Bab 5. APLIKASI TURUNAN

11 IPA/IPS

Materi:

- Garis singgung & normal- Fungsi naik-turun & nilainya- Menggambar kurva- Optimasi (maks/min)

A. PERS. GARIS SINGGUNG & NORMAL

!, #) = ('()*), +,-*.'/

01 !

2 = 01 '()*)

# − #4 = 2 ! − !4

)/56): ✓

✓

✓

1) #$%&'()% *$+, -)+., ,.%--'%- *)/) ('+0)

1(3) = 35 − 3 + 1 /. &.&.( (−8, 8)

:$(: 3 = −1 → = = −1 + 1 + 1 = 1 → &.&.( *)/) ('+0)

1> 3 = 33@ − 1 → A = 3 . (−1)@ − 1 = 2

= − 8 = 2 3 + 8= − =D = A 3 − 3D →

= = 23 + 3

2) #$%& '(%)& *+%,(- ?

,/ = 2 →

2 − 4 = −12 6 + 42 − 2/ = , 6 − 6/ →

22 − 2 = −6 − 1

,/.,9 = −1 → ,9 = −12:):); (−4, 4)

22 + 6 = 1

!"#$%&'#(")

!"#$%%$&!!*&!

3) #$%&'()% *+, -).) / = 312 + 81 + 2 .6 )7868 − :

1 = −: → / = 3 − 8 + 2 = −3 → &6&6(%/) (−:,−>)

/? = 61 + 8 → A = −6 + 8 = 2

/ + > = 2 1 + :/ − /B = A 1 − 1B →

/ = 21 − 1

4) #$% &'(' ) = −,- + 2, + 5 (1 23(14'5 6

) = 6 →

−8, 6 & ;, 6

<= , = −2, + 2 → >? =

2 = −,- + 2, + 5,- − 2, − 3 = 0 → (, + 1)(, − 3) = 0

, = −8) = 6

, = 3) = 6

−2 −8 + 2 = 4>- = −2 . ; + 2 = −4

−8, 6 → ;, 6 →) = 4, + 6 ) = 14 − 4,

5) #$% & '()*+, -+.+ / = −23 + 52 + 7 .6 (8, 9)

/; = −32= + 5 → *? =

/ − 9 = −7 2 − 8 →@AB → / = −72 + 23

−3 . 8= + 5 = −7

→ E6E6F -+.+ FG)H+

*?.*= = −1 → *= =17

JKLMNO →

/ − 9 =172 − 8 → 7/ − 63 = 2 − 2 → 7/ = 2 + 61

6) #$%

→ ' (, * +, +-.-/ 0-1-23.-

4(() =

4 7 = 78 − 27 − 3 +, ' (, *

4 − 4 − 3 = −3 =>?-@,4

→ @-A -+- ?1B B,=??C=?

B. FUNGSI NAIK - TURUN

!"#$" %"&'!( #)#& &#() *()# +, > 0

& $"0"& *()# +, < 0

*()# +, = 0 → !$#!(4&50/5)!$0(7

1) #$%& ' = )* − 3) + 4 /01/2%&1:

&) /$/$% 4/&4$51067) $1/068&9 1&$%:) $1/068&9 /2621;) 1$9&$ <&%4 &<$1 >&;& − 1 ≤ ) ≤ 3

!) #$ = 0

# = '( − 3' + 4

2' − 3 = 0 → ' = 32

# = 32

(− 3 . 32 + 4 = 7

4

12123 41!425678: 32 ,74

;2 <!8!=5>!, 12123 41!425678 = 12123 <?6@!3

!) #$%&'( )*)% %)(* +(*) ,- > 0

, = 12 − 31 + 4

21 − 3 > 0 → 1 > 32

9) :$;$% +(*) ,- < 0

21 − 3 < 0 → 1 < 32

':)'(=%>;

%)(*:$;$%

32

!) !# #$%&'()* − 1 ≤ . ≤ 3

0 = .2 − 3. + 4

. = −1 → 0 =

. = 32 → 0 =

. = 3 → 0 =

1 + 3 + 4 = 8744

$#*)# 9):; = 8 & 9#$#9=9 = 7/4

2) # $ =13 $

( + 4$ − 5

-) ./ = 0 → $2 + 4 = 0 → ∅

$ ∈ 56-7

8-9 -:- 8;8;9 <8-<;=>6?

