E e 0 h e 0 f 0 lim h 0 h eh 1 lim h 0 h 1 y f

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Unformatted text preview: ing
to

 M (W ) = 5W + 2. The
number
of
bites
(B)
depends
on
the
number
 of
mosquitoes
according
to

 B( M ) = 0.5 M . € Find
the
deriva0ve
of
B
as
a
func0on
of
W.
 Deriva0ve
of
the
Natural

 Exponen0al
Func0on
 Defini&on:
 The
number
e
is
the
 number
for
which
 h e −1 lim =1 h →0 h Natural
Exponen0al
 Func0on:
 f ( x) = e x y f ( x) = ex € x Deriva0ve
of
the
Natural

 Exponen0al
Func0on
 Note:

 This
defini0on
states
that
 the
slope
of
the
tangent
to
 the
curve
at
(0,1)
is
exactly
 1,
i.e.
 e 0+ h − e 0 f ' (0) = lim h →0 h € eh − 1 = lim h →0 h =1 y f ( x) = ex € m =1 € x Deriva0ve
of
the
Natural

 Exponen0al
Func0on
 y If



















then
 f ' ( x ) = e . f ( x) = e , x x f ( x) = ex ( x, e x )... slope = e x Proof:
 (1, e)... slope = e € € € (0, 1)... slope = 1 € € Note:

 The
slope
of
a
tangent
line
to
the
curve
 Note:

 x f ( x) = e 

















is
equal
to
the
val...
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