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Transcendental Function

By Danang Mursita


The Inverse Function

• Let f and g are continuous function

• y = f(x) = f( g(y)) = (fog)(y) for all y in Dg = B• x = g(y) = g( f(x) ) = (gof)(x) for all x in Df = A• If it is obtained the above compositions then g is said as an

inverse of f. Notation g = f-1.• So, (fof-1) = (f-1of) = I ( identity function)

B

A

x y

f

g

f(x) = y and g(y) = x


The Inverse Function

• How to find the inverse of y = f(x) ?1. Substitute x by y, so that x = f(y)2. Arrange this form, x = f(y) into y = g(x)3. So, g(x) = f-1(x)

• Example : find the inverse of 1. f(x) = x – 5 2. f(x) = (x – 2 )/(x + 3) 3. f(x) = x2 + 4


The existence of Inverse

• If the function f(x) is increasing neither decreasing on interval I then f(x) has inverse on interval I.

• The function f(x) has an inverse if and only if f(x) is one – to – one function

• Definition : f(x) is one – to – one function– if f(x1) f(x2) then x1 x2 or – This function f(x) has only one point of intercept with

the any line y = b


The Properties of the function and its Inverse

• The domain of f(x) equal with the range of f-1(x) and the range of f(x) equal with the domain of f-1(x)

• The graph of f(x) and f -1(x) are symmetry by the line y = x.

f

1f

y = x


Problems

• Find the inverse of these functionsf x x

xx( ) ,

10

f x x( ) 2 13

f x x( ) 4 25

f xx

x( ) ,

5

102

f xx

x( )

1

1

2

32)(

x

xxf

1.

2.

3.

4.

5.

6.


The logarithm and the Exponential Function

• The Logarithm function is inverse of the exponential function or in the contrary.

• Notation :– y = blog x x = by with x and b are real positives


The Properties of logarithm

• blog b =• blog 1 =• blog (ac) = • blog (a/c) =• blog (ac) =

• blog

alogalog c

cb


The Natural Logarithm Function

• Notation and Definition :– elog x = ln x with natural number e = 2,718…–

• Differentiation and integration:– f ‘(x) = 1/x

– if f(x) = ln u then f ‘(u) = u’/u

x

dtt

xln)x(f1

1

Cxlndxx

1

Culndxu'u


Problems

Find dy/dx 1. y = ln(2x)2. y = ln (sin x)3. y = ln ( ln x )4. y = x2 ln x5.

6. Evaluate

2

1

xx

lny

xsinxx

lny3 31

0

12

3

2

510

19

8

47

dxx

x.

xxdx

.

xlnxdx

.

dxx

x.


The Natural Exponential Function

• If y = ex = exp(x) then x = ln y• The properties of exponential function

– ln ex = x for all x– elny = y for all y > 0

• Differentiation and integration– f(x) = eu f ‘(u) = eu u’

– Cedx'ue uu


The General Exponential Function

• Notation– f(x) = ax = elnax = exlna – f(x) = au = elnau = eulna

• Differentiation–

• Integration – Ca

alndx'ua uu 1

'ua)a(ln)x('f u


The General Logarithm Function

• Notation – f(x) = blog x = elog x/ elog b = ln x / ln b– f(x) = blog u = elog u / elog b = ln u / ln b

• Differentiation–

– blnu

'u)x('f

blnx)x('f

1


Problems

Find dy/dx Evaluate

x

ex

xx

xx

x

x

xelny.

y.

ee

eey.

ey.

xlne

y.

x

15

54

3

12

1

3

2

3

3

2

1

2

410

49

48

17

6

2

3

ln

lnx

x

x

x

xx

x

dxe

e.

dxx.

x

dx.

dxesine.

dxex.


