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VI .Calculus
: I T R A
P Calculus
s d r o
W Pronunciation Indonesian
n o i t c n u
f /ˈfʌŋ(k)ʃ(ə) n/ fungs i n
i a m o
d /də(ʊ)ˈmeɪn / daerahasa l e
g n a
r /reɪn(d)ʒ/ daerahhasi l n
o i t c n u f e s r e v n
i /ɪnˈvəːs ˈfʌŋ(k)ʃ(ə) n/ fungsii nvers n
o i t c n u f e t i s o p m o
c /ɪnˈvəːs ˈfʌŋ(k)ʃ(ə) n/ fungs ikomposis i g
n i p p a
m /ˈmapɪŋ / pemetaan n
o i t c n u f s u o u n i t n o
c /kənˈtɪnjʊəs ˈfʌŋ(k)ʃ(ə) n/ fungs ikontinu f
o n i a m o
d deifnition /də(ʊ)ˈmeɪn ɒv dɛfˈɪnʃɪ(ə) n/ daerahdeifnis i /
n o i t c n u f e v i t c e j n i
n o i t c n u f o t n
i //ɪnɪˈnˈdtʊʒɛˈkfʌtŋɪv (kˈ)fʃʌ(ŋə) (nk/)ʃ(ə) n/ fungs isatu-satu
/ n o i t c n u f e v i t c e j r u s
n o i t c n u f o t n
o //sˈɒənˈːtduʒːˈɛkfʌtŋɪv (kˈf)ʌʃ(ŋə) (nk)/ʃ(ə) n/ fungs ipada
y t i n if n
i /ɪnf'ɪnətɪ/ takhingga l
a v r e t n i n e p
O /'ɪntəv / l Interva lbuka l
a v r e t n i d e s o l
C /'ɪntəv / l Interva ltutup n
o i t c n u f a f o t i m
il /l'ɪmɪt ɒv ˈfʌŋ(k)ʃ(ə) n/ ilmit f ungsi e
v i t a v i r e
d /dɪ'rɪvətɪv / turunan s
e u l a v e m e r t x
e /ɪk'str:imˈvajluːz/ nliai-nlia iekstrim m
u m i x a
m value /'mæksɪməm ˈvajluː/ nlia imaksimum e
u l a v m u m i n i
m /ˈmɪnɪməm ˈvajluː/ minimum t
n i o p m u m i x a
m /'mæksɪməm pɔɪn / t titik maksimum t
n i o p m u m i n i
m /ˈmɪnɪməm pɔɪn / t titik minimum
n o i t c e lf n i f o t n i o
p / pɔɪn t ɒv ɪn'lfekʃn / titik belok e
l u r n i a h
c /tʃeɪn ur ːl / aturanranta i m
r o f e t a n i m r e t e d n
i /ɪˌndˈɪtəːmɪnət fɔːm / bentuktaktentu l
a r g e t n i e t i n if e
d /'defɪnət ' ɪntɪgrəl / integra ltentu l
a r g e t n i e t i n if e d n
i /ɪnˈdɛfɪnɪt 'ɪntɪgrəl / integra ltaktentu d
n a r g e t n
i /ɪˈntɪgrand/ fungs iyangd iintegralkan n
o i t a r g e t n i f o e l b a i r a
v /ˈvɛːrɪəb(əl) ɒv ɪntˈɪgreʃɪ(ə) n/ variabel i ntegrasi /
t i m il r e w o l
r e w o
l bound //ˈˈlləə ʊʊəəbl'ɪamʊn ɪt/ d/ batasbawah
r e p p
u ilmit/ r
e p p
u bound //ˈʌˈʌppə ə bl'ɪamʊn ɪt/ d/ batasatas o
w t n e e w t e b a e r a
s e v r u
c /ˈɛːrɪə bˈɪtwiːn ut ː kəːv /z lkuuarsvadaerahdiantara2 n
o i t a r g e t n i l a i t r a
p /ˈpɑ ʃː(əl) ɪntˈɪgreʃɪ(ə) n/ integra lparsial y
b n o i t a r g e t n i
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l a r g e t n i e l p i t l u
m /ˈmʌltɪp(əl) 'ɪntɪgrəl / integra lilpat l
a r g e t n i e l b u o
d /ˈdʌb(əl) 'ɪntɪgrəl / integra lganda f
o d il o s a f o e m u l o v
n o i t u l o v e
r r/ɛˈvvəɒˈljluuːʃːm (ə) nɒv /ˈsɒlɪd Volumbendaputar
Example:
) x
(f “ fx” or fo fx x f o f n o i t c n u f e h t
g d n a f f o n o i t c n u f e t i s o p m o c / f e l c r i c g )
b , a
( Openi nterva lo facommab ]
b , a
[ Closedi nterva lo facommab ]
b , a
( H -afl openi nterva lo facommab ,openonthel eft and closed t
h g i r e h t n o
s t i m i L y a s o t w o H
∞ →
xilm (fx) TThhee ililmmiittoas f fxxgaosexs/atpepnrdosatcohei nsifinnitifynioty f fx //əl'ɪ'pmrəɪt//ʊtʃ/ tend/
+ →a
xilm (fx) TThheerilimghitt-oha fn fxdasilmxitapop fr foxachesaf romabove/right
− →a
xilm (fx) TThheel eilmft-ithoan fd fxilamsitxoap fp fxroachesaf rombelow
s e l p m a x
E :
e n i m r e t e
D 2
0 1 x
0 0 1 x m il
0 1 x
→
− − r
e w s n A
n o i s s e r p x e e h t y fi l p m i S : 1 p e t
S .
