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1

lim

...

3

1

x

x

x

x

=

+ − −

-1

lim

(

)(

)

...

x

x

+

a

x b

+

− =

x

0 4

3

lim

4

4

x

ax b

x

x

+ −

=

&

a

+ =

b

...

0

2

1 2 ...

lim

...

1

1

2

x

x

x

+ +

+

=

+

0 3 8

8

lim

...

2

x

x

x

=

0$

(

)

(

)

3 3

3

2

lim

...

4

3

x

x

x

=

+

0( 3 2 2 4

3

3

4

lim

...

4

x

x

x

x

x

x

=

0*

lim1

1

1

...

1

...

2

4

2

n

x

+

+

+

+

=

(22)

0- 2 0

tan

lim

...

2

x

x

x

x

+

=

00

(

)

(

)

(

)

2

2

2 2

5

6 sin

2

lim

...

2

x

x

x

x

x

x

+

=

− −

01 2 0

cos 2

1

lim

...

x

x

x

=

1 % 0

sin

lim

1

x

x

x

=

&

(

)

1

sin

lim

...

1

x

x

x

π

π

=

1 0

sin 4

sin 2

lim

...

3 cos

x

x

x

x

x

+

=

1 0

tan

lim

...

1 cos 2

x

x

x

x

=

1$ 0

cos

cos 3

lim

...

1 cos 2

x

x

x

x

=

1( 2 0

1 cos

lim

...

tan 2

x

x

x

=

1* 0

cos 4

1

lim

...

tan 2

x

x

x

x

=

1+ 0

sin

lim

...

tan

x

a

x

b

cx

=

1- % 0

sin

lim

1

x

x

x

=

&

2

lim

tan

...

2

x

x

x

π

π

=

10 % 0

sin

lim

1

x

x

x

=

&

(

)

2 1

1

lim

...

sin

x

x

x

π π

=

11

(

)

(

)(

)

0

2 tan

lim

...

1 1 cos 2

x

x x

x

x

x

+

=

+

$ % 0

tan

lim

1

x

x

x

=

&

lim

x a

3

3

tan

(

)

...

x

a

x

a

x

a

=

(23)

$

2 2

2

2

8

2

lim

...

2

2

4

x

x

x

x

x

x

+

=

$ 2 2

2

2

3

2

2

3

lim

...

2

x

x

x

x

x

+

− −

=

$ $ 2

3

2

3

lim

...

9

x

x

x

x

+

=

$ (

(

)(

)

(

)(

)

4 5

2

lim

...

2 1

x

x

x

x

x

+

=

+

$ *

(

)

2 3 2 0

1 sin 6

lim

...

3

2

x

x

x

x

x

x

=

+

+

$ +

(

)

lim

...

sin

2

2

x k

x

k

x

k

k

x

=

+

$ - 2

0

tan 2 .tan 3

lim

...

5

x

x

x

x

=

$ 0 2 0

2

lim

...

sin

x

x

x

x

+

=

$ 1

(

)

(

)

2 2 1

1 cos

1

lim

...

4

2

1

x

x

x

x

=

+

$ 2 0

cos

cos 2

lim

...

x

x

x

x

=

$ 2 3 2

6

lim

...

8

x

x

x

x

+ −

=

$ 3 1

1

3

lim

...

1

1

x

x

x

=

$ $ 2 3

9

lim

...

3

1

x

x

x

=

$ (

(

)

(

)

2 2 2

2

6

lim

...

4

4

x

x

x

x

x

x

→−

+

− −

(24)

$ *

2 3

2 0

16

lim

...

4

x

x

x

x

x

=

+

$ + 0

2

2

lim

...

2

2

x x

x x x

− − →

=

+

$ -0

1

1

lim

1

...

1

t

t

t

+

=

$ 0 0

10

lim

...

9

5

81

x

x

x

+

=

$ 1 0

6

sin 2

lim

...

2

3 tan 4

x

x

x

x

x

=

+

$ 2

0

2

1 cos 2

lim

...

sin

x

x

x

+

=

$

0

sin 2

lim

...

3

2

9

x

x

x

+

=

$

4

sin

cos

lim

...

cos 2

x

x

x

x

π →

=

$ $

2 0

5

lim

...

sin 2

x

x

x

x

=

$ ( 0

sin 6

lim

...

tan 2

x

x

x

=

$ * 0

sin 2

lim

...

1 cos

x

x

x

x

=

$ + 0

1 cos

lim

...

