Introduction Calculus

8/19/2019 Introduction Calculus

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Chap. 0Chap. 0CALCULUS &CALCULUS &

LINEAR ALGEBRA ILINEAR ALGEBRA I(4 credits)(4 credits)

Agoes SoehianieAgoes SoehianieSiss Ger!an Uni"ersit#Siss Ger!an Uni"ersit#

Serpong $ IndonesiaSerpong $ Indonesia


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R%es o' the Ga!e

1. Learn the regulation of SGU2. Some important points:

NO FOOD and LITT!ING in "L#SS!OO$S

%& must 'e off during le(tures

). Grading:*+, -eel/ 0uies

+, Final 3amination

if Fail after Final 3am !epetition

Grading Scae *$+00 A E,ceent-*$4 B er# Good/0$-4 C Good*0$* 1 2air

Beo *0 2 2ai


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C3URSE LAN

Introdu(tion: !e4ie5 of num'ers 6 fun(tionsLimits and "ontinuit/

Differentiation

#ppli(ation of Differentiation

Integration

#ppli(ation of Integration

3ponential7Logarithmi(s and trans(endental fun(tions

Integration Te(hni8ues

$ain Te3t'oo:

!o'ert T. Smith and !olland 9. $inton7 "al(ulus 2nd dit

ion7 $( Gra5 %ill7 IS9N: ++112*;1+

http://highered.mcgraw-hill.com/sites/0072398485/student_view0/http://highered.mcgraw-hill.com/sites/0072398485/student_view0/http://highered.mcgraw-hill.com/sites/0072398485/student_view0/http://highered.mcgraw-hill.com/sites/0072398485/student_view0/


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Acade!ic Caendar

5 Se!ester +6*6-Co%rse +/ ee7sSient ee7 + ee72ina E,a! 8 ee7s

Repetition + ee7

9oida# Id% 2itri + ee7

9oida# Christ!as$Ne :ear+ ee7


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I!portant 1ates

5 Se!ester +6*6- Start ;on6 88 A%g%st S%

S% ?%e6 +- @an%ar# S% 2ri6 0 2e


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Chap. 00Chap. 00Introd%ctionIntrod%ction

A.A. SoehianieSoehianