文档介绍：Section
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Best Affine Approximations,
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In this section we will take up the general question of how to find best affine approximations
and also discuss an interpretation of the derivative of a function as an instantaneous rate
of change. We will consider putational procedures for finding derivatives in
Sections through .
To begin, suppose f is a function defined on an open interval containing the point c
and let T be an affine function with T (c) = f(c). As in Section , we may write T in the
form
T (x) = m(x − c) + f(c) ()
for some constant m. Let
R(h) = f(c + h) − T (c + h) = f(c + h) − mh − f(c). ()
Then
R(h) f(c + h) − T (c + h)
lim = lim
h→0 h h→0 h
f(c + h) − mh − f(c)
= lim ()
h→0 h
f(c + h) − f(c)
= lim − m .
h→0 h
Hence R(h) is o(h), and T is the best affine approximation to f at c, if and only if
f(c + h) − f(c)
lim − m = 0, ()
h→0 h
which is true if and only if
f(c + h) − f(c)
lim = m. ()
h→0 h
In particular, if
f(c + h) − f(c)
lim
h→0 h
exists, then f has a best affine approximation at c and
f(c + h) − f(c)
f 0(c) = lim . ()
h→0 h
1
2 Best Affine Approximations, Derivatives, and Rates of Change Section
Conversely, if T (x) = m(x − c) + f(c) is the best affine approximation to f at c, then it
follows that
f(c + h) − f(c)
m = lim . ()
h→0 h
Definition We say a function f is differentiable at a point c if
f(c + h) − f(c)
lim ()
h→0 h
exists.
In summary, if we are given a function f which is differentiable at c, then the best
affine approximation to f at c exists and is given by
T (x) = f 0(c)(x − c) + f(c), ()
where
f(c + h) − f(c)
f 0(c) = lim . ()
h→0 h
Conversely, if f has a best affine approximation at a point c, then f is differentiable at c
and the best affine approximation is given by ()