11.
}
d
d
y
x
}
5
lim
h
→
0
}
y
(
x
1
h
h
)
2
y
(
x
)
}
5
lim
h
→
0
5
lim
h
→
0
5
lim
h
→
0
5
lim
h
→
0
(4
x
1
2
h
2
13)
5
4
x
2
13
At
x
5
3,
}
d
d
y
x
}
5
4(3)
2
13
5 2
1, so the tangent line has
slope
2
1 and passes through (3,
y
(3))
5
(3,
2
16).
y
5 2
1(
x
2
3)
2
16
y
5 2
x
2
13
12.
Let
f
(
x
)
5
x
3
.
f
9
(1)
5
lim
h
→
0
}
f
(1
1
h
h
)
2
f
(1)
}
5
lim
h
→
0
}
(1
1
h
h
)
3
2
1
3
}
5
lim
h
→
0
5
lim
h
→
0
(3
1
3
h
1
h
2
)
5
3
(a)
The tangent line has slope 3 and passes through (1, 1).
Its equation is
y
5
3(
x
2
1)
1
1, or
y
5
3
x
2
2.
(b)
The normal line has slope
2
}
1
3
}
and passes through
(1, 1). Its equation is
y
5 2
}
1
3
}
(
x
2
1)
1
1, or
y
5 2
}
1
3
}
x
1
}
4
3
}
.
13.
Since the graph of
y
5
x
ln
x
2
x
is decreasing for
0
,
x
,
1 and increasing for
x
.
1, its derivative is
negative for 0
,
x
,
1 and positive for
x
.
1. The only
one of the given functions with this property is
y
5
ln
x
.
Note also that
y
5
ln
x
is undefined for
x
,
0, which
further agrees with the given graph. (ii)
14.
Each of the functions
y
5
sin
x
,
y
5
x
,
y
5
ˇ
x
w
has the
property that
y
(0)
5
0 but the graph has nonzero slope (or
undefined slope) at
x
5
0, so none of these functions can be
its own derivative. The function
y
5
x
2
is not its own
derivative because
y
(1)
5
1 but
y
9
(1)
5
lim
h
→
0
}
(1
1
h
h
)
2
2
1
2
}
5
lim
h
→
0
}
2
h
1
h
h
2
}
5
lim
h
→
0
(2
1
h
)
5
2.
This leaves only
e
x
, which can plausibly be its own
derivative because both the function value and the slope
increase from very small positive values to very large
values as we move from left to right along the graph. (iv)
15. (a)
The amount of daylight is increasing at the fastest rate
when the slope of the graph is largest. This occurs
about one-fourth of the way through the year, sometime
around April 1. The rate at this time is approximately
}
2
4
4
h
d
o
a
u
y
rs
s
}
or
}
1
6
}
hour per day.
(b)
Yes, the rate of change is zero when the tangent to the
graph is horizontal. This occurs near the beginning of
the year and halfway through the year, around
January 1 and July 1.
(c)
Positive: January 1 through July 1
Negative: July 1 through December 31
16.
The slope of the given graph is zero at
x
<
2
2 and at
x
<
1, so the derivative graph includes (
2
2, 0) and (1, 0).
The slopes at
x
5 2
3 and at
x
5
2 are about 5 and the
slope at
x
5 2
0.5 is about
2
2.5, so the derivative graph
includes (
2
3, 5), (2, 5), and (
2
0.5,
2
2.5). Connecting the
points smoothly, we obtain the graph shown.
17. (a)
Using Figure 3.10a, the number of rabbits is largest
after 40 days and smallest from about 130 to 200 days.
Using Figure 3.10b, the derivative is 0 at these times.
(b)
Using Figure 3.10b, the derivative is largest after
20 days and smallest after about 63 days. Using
Figure 3.10a, there were 1700 and about 1300 rabbits,
respectively, at these times.
18. (a)
The slope from
x
5 2
4 to
x
5
0 is
}
0
2
2
2
(
2
0
4)
}
5
}
1
2
}
.
The slope from
x
5
0 to
x
5
1 is
}
2
1
2
2
2
0
2
}
5 2
4.
The slope from
x
5
1 to
x
5
4 is
}
2
2
4
2
2
(
2
1
2)
}
5
0.