@) ./ > 0 → $2 + 4 > 0 →

8-9 BC>D9;>E) ./ < 0 → $2 + 4 < 0 →

:) G-:- − 1 ≤ $ ≤ 3

# −1 =

# 3 =

−283

16

B;>

B-9<

9C?K- -9-> >-;9 :;B->- GC>

3) # $ = $& − 3$ + 2

*) +, = 0 →

3($ + 1)($ − 1) = 0

3$1 − 3 = 0

$ = −1 $ = 1 →

# −1 = −1 + 3 + 2 = 4

# 1 = 0

34345 63*64789:: −1 , 4 & (1 , 0)

! " = "$ − 3" + 2

)) +, > 0 → 3 " + 1 " − 1 > 0

−1 1− ++

1234 53 " < −1 2728 " > 1

9) 78:81 → 53 − 1 < " < 1

! " = "$ − 3" + 2

)) +,-./012 − 3 ≤ " ≤ 3

! −3 =! −1 =! 1 =! 3 =

−27 + 9 + 2 = −16

4

0

20

,+21+ :1;< = 20 & :+, = −16 naik turun naik

4) #$%&'& ( ) = 3), + 8)/ −18)2 + 6 4&5 678'&ℎ '&$5 ?

4&5 678'&ℎ '&$5 → (<()) ≤ 0

12)/ + 24)2 −36) ≤ 0

12) )2 + 2) − 3 ≤ 0

12)() + 3)() − 1) ≤ 0

(A$B& 4C8C', A$B& 474&6)

−3 0 1++ −−

−3 105C8E& 4$F&5 '&$5 F$:

) ≤ −3 & 0 ≤ ) ≤ 1

turun turun

IJKLIM:

5) #$%&$'($) * =12 .

/ −3. + 4

4567 878792 :85:7;<=6<*5

*> = 0 → . − 3 = 0. = 3

A(BCA$:

3+−

−1/2

4

0

0

4F G0

2H

46

J$%&$':

4=9 9= 8K6K<5<

6) #$%&$'($) * = −13 /

0 + /2 + 1

3456 767682 :74:6;<=5<*4

*> = 0 → −/2 + 2/ = 0

−/(/ − 2) = 0

B(CDB$:

0 2−− +

1

7/3

5/3

1−13/3

7/323/3

I J−2−1K1L34

N$%&$':

7) #$%&$'($) * = 2-. + 3-1 −12- − 4

565672 85986:;<=:

*? = 0 → 6-1 + 6- − 12 = 0

(- + 2)(- − 1) = 0

D(EFD$:

−2 1++ −

9

−11

−4

0

165

I J−3−K−10L2

M$%&$':

-1 + - − 2 = 0

11 IPS

Latihan 5 hal. 163 no. ganjil

Latihan 8 hal. 174 no. ganjil

Latihan 9 hal. 179 no. ganjil

Latihan 10 hal. 182 no. 1a, 2a, 3

C. OPTIMASI FUNGSI

!"#$%&% → %()* (#(& %$+$%&%

+$,($ !"#$%&% - . ()(+ #/01(2$ *((# -3(.) = 0

1) #$%&'()% %*+)* ,)(- . = −223 + 122 + 7

.6 = 0 → −42 + 12 = 0 → 2 = 3

. = −2 . 33 + 12 . 3 + 7 = 25

%*+)* ,)(- = 25

2) #$%&'()% %*+)* ,)(- & ,*%*,', /$+)&*0

12 = 0 → 378 − 67 − 45 = 0

7 = −3 =/ 7 = 5

1 = 7> − 378 − 457 + 1

78 − 27 − 15 = 0 → 7 + 3 7 − 5 = 0

ABCDAE:

−3 5++ −

&),G)(: ,)(- &$/H)I* -))& 7 = −3

%*+)* ,)(- = −27 − 27 + 135 + 1 = 82

%*+)* ,*% = −174

3) #$%&'()% *')+ ,)(+ -$.+$/0-)%1)%/2/ ($*0*0%/%2) 72 5,.