. d e s i r o t c a f e b n a c r o t a r e m u n e h T
2 100 (x 10 ()x 10)
x
0 1 x 0
1
x−− = − − +
2 p e t
S :Cance lal lcommonterms
x −10 canbecancelledf romthenumeratoranddenominator.
)
0
1
x
()
0
1
x
(
x
1
0
0
1
x
+
−
=
+
−
t e L : 3 p e t
S x→ 10 and write ifna lanswer f
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20 1 x 0
1 x
0 0 1
x ilm x 10 20
m il
0 1
x →
→
− = + =
− e
s i c r e x E
g n i w o ll o f e h t f o t i m il e h t d n i F
s e v i t a v i r e D
f o e v i t a v i r e d e h
T fat point awith respectto xisthe ilmit o fchanging rate f
o fnear pointa.
e h
T derivativeo faf unction f a at ,denoted by `f a( ,) i s
'
0 h
) a ( f ) h a ( f m il ) a ( f
h →
− + =
. s t s i x e t i m il s i h t f i
t n i o p a t a e v i t a v i r e d a s a h n o i t c n u f
A i fand onlyi fthef unction’sright -t
f e l d n a d n a
h -hand derivativesexistandareequa.lInthiscase ,wesaythe s
i n o i t c n u
f differentiable.
t h g i r e h t f
I -hand ilmitdoesnot equa lto thel eft-hand ilmit ,then the ilmit t
a h t d i a s e w d n a t s i x e t o n s e o
d f is notdifferentiable at a.
e v i t a v i r e d t u o b a s e l u r h c u m e r a e r e h
T (rulesof differentiation)
¬ Factorrule
¬ Sumrule
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¬ Quotientrule
¬ Chainrule f(or compositef unction)
¬ Rulef ortrigonometryf unction
e s i c r e x E
.
1 Findthederivativeo fthef unction f(x ) x= 2-8x 9+ a a t using deifnitiono f
e v i t a v i r e
d .
.
2 I f(x f )=3x-2x2 ,if f`(xnd )f romdeifnitionandhenceevaluatef` )( . 4
y a s o t w o
H Derivatives
) x (`
f T fhpreim( ifrestx/ )d fedraivsaht x iveo f fwithrespect xto /praɪm//dæʃ/ r
'
/ ɪspekt/ )
x (` `
f T fhdoeusbelceo-npdrimdeerixv/at fidvoeuob fle f-wdiatshhrexspect tox
) x (` ` `
f T fhtreiptlhei-rpdridmeerixva/ti fvtreipole f- fdwaisthhrxespect to x
fI(V)x f f our x/f f ourprimex
h t r u o f e h
t derivativeo f fwithrespect tox df
dx “TDheFdDerXiv”ativeo f fwithrespect to x d2f
dx “thDe”sseqcuoanrdedd“eFrivDatXiv”esqou fa frweidthrespect tox ∂y
∂x
” X y b y ll a i t r a p Y D “
x o t t c e p s e r h t i w y f o e v i t a v i r e d l a i t r a p ) t s r if ( e h T
Deltay by deltax ∂2y
∂x2
” d e r a u q s X y b y ll a i t r a p Y d e r a u q s D “
x o t t c e p s e r h t i w y f o e v i t a v i r e d l a i t r a p d n o c e s e h T
Deltatwo y bydeltax squared
a m i x a
M /ˈmaksɪmə/andminima/ˈmɪnɪmə/
• Find f(’x ) and solve f(’x ) = 0 . This value o f x (say x* ) is the t
n i o p l a c i t i r c / e m e r t x e / y r a n o i t a t
s ; probably maxima or minima wli l .
t n i o p s i h t t a r u c c o
• Find ’f(’x) .I f ’f(’x )> 0 then x is a loca lminima .I f ’f(’x )< 0 then x is a a
m i x a m l a c o
l .Wecansaythat xi samaximum/minimumpoint.