The Inverse Trigonometric Function

• Trigonometric function, f(x) = sin x is not one-to-one function, so it has not an inverse function

• If it has an inverse function, then its domain will be bounded on some interval

1

-1

X

Y


The arcs Sinus

• Domain of Sinus Function, f(x) = sin x

• Notation

• Relation between the function and its inverse

1)x(f1;

2

πx

2

π

2

πy

2

π;1u1usiny 1

uysinusinarcusiny 1


The Arc Sinus

• Differentiation, y = sin-1u

• Integration

ycosdudy

dyycosduysinuusiny11

y

u1

21 u

2u1

'u'y

2u1ycos

Cusinu

dx'u 121


The others of Inverse Trigonometric

y;uucosy).( 0111 1

Cucos

u1

du

u1

'u'y 1

22

21

1222

u

'u'yy;uutany)(

21

1200

203

u

'u'yyy;uucoty).(

Cucot

Cutan

u1

du1

1

2

122014

21

uu

'u'yyy;|u|usecy).(

1200

215

21

uu

'u'yyy;|u|ucscy).(

Cucsc

Cusec

1uu

du1

1

2


Problems

Find dy/dx Evaluate

)x(csc)x(secy).(

xcoty).(

xtany).(

xcoslny).(

xsiny).(

225

4

13

2

11

11

1

1

1

1

2

22

1

02

3

1

02

19

18

17

16

21

xx

dx).(

dxe

e).(

xxdx

).(

x

dx).(

x

x


The Hyperbolic Function

• Sinus Hyperbolic and Cosinus hyperbolic

22

xxxx eexcoshand

eexsinh

xx

xx

ee

eexcoshxsinh

xtanh).(

1

xx

xx

ee

eexsinhxcosh

xcoth).(

2

xx eexcoshxhsec).(

21

3

xx eexsinhxhcsc).(

21

4


Differentiation and Integration of Hyperbolic Function

Cucoshduusinh'uusinh'yucoshy).(2

Cutanhduuhsec'uuhsec'yutanhy).( 223

Cucothduuhcsc'uuhcsc'yucothy).( 224

Cuhsecduutanhuhsec'uutanhuhsec'yuhsecy).(5

Cuhcscduucothuhcsc'uucothuhcsc'yuhcscy).(6

'uucosh'ueeee

D'yxsinhy).(uuuu

x

221


Problems

Find dy/dx Evaluate

2

2

5

4

23

2

1

xcoshtany).(

xcoshxsin

y).(

xtanhlny).(xcoshey).(

esinhy).(x

x

dxexsinh).(

dxehsece).(

dxxtanh).(

dxxcosh).(

xcosh

xx

9

8

7

26

2


The Power of Function

• Let y = f(x)g(x). Then ln y = g(x) ln f(x)• The derivative, dy/dx :

• Example : find dy/dx

)x('f)x(f)x(g

)x(fln)x('gdxdy

y

1

)x(g)x(f)x('f)x(f)x(g

)x(fln)x('gdxdy

xcosh

xlnx

xsinxy).(

ey).(22

11


Problems : Find dy/dx

xsecx eesecy 22

xlnexy 35 xetany

)yxln(ey 33

1322 xyey x

652 xxlny

xcoslny 3

2x

xlny

xsinlny

))xsin(ln(y 12

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


The Indeterminate Form of Limit

• Let y = f(x)g(x). Then lim y = lim f(x)g(x) has indeterminate form : 00, 0 and 1.

• How to solve this limit ?– ln lim y = ln lim f(x)g(x) lim ln y = lim g(x) ln f(x)– Let lim ln y = A. Then lim y = eA.

• Examples :

xxlim).(

xtanlim).(

xlim).(

x

xcos

x

x

x

12

13

2

1

0

0


Problems

xx

xlim

11

1

xxsinlimx

121

0

2

2 x

x xcoslim

xlnx

xelim

12

01

xlnx

xlim1

21

xx

xlnlim1

xxxxlim

1

53

x

x xx

lim

2

1

1.

2.

3.

4.

5.

6.

7.

8.


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Http://danangmursita.staff.telkomuniversity.ac.id/ Transcendental Function By Danang Mursita