Unformatted text preview: l to ∆
y
2 ∆
x = ∆
x Let us try to figure out its value when ∆x is small:
6 x0 ∆ +3( ∆ )
x
x
∆
x 2 =6 x0 +3∆
x ∆x → 0 6 x0 Clearly, as ∆x is getting arbitrarily small, the above
Clearly,
expression converges to just 6 x0
7 Definition of a Derivative
The derivative of a function x0 is defined as follows: dy
dx x0 ∆y
≡ f ′( x0 ) ≡ lim
∆x → ∆
0
x x0 f ( x0 + ∆x ) − f ( x0 )
≡ lim
∆→
x0
∆x Read as: a derivative of a function at a given x0 is the limit of its
Read
difference quotient at x0 as ∆x approaches zero.
1) A derivative is a function of just x0 since ∆x is
1)
assumed to be infinitely close to zero. The difference
quotient is a function of both.
quotient
2) Derivatives may not be computed for certain functions.
2)
For example, functions with kinks do not have derivatives
everywhere in their domain.
everywhere
3) When computing a derivative, remember it only makes
3)
sense to do so with a particular point in the function’s
domain in mind—”loose” derivatives are nonsensical.
domain
8 Derivatives and Slopes Consider the following cost function: C ( Q ) =1 +Q 2
Consider
C ∆C
∆Q → 0 ∆ Q
Q0 f ′( Q0 ) = lim A straight line passing
straight
through A and B
through C1 is crossing the
is
horizontal axis
at angle α
at B
∆C
∆Q → ∆
0
Q = f ′( Q0 ) tan β = lim ∆C whose tangent
whose
is equal to tan α = C
is Q0 − C0 ∆C
=
Q1 − Q0 ∆Q
1 A C0
β α
Q0 As ∆ Q is getting smaller, this straight
As
line will be approaching the tangent line
tangent
to C(Q) at Q0 ∆
Q Q1 Q
9 Derivatives and Increasing/Decreasing Functions
Since a function’s derivative is representing its slope, we have two
Since
We shall later study rules of differentiation that will tell us exactly how
We
handy rules for analyzing the function’s behavior in some set S in its
to take derivatives of various functions.
to
domain:
domain:
1) If f ′( x ) > 0 for x ∈ S , function f ( x ) is positively sloped for x ∈ S
If
df
Consider the function f ( x ) = x : its derivative dx = 2 x0 , so it is positive
Consider
x0
for every x0 > 0 . Hence, function f ( x ) has a positive slope for that range
2 of x. 2) If f ′ ( x ) < 0, x ∈ S , function f
2) ( x) is negatively sloped if x ∈ S For function f ( x ) its derivative f ′( x ) = 2 x is negative for x<0.
For
Hence, this function has a negative slope for x<0
10 A Note on Kinky Functions
The geometric interpretation of the derivative concept makes it clear why
The
the latter is not defined for functions with the kinks (where the kinks
occur).
occur).
If we only increase x,
If
However, if we
However,
this will be the slope.
this
y decrease x, the
decrease the
slope line will be
quite different
quite A At point A this
At
function has a kink,
which makes it
problematic to
establish the location
of the slope line at
this point.
this x
11 LeftSide and RightSide Derivatives
As the previous example demonstrated, it can sometimes be important
As
whether we are taking the increments...
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 Fall '11
 Kim
 Derivative, lim g

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