(b)
lim
x
→
‘
}
g
f
(
(
x
x
)
)
} 5
lim
x
→
‘
5
lim
x
→
‘
}
ˇ
x
ˇ
2
w
2
w
x
2
w
a
w
2
w
}
5
lim
x
→
‘
!
1
§
2
§
}
a
x
§
2
2
}
§
5
1
(c)
lim
x
→
‘
}
g
f
(
(
x
x
)
)
} 5
lim
x
→
‘
5
lim
x
→
‘
}
ˇ
x
ˇ
2
w
2
w
x
2
w
a
w
2
w
}
5
lim
x
→
‘
!
1
§
2
§
}
a
x
§
2
2
}
§
5
1
■
Section 3.8
Derivatives of Inverse
Trigonometric Functions (pp. 157–163)
Exploration 1
Finding a Derivative on an
Inverse Graph Geometrically
1.
The graph is shown at the right. It appears to be a oneto
one function
[
2
4.7, 4.7] by [
2
3.1, 3.1]
2.
f
9
(
x
)
5
5
x
4
1
2. The fact that this function is always
positive enables us to conclude that
f
is everywhere
increasing, and hence onetoone.
3.
The graph of
f
2
1
is shown to the right, along with the
graph of
f
. The graph of
f
2
1
is obtained from the graph of
f
by reflecting it in the line
y
5
x
.
[
2
4.7, 4.7] by [
2
3.1, 3.1]
4.
The line
L
is tangent to the graph of
f
2
1
at the point (2, 1).
[
2
4.7, 4.7] by [
2
3.1, 3.1]
5.
The reflection of line
L
is tangent to the graph of
f
at the
point (1, 2).
[
2
4.7, 4.7] by [
2
3.1, 3.1]
6.
The reflection of line
L
is the tangent line to the graph of
y
5
x
5
1
2
x
2
1 at the point (1, 2). The slope is
}
d
d
y
x
}
at
x
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 Fall '08
 GERMAN
 Slope, lim, Inverse function, a2 x2 x2

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