7

22 7 + 2 = 72 → 7 + 2 = 36

2 = 36 − 7

=')+ = 7 2 = 7 36 − 7 = 367 − 7>

?/). ,)(+/,0% → = A = 036 − 27 = 0 → 7 = 18 → 2 = 18

=')+ ,)(+ = 18 . 18 = 324 5,>

4) #$%&'ℎ )$' *+& ,-.+/+0 ')'&'ℎ 36. 456/$7'6 6+&'+ %'7.)'8+ 9'.+& 7'&+ *+& ,58/'%' ):6 7$')8'/ *+& 75)$'.

; + = = 36 → ; = 36 − =

9 = ; . =A = 36 − = . =A =

*+& B = ; & *+& BB = =

−=D + 36=A

E:'8 %'7./%+6 → G H = 0 → −3=A + 72= = 0

−3= = − 24 = 0

L = MN →0 (85P5Q/)9 = 12 . 24A = 6912 ; = 12

5) #$%& '(%)$* 48 × 60 01, $'$3 4&56$) '7)$' )$38$ )6)68

493 1(17)739 '((18$) 6:6393;$. =7> 1$'* '7)$' = ?

A A60 − 2A48−2A

=7> = 60 − 2A 48 − 2A A

= 4AD − 216AF + 2880A

=7> H = 0

12AF − 432A + 2880 = 0

AF − 36A + 240 = 0

! !60 − 2!48

−2!

!( − 36! + 240 = 0

! =−, ± ,( − 4 . /

2 .

! =−(−36) ± (−36)( − 4 . 1 . 240

2 . 1

! =36 ± 336

2 = 18 ± 2 21 = 18 ± 9,2

!6 = 27,2 89 !( = 8,89:;:/<

=.>;.>? = 60 − 2 ∗ 8,8 = 42,4

A:,.9 = 48 − 2 ∗ 8,8 = 30,4

B8A C.DE = 42,4 ∗ 30,4 ∗ 8,8 = FGHI /CJ

6) #$%$&'( )*. 5, ./.&0) /$&'01 15 × 7 56

7 715 − 277−

27

:*; = 15 − 27 7 − 27 7= 47> − 447? + 1057

:*; B = 0

127? − 887 + 105 = 0

! !15 − 2!7−

2!

12!' − 88! + 105 = 0

6! − 35 2! − 3 = 0

!. =356 /0 !' =

32 ? ?

maks

min

2/3 = 15 − 2 . 32 7 − 2 . 32 . 32

= 12 . 4 . 32 = 72 678

7) #$%&%'(: *+,-.. 01234+ 5-67+2.882 5: + 2= = 18 A32

= = ℎ8638/12+.. D-2.1,82 01234+ 5-2A858.82 786E+28F.

5: + 2= = 18 → = = −5

2: + 9

J : = : . = = : −5

2: + 9 = −

5

2:K+ 9:

JL : = −5: + 9

8) #$%&%'(: *+,-, ./.,0 1 2 3,0,4 5,.67,8 5+9668.6: 4;4<5/36:7+ 76,.6 9,5,8=/960,8 3+8-,.,:,8 3=8

1 2 = 302 + 750 − 0,12H. J;8.6:,8 9+,-, 4,5K+8,0

7,,. <,95+: 4;4<5/36:7+ 50 9,5,8=.

