differentiating exponentials and logarithms
DESCRIPTION
Differentiating exponentials and logarithms. A geometric approach to f(x)=e x. A geometric approach to f(x)=e x. A geometric approach to f(x)=e x. Do Q1, Q2, Q3, Q4, p.54. An algebraic approach to f(x)=e x. A definition for f(x)=e x. Calculating e. Integrating e x. Do Q5-Q11, p.54. - PowerPoint PPT PresentationTRANSCRIPT
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Differentiating exponentials and logarithms
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A geometric approach to f(x)=ex
h
xfhxfxf
h
)()(lim)(
0
.12
lim2)( then ,2)( If0 h
xfxfh
h
xx
.2 )0()( then ,2)( If xx fxfxf .3 )0()( then ,3)( If xx fxfxf
. )0()( then ,)( If xx bfxfbxf
)( )0()( then ,)( If xffxfbxf x
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A geometric approach to f(x)=ex
.1
lim)(Let 0 h
bbL
h
h
?1
1lim)( doesWhen
0
h
bbL
h
h
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A geometric approach to f(x)=ex
?11
lim)( doesWhen 0
h
bbL
h
h
.84590452352.71828182When eb
)()( then ,)( If xfxfexf x
.)( then )( If xx exfexf
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An algebraic approach to f(x)=ex
n
n
x
n
xe )1(lim
xen x 1 :1
41 :2
2xxen x
2731 :3
32 xxxen x
!!3!2!1
1 :32
n
xxxxen
nx
Do Q1, Q2, Q3, Q4, p.54
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A definition for f(x)=ex
n
n
x
n
xe )1(lim
xx
en x 1)1
1( :1 1
41)
21( :2
22 x
xx
en x
2731)
31( :3
323 xx
xx
en x
256168
31)
41( :4
4324 xxx
xx
en x
312512525
2
5
21)
51( :5
54325 xxxx
xx
en x
0
32
!!!3!2!11)(1 lim :
n
nnn
n
x
n
x
n
xxxx
n
xen
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Calculating e
!!3!2!1
1 If32
n
xxxxe
nx
!
1
!3
1
!2
1
!1
11 then 1
nee
720
1
120
1
24
1
6
1
2
111
n
n
n
n
x
ne
n
xe )
11(lim then ,)1(lim since Also,
.100
101Try
100
.10000
10001Try
10000
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Integrating ex
. have we Since cedxeeedx
d xxxx
.1
have ly weConsequent cea
dxe baxbax
Do Q5-Q11, p.54
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The natural logarithm
log ln , , 0.exp for xey e x x y y x y y
01ln
1ln e
nnen ln
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Derivative of the natural logarithm
.1
ln ,0For x
xdx
dx
The proof is a consequence of the ‘mini-theorem’ outlined on p.55.
Do Exercise 4B, p.57
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The reciprocal integral
.ln1
,0For cxdxx
x
This plugs a gap!!!
Do Exercise 4C, pp.58-59
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Extending the reciprocal integral
?0 when 1
isWhat xdxx
0 when defined is )ln( that Note xx
.11
)ln( that andxx
xdx
d
.ln1
,0For cxdxx
x
Do Q1, p.62Do Misc. Exercise 4, Q1-Q18, pp.62-64