Differentiating exponentials and logarithms

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

A geometric approach to f(x)=ex

.1

lim)(Let 0 h

bbL

h

h

?1

1lim)( doesWhen

0

h

bbL

h

h

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

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

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

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

Integrating ex

. have we Since cedxeeedx

d xxxx

.1

have ly weConsequent cea

dxe baxbax

Do Q5-Q11, p.54

The natural logarithm

log ln , , 0.exp for xey e x x y y x y y

01ln

1ln e

nnen ln

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

The reciprocal integral

.ln1

,0For cxdxx

x

This plugs a gap!!!

Do Exercise 4C, pp.58-59

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