1ls3_lectures1617

# 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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