is an example of one which appears to be tangent
to the line y = 0 at the point
(0,0)
.
PICT
You see in this picture, the graph of the function y = ε
|x|
also where ε > 0 is just a
positive number. Note there exists δ > 0 such that if
|x|
< δ, then
|o(x)|
< ε
|x|
or in other
words,
|o(x)|
-|x|--< ε.
You might draw a few other pictures of functions which would have the appearance
of being tangent to the line y = 0 at the point
(0,0)
and observe that in every
case, it will follow that for all ε > 0 there exists δ > 0 such that if 0 <
|x|
< δ,
then
|o(x)|< ε. (7.8)
|x|
(7.8)
In other words, a reasonable way to say a function is tangent to the line y = 0 at
(0,0)
is to
say for all ε > 0 there exists δ > 0 such that 7.8 holds. In other words, the function y = o
(x )
is tangent at
(0,0)
if and only if
|o(x)|
lix→m0 |x| = 0.
Definition 7.3.1A function y = k
(x)
is said to be o
(x)
if
|k(x)|
lxim→0--|x|- = 0 (7.9)
(7.9)
As was just discussed, in the case where x ∈ ℝ and k is a function having values in ℝ this
is geometrically the same as saying the function is tangent to the line y = 0 at the point
, etc. The usage is very imprecise and sloppy, leaving out
exactly the details which are of absolutely no significance in what is about to be discussed. It
is this sloppiness which makes the notation so useful. It prevents you from fussing with things
which do not matter.
Now consider the case of the function y = g
(x)
tangent to y = b + mx at the point
(c,d)
.
PICT
Thus, in particular, g
(c)
= b + mc = d. Then letting x = c + h, it follows x is close to c if
and only if h is close to 0. Consider then the two functions
y = g(c+ h),y = b+ m (c+ h).
If they are tangent as shown in the above picture, you should have the function
The above definition is more general than what will be extensively discussed here. I will
usually consider the case where the function is defined on some interval contained in ℝ. In
this context, the definition of derivative can also be extended to include right and left
derivatives.
Definition 7.3.3Let g be a functiondefined on an interval, [c,b). Then g_{+}^{′}
(c)
is thenumber, if it exists, which satisfies
′
g+(c+ h)− g+ (c)− g+ (c)h = o(h)
where o
(h)
is defined in Definition 7.3.1except you only consider positive h. Thus
lim |o-(h)|= 0.
h→0+ |h|
This is the derivative from the right. Let g be a function defined on an interval, (a,c]. Theng_{−}^{′}
(c)
is the number, if it exists, which satisfies
g− (c+ h)− g− (c)− g′− (c) h = o(h)
where o
(h)
is defined in Definition 7.3.1except you only consider negative h. Thus
lim |o-(h)|= 0.
h→0− |h|
This is the derivative from the left.
I will not pay any attention to these distinctions from now on. In particular I will not
write g_{−}^{′} and g_{+}^{′} unless it is necessary. If the domain of a function defined on a subset of ℝ is
not open, it will be understood that at an endpoint, the derivative meant will be the
appropriate derivative from the right or the left. First I need to show this is well defined
because there cannot be two values for g^{′}
(c)
.
Theorem 7.3.4The derivative is welldefined because if