1L 2 = 30 − 0,22

1L 50 = 30 − 0,2 . 50 = 20

*+,-, = 20 . 100.000,− = N< 2 K6.,

9) #$%$&': )*+, -,.,+ /0,10 1*1*+ /,,1 1 231*+ 2*45,1,+,4

/ 1 = 218 − 31; − 201 + 15 13410+,4 1 /,,1

+3@3A,1,445, BC D//

F 1 = /G 1 = 61; − 61 − 20

BC = 61; − 61 − 20 → 61; − 61 − 36 = 0

6 1 + 2 1 − 3 = 0

1J = −2 K. 1; = 3 231*+.3-3@1

10) $%&ℎ: )*+% ,%-*./ 0*12+%,%1 341*12+%& -412%1 0%5.

67 83-4&*+ 3%+% &41&.+%1 0%5. 941*12+%&%1 0.%/ 0*12+

/%%& 0.%/1:% ;<=

>%5. ,%-*./: -,-& = 67 83-4&*+

>%5. 0.%/: ->-& =

>.%/ = @,A →

36@ = @,A → , = <

->-, .

-,-& =

->-, = 2@,

>.%/ = ;<= →

2@G . 10 =120@ 83A/-4&*+

11) #$%ℎ: ()*$ *+ ,$-$. /0-$ *$1+% ,).$23**$4 3,$1$

2+/$45$* 678.9

,+%)*.$*$ %+4%3*$4 -$:3 ;+4)4<*$%$4

-3$2 /0-$ 2$$% 1$,)3245$ = 8.

,>,%

= 678.9

,+%)*> =

43B19 →

,>,1

= 4B . =E = 144B

,>,%

=,>,1

.,1,%

→

4B1E =

67 = 144B .,1,%

→,1,%

=29B

H3$2 = 4B1E → ,H,1

= 8B1,H,%

=,H,1

.,1,%

= 8B . = .29B

=323

12) $%&'( ) 15 + & -%&'( . 12 + -/01%2&ℎ 2/4&5ℎ 16 +.8/95& 545'( -%&'( 9%%:&- :/ -&'&ℎ 9% -%-%: ;.$/'-5:&' 4&0&: ); &(&0 1&'4&'( :&<&- +%'%+5+.

1512

16

= 16 − =

+'

+ = =@ + 15@

= =@ + 225B@

' = (16 − =)@ + 12@

= =@ − 32= + 400B@

;) .

! = #$ + 225($ ) = #$ − 32# + 400

($

./012 = ! + ) .3 = 0

12 #$ + 225 5 ($ . 2# + 1

2 #$ − 32# + 400 5 ($ . 2# − 32 = 0

##$ + 225

+ # − 16#$ − 32# + 400

= 0

##$ + 225

= 16 − ##$ − 32# + 400

89

! !" − 32! + 400 = (16 − !) !" + 225

!" !" − 32! + 400 = (16 − !)" !" + 225

!/ − 32!0 + 400!" = !" − 32! + 256 !" + 225

!/ − 32!0 + 400!" = !/ − 32!0 + 481!" − 7200! + 57600

0 = 81!" − 7200! + 57600

9!" − 800! + 6400 = 0 → ! = 80 56 ! = 80/96898:;

13) $%&'()*& +,- .*)/ )%0(1(' 2*-*. 3,-* 0*24(/ 4 1.

30

*

3

* = 9 − 09 3 = 3

:; =13<0

9' = 13<=

> 9 − 09 + 3

= 13< =@ 9 − 09 + 9=>

= ABC AD=@ − =D

A> + @=>

' = * + 3

!" =13& 16() − (+

,- + 4(-

!"0 = 0 →13&

1216() − (+ 4 ,- . 64(6 − 6(7 + 8( = 0

32(6 − 3(7

16() − (++ 8( = 0 → 8( =

3(7 − 32(6

16() − (+

6 36(- − () = (- − 189:;< 2(, >:?< @<?:A;

36 36(- − () = () − 36(- + 324

BC B

D<@:? (- = : → 1296: − 36:- = :- − 36: + 324

>F:G(:H>:A →

1296% − 36%( = %( − 36% + 324

37%( − 1332% + 324 = 0

% = 35,8 12 0,24