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
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