.
4.
x
x
x
−
−
=
−
28
2
2
2
=
+
=
→
lim (
)
(
)
h
x
4
2
4
2
8
5.
The vertex of the parabola is at (0, 1). The slope of the line
through (0, 1) and another point (
h
,
h
2
+
1) on the parabola
is
.
h
h
h
2
11
0
−
=
Since lim
,
h
h
→
=
0
0 the slope of the line
tangent to the parabola at its vertex is 0.
6.
Use the graph of
f
in the window [
−
6, 6] by [
−
4, 4] to find
that (0, 2) is the coordinate of the high point and (2,
−
2) is
the coordinate of the low point. Therefore,
f
is increasing
on (
−∞
, 0] and [2,
∞
).