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CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
3.2
Linear and Affine Functions
So far we have seen the derivative in two settings. For a realvalued
function
f
(
x
) of one variable, the derivative
f
′
(
x
0
) at a point
x
0
first
comes up as a number, which turns out to be the slope of the tangent line.
This in turn is the line which best approximates the graph
y
=
f
(
x
) near
the point, in the sense that it is the graph of the polynomial of degree one,
T
x
0
f
=
f
(
x
0
) +
f
′
(
x
0
) (
x
−
x
0
), which has
firstorder contact
with the
curve at the point (
x
0
,f
(
x
0
)):
vextendsingle
vextendsingle
f
(
x
)
−
T
x
0
f
(
x
)
vextendsingle
vextendsingle
=
o
(

x
−
x
0

)
or
vextendsingle
vextendsingle
f
(
x
)
−
T
x
0
f
(
x
)
vextendsingle
vextendsingle

x
−
x
0

→
0 as
x
→
x
0
.
If we look back at the construction of the derivative
vector
p
′
(
t
) of a vectorvalued
function
−→
p
:
R
→
R
3
in
§
2.3
, we see a similar phenomenon:
vector
p
′
(
t
0
) =
−→
v
(
t
0
)
is the direction vector for a parametrization of the tangent line, and the
resulting vectorvalued function,
T
t
0
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 Fall '08
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 Calculus, Derivative, Continuous function, Vectorvalued function, ﬁrstorder contact

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