the natural log function
DESCRIPTION
A Brief History of e 1616-1618 John Napier, Scottish, Inventor of Logarithms, e implied in his work—gave table of natural log values (although did not recognize e as base) 1661 Christian Huygens, Dutch, studies relationship between the area under a rectangular hyperbola and logarithms, but does not see connection to e (later he does evaluate log e to 17 decimal places, but does see that this is a log) 1668 Nicolaus Mercator, German, names the natural logarithm, but does not discuss its baseTRANSCRIPT
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The Natural Log Function
5.1 Differentiation5.2 Integration
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A Brief History of e• 1616-1618 John Napier, Scottish, Inventor of
Logarithms, e implied in his work—gave table of natural log values (although did not recognize e as base)
• 1661 Christian Huygens, Dutch, studies relationship between the area under a rectangular hyperbola and logarithms, but does not see connection to e (later he does evaluate log e to 17 decimal places, but does see that this is a log)
• 1668 Nicolaus Mercator, German, names the natural logarithm, but does not discuss its base
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A Brief History of e• 1683 Jacques Bernoulli, Swiss, discovers e
through study of compound interest, does not call it e or recognize its connections to logs
• 1697 Johann Bernoulli (Jacques brother) begins study of the calculus of exponential functions and is perhaps the first to recognize logs as functions
• 1720s Leonard Euler, Swiss, first studied e, proved it irrational, and named it (the fact that it is the first letter of his surname is coincidental).
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A Brief History of Logs
• Napier: studies the motion of someone covering a distance d whose speed at each instant is equal to the remaining distance to be covered. He divided the time into short intervals of length , and assumed that the speed was constant within each short interval. He tabulated the corresponding values of distance and time obtained in this way.
• He coined a name for their relationship out of the Greek words logos (ratio) and arithmos (number). He used a Latinized version of his word: logarithm.
• In modern terms, we can say that the base of the logarithm in Napier's table was
• The actual concept of a base was not developed until later.
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Definition of Natural Log Function
• 1647--Gregorius Saint-Vincent, Flemish Jesuit, first noticed thatthe area under the curve from 1 to e is 1, but does not define or recognize the importance of e.
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Definition of Natural Log Function
0 ,1ln1
xdtt
xx
x
y
x
y
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Definition of e
67182818284.2
11ln1
e
dtt
ee
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Review: Properties of Logs(i.e. Making life easier!)
ana
baba
baab
n lnln
lnlnln
lnln)ln(0)1ln(
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Practice: Expand each expression
1
3ln
56ln
23ln
910ln
23
22
xx
x
x
x
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Derivatives of the Natural Log Function
uu
dxdu
uu
dxd
uuu
dxdu
uu
dxd
xx
xdxd
'1ln
0 ,'1ln
0 ,1ln
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Practice: Find f ’ (Don’t forget your chain rule!!)
)1ln()( )2
)2ln()( )1
2
xxf
xxf
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3ln)( )4
ln)( )3
xxf
xxxf
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12
1ln)( )6
1ln)( )5
3
22
x
xxxf
xxf
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1 ,
231 )8
2 ,1
2 )7
2
2
2
3
xxxxy
xx
xy
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32ln of extrema relative theFind )10
cosln )9
2
xxy
θf(x)
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Log Rule for Integration
Cuduu
Cxdxx
ln1
ln1
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Practice: Don’t forget to use u-substitution when needed.
dx
x
dxx
141 )2
2 )1
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xdxx
x
dxx
x
3
2
2
13 )4
1 )3
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dx
xxx
dxxx
21 )6
tansec )5
2
2
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dxx
xx
dxx
11 )8
231 )7
2
2
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xdxx
dxx
x
ln1 )10
12 )9 2
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xdx
xdx
sec )12
tan )11
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Integrals of the Trig Functions
**cotcsclnsec
tanseclnsec
**sinlncot
coslntan
sincos
cossin
Cuuudu
Cuuudu
Cuudu
Cuudu
Cuudu
Cuudu
** Prove these!
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40,on tan)( of valueaverage theFind )14
tan1 )13 2
xxf
dxx