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e l p m a x E ff o e c n e f o t s e d i c e d e h h c i h w , n e d r a g e l b a t e g e v a t r a t s o t s t n a w l e a h c iM in
e h
t shape o fa rectangle from the rest o fthegarden .Michae lonly has 160 f
o
m fencing ,so he decides to use a wal las one border o fthe vegetable e t a l u c l a C . n e d r a
g the width and length o fthe garden that corresponds to t a h t a e r a e l b i s s o p t s e g r a
l Michae lcanfenceof.f r e w s n A d e r i u q e r e r a t a h t s n o i t a u q e e h t e t a l u m r o f d n a m e l b o r p e h t e n i m a x E : 1 p e t S d n a a e r a e h t o t d e t a l e r e r a n e v i g n o i t a m r o f n i f o s e c e i p t n a t r o p m i e h T d e if i d o
m perimeter o fthe garden .We know that the area o fthe garden :
s i
A = W × L (Equation 1)
s e d i s e e r h t e h t d n a s e d i s 3 y l n o s r e v o c e c n e f e h t t a h t d l o t o s l a e r a e W d d a d l u o h
s up to 160 m .Thiscanbewrittenas: =
0 6
1 W + L + L e s u n a c e w , r e v e w o
H thel ast equationtowriteW in termso fL: W =160− 2L (Equation 2)
e t u t i t s b u
S Equation2 intoEquation1 to get:
A= ( 160 − 2L)L =160L − 2L2 (Equation 3)
e t a i t n e r e ff i D : 2 p e t S i m i x a m n i d e t s e r e t n i e r a e w e c n i
S zing the area , we di fferentiate 3 n o i t a u q
E to get:
A′(L )=160− 4L t n i o p y r a n o i t a t s e h t d n i F : 3 p e t S t e s e w , t n i o p y r a n o i t a t s e h t d n if o
T A′(L )=0 and solve f or thevalueo fL t a h t i m i x a
m zesthearea.
A′(L )=160 − 4L
0=160− 4L
4L =160
L =40 metres o t n i e t u t i t s b u
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W =160 − 2 L=160− 2(40) = 160− 08 =80m r
e w s n a l a n if e h t e t i r W : 4 p e t S
a e r a l a m i x a m e h t d l e i y l li w m 0 4 f o h t g n e l a d n a m 0 8 f o h t d i w A
.f f o d e c n e f
s e s i c r e x E
.
1 Thesumo ftwopositivenumbersi s20 .Oneo fthenumbersi smultipiled k
a m t a h t s r e b m u n e h t d n i F . r e h t o e h t f o e r a u q s e h t y
b e thisproductsa
m u m i x a m .
2 Afterdoingsomeresearch ,atransportcompanyhasdetermined that the t
a e t a
r whichpetrol i sconsumed byoneof i tsl argecarriers ,travelilngat d
e e p s e g a r e v a n
a fo xkmperhour, i sgivenby P(x = ) ( 55 /2x ) x/ 0+( 20 ) r
e p s e r t
il k liometre.
.i Assumethat thepetro lcostsRp.4,000 per iltreand thedriver earns R .p 18,000 rp e hour ( travelilngtime) .Now deducethat thetota lcost , C,i n Rupiahs,f ora2, 00 0 kmtrip i sgivenby:
C(x = ) (256000) x / +40x .i
i Hencedeterminetheaveragespeed to bemaintainedtoeffect a t
s o c m u m i n i
m fora2,000 kmtrip.
l a r g e t n I
f i e s u a c e b , s e v i t a v i r e d i t n a l l a f o t e s e h t s a d e b i r c s e d e b d l u o c s i l a r g e t n I
. t n a t s n o c a s u l p F s i f f o l a r g e t n i e h t n e h t , f s i F f o e v i t a v i r e d e h t
c F x d
f = +
∫
d e ll a c s i F g n i d n if f o s s e c o r p e h
T integration ,thef unctionf i scalled d
n a r g e t n
i ,and thedifferentia ldx i ndicatesthat xi sthevariable of n
o i t a r g e t n
i .
e h t f
I bounds or ilmitsofi ntegrali sgiven ,wesaid thei ntegra lasdefinite l
a r g e t n
i ,butwhenthei ntegra lhasno boundsnor ilmits ,wesaidthe s
a l a r g e t n
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s l a r g e t n I y a S o t w o H∫
b
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d
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x
(
f
Thei ntegra lo f fx f romatob x f f o b o t a m o r f l a r g e t n i e h Ty t i n if n i o t o r e z m o r f l a r g e t n i