1 cos 2

x

x

x

(25)

PELUANG

$ - ; " " & $& *& +& -& 1 ; '

! ' ' 3 (

$ 0 2 . . , " - = " & " = "

' 3

$ 1 ; " " ' " - " " " " " " " "

$$ . " , 1 . . . . !* = , ' ,

; ' '

$$ . " , * + . . . . = , ' , ;

' '

$$ 2 $& *& +& - 1 ' " ' " " " ' '

; ' !' " ' " (

$$$ 2 . " 0 ( = " ' " . . ,

' ." ( . % . . , " " " &

' 3 ' " . .

$$( " = " , 3 . = " +

. ; 3 ' ! " $

$$* ' " ' ." ( . ( - = " ; !

= " ; ' 3 " ' " " " ' ">

Program Linier

$$+ "

(

x

,

y

)

4

x

+

y

4

,

2

x

+

3

y

6

4

x

+

3

y

12

& "

y

x

y

x

F

(

,

)

=

+

$$- "

(

x

,

y

)

x

0

,

y

0

&

2

x

+

5

y

10

4

x

+

3

y

12

&

2

2

+

=

y

x

(26)

$$0 %

x

0

,

y

0

&

2

x

+

y

6

x

+

2

y

6

F

(

x

,

y

)

=

x

+

y

&

$$1

4

x

+

5

y

x

0

,

y

0

&

x

+

2

y

10

x

+

y

7

$(

2

x

+

5

y

x

0

,

y

0

&

x

+

y

12

x

+

2

y

16

$(

8

x

+

6

y

x

0

,

y

0

&

4

x

+

2

y

60

2

x

+

4

y

48

$(

4

y

x

y

2

x

&

3

y

2

x

,

2

y

+

x

20

x

+

y

3

$($

3

x

+

6

y

4

x

+

y

20

,

x

+

y

20

&

x

+

y

10

&

x

0

,

y

0

$((

2

x

+

3

y

3

x

+

2

y

28

&

x

+

2

y

8

&

x

0

,

y

0

$(*

3

x

+

y

x

0

,

y

0

&

2

x

+

y

4

,

x

+

y

3

$(+

4

x

+

3

y

x

0

,

y

0

&

x

+

y

4

&

2

x

+

3

y

9

$(-

6

x

10

y

x

+

y

10

,

x

+

2

y

10

&

x

2

&

y

0

$(0

4

x

+

5

y

x

+

y

8

&

3

x

6

,

x

+

y

5

y

0

STATISTIKA

$(1 2 . . ! " ( & " ! " ' " ' $

$$ . ! . . " & " ! "

' " ' , ' " ' "

$* ' " ! " * ? $* &! % ' ' .

= ' ; ' . . = " ' "& ' " !

" , ? $+ &! ' " ' = ' ; '

$* " ! " ' (& % " ! " ' " &+ " ! " + ' ' " 0& & " ! " ' "

$* ) "& " ! " *0 % " ! " " " = +*

(27)

$*$ " " ' " = ' , . " ! " " & " -& 0& -&* % ' = " * .

" * . ' ' " ! " =

$*( % $ = " ! " +&* * = ; " ! "

- " = 8 " ! " 0& " ! " =

$** " . . = ! " *& 0& - . ' . '

' 3 ? " ! " ' ! . . ? ( &! & ? * &!& ? &! ? &! " ! " ' ( =

$*+ . = & ;& 8& 2 " " ' " :

• ) " '

1

2

"

• ) " ; ' ? &!

• ) " 8 ' ? * &!

• ) " 2 ? 0 &!

; " ! " " = ? * * &! " = 2

$*- , * . = ' ' ' " : (0 & $+ & +* &

- & + , * @ ' ' ' , ? * &!

@ ' ' ' , ' ? * &! ? " ! " ' ,

'

$*0 ) " " ! " ? $ &! % " " ! " = ?

$ * &! = = " ? 0* &! ' , =

= "

$*1 ) " , " " " ( = & $ = ; $ = 8 " ! " = - " ! " = ; 8 - " ! "

$+ " ! " , " " $1 . (* % 4 ' & =

(28)

$+ . ' ' % " , 1 & ' 6 ' & "&

" " " ' " ' ' ' % "

? $ &! " " ! " " '

$+ ? " ! " " ' $ = " *+ 3 " ! " " +0 3 ? " ! " " * . " ' "

$+$

& ;& 8 ' "

"!"

" "

&

*

=

? " !

"

'

"

** %

" !

"

8 '

"

"!"

" *+

+*&

" !

"

;

DIFFERENSIAL

$+( % ' 0 ) " ' " ' " 3 5

' " ' " 3

$+* % "

2

1

;

1

2

1

3

cos

)

(

+

=

x

x

x

x

f

&

f

'

( )

x

=

...

$++

y

=

(

1

x

) (

2

2

x

+

3

)

$+- /

y

=

4

(

2

x

2

3

)

3

$+0 " . ' "

x

' .

+

x

x

900

120

3

" ' ' . . " ' "

= "

$+1 %

f

(

x

)

=

x

2

.

4

6

x

&

f

'

( )

2

=

...

$- ) " "

x

=

2

#

1

5

27

=

x

y

$- %

f

(

x

)

=

6

x

+

7

f

'

( )

3

=

...

(29)

$-$ 2 " #

y

=

x

3

6

x

2

+

9

x

+

1

# " " # $-( "

y

=

cos

4

x

$-*

;

sin

0

sin

cos

sin

)

(

=

+

x

x

x

x

x

f

&

...

2

'

π

=

f

$-+ 2 " " #

y

=

4

+

3

x

x

3 "

x

0

5 # " ' "

$-- 4 /

y

=

5

x

3

3

x

2 .". '

x

" "

P

% " "

P

m

&

2

m

+

1

=

...

$-0 % " .

f

(

x

)

=

x

3

px

2

px

1

x

=

p

&

p

=

...

$-1

f

( )

x

=

px

2

(

p

+

1

)

x

6

3 " " "

x

=

1

&

p

=

...

$0 ) #

y

=

2

x

3

4

x

+

3

" " '

1

$0

f

( )

x

=

a

tan

x

+

bx

3

4

'

π

=

f

%

9

3

'

π

=

f

&

a

+

b

=

...

$0 ) /

f

( )

x

=

x

3

+

x

2

+

x

3

2

" "

(

1

,

0

)

$0$ 4 /

y

=

x

4

8

x

2

9

" " #

$0( " ' . /

y

=

x

3

+

6

x

2

+

9

x

+

7

$0*

f

(

x

)

=

3

x

2

5

x

+

2

&

g

(

x

)

=

x

2

+

3

x

3

%

h

(

x

)

=

f

(

x

)

2

.

g

(

x

)

&

h

'

(

x

)

=

...

$0+ ' " ' ' " " ' "

P

" 5&

x

=

...

$0- %

2 3

3

2

)

(

=

+

x

x

x

f

&

f

'

( )

x

=

...

$00 4 /

y

x

x

2

x

2

3

3

1

3 2

+

=

" " "
(30)

$1 #

f

(

x

)

=

x

3

+

3

x

2

9

x

+

7

" #

$1 ) #

y

=

x

3

+

2

x

2

5

x

" "

(

1

,

2

)

$1 ) " #

y

=

tan

x

" "

,

4

(

π

1

)

$1$ 2 "

f

(

x

)

=

2

x

3

+

9

x

2

24

x

+

5

%

f

'

(

x

)

<

0

x

' " #

$1( ) " "

(

1

,

1

)

#

x

x

y

=

2

2

$1* 6

y

=

4

x

3

18

x

2

+

15

x

20

3 5 "

x

=

...

$1+ " /

f

(

x

)

=

(

x

2

)(

x

1

)

2 3

x

=

...

x

=

...

$1- %

f

(

x

)

=

(cos

2

x

sin

2

x

)

&

f

'

( )

x

=

...

$10 %

y

=

3

x

4

+

sin

2

x

+

cos

3

x

&

=

...

dx

dy

$11 4 /

f

( )

x

=

5

+

15

x

+

9

x

2

+

x

3 " #

(

( )

6

5

3

2

+

=

x

x

x

f

&

f

(

0

)

+

6

.

f

'

( )

0

=

...

( 5 /

( )

2

5

2

1

3

1

3 2

+

+

=

x

x

x

x

f

( %

( )

x

x

x

x

f

=

1

+

3

2

&

(

)

=

...

dx

x

df

( $ 6

( )

x

x

x

f

sin

cos

2

+

=

4 /

2

π

=

x

.". '

y

(

0

,

b

)

...

=

b

( ( " #

y

=

2

x

3

4

x

2

5

x

+

8

) #

x

=

2

(31)

( + % /

y

=

x

3

+

6

x

2

+

15

x

2

2

<

x

<

6

a

b

&

a

b

=

...

( - 4 " "

(

2

,

8

)

#

y

=

2

x

x

+

2

.". '

x

'

y

" "

(

a

,

0

)

(

0

,

b

)

&

a

+

b

=

...

( 0 %

x

1

x

2 !

x

2

+

kx

+

k

=

0

&

x

12

+

x

22 3 "

k

=

...

( 1 /

y

=

2

x

3

6

x

2

48

x

+

5

" #

3

x

4

(

( )

x

x

x

f

2

1

2

=

f

'

(

x

)

( " /

2 3

)

3

1

(

)

2

(

x

x

y

+

=

"

x

=

3

( 4 /

y

=

x

x

2

" #

( $ " /

y

=

e

3x+5

+

ln(

2

x

+

7

)

TRIGONOMETRI

( ( )

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