Graphics Reference
In-Depth Information
Figure 4.7.
The geometry of the
derivative.
x
n+ 1
f(p) + T(h)
f(p + h)
f(p)
T(h)
R
n
p
p + h
()
-
()
=
()
-+
(
(
)
-
()
)
++
(
)
-
()
-
()
ST
hh
h
S f
hphpphph
h
hphp
h
f
f
f
T
lim
lim
h
Æ
0
h
Æ
0
()
-+
(
(
)
-
()
)
(
)
-
()
-
()
S
f
f
f
ph
+
f
p
T
h
£
lim
+
lim
h
h
Æ
0
h
Æ
0
=
0
.
Now t
p
Æ
0
as t Æ 0, for any
p
Œ
R
n
. Therefore, if
p
π 0, then we can let t
p
play the
role of
h
above to get
()
-
()
=
()
-
()
St
pp
p
Tt
ST
pp
p
0
=
lim
,
t
t
Æ
0
so that S(
p
) = T(
p
).
One should think of the derivative as the linearization of a function. No
one
linear
transformation approximates f. Instead, there are lots of linear transformations, one
at every point, which locally approximate f. The linear map T
p
defined by
()
=
()
+
()
-
(
)
,
T
p
qp pqp
f
f
(4.4)
whose graph is a plane, is what approximates the graph of f in a neighborhood of
p
.
Definition.
The graph of the linear transformation T
p
:
R
n
Æ
R
m
defined by equation
(4.4) is called the
tangent plane
to the graph of f at the point (
p
,f(
p
)) in
R
n
¥
R
m
. If
n = m = 1, then it is usually called the
tangent line
.
The current notion of a tangent plane is rather special. We shall have much more
to say about tangent planes and begin to see their importance in Chapter 8.
The derivative of f(x) = x
2
at a is defined by Df(a)(x) = 2ax.
4.3.2. Example.
Proof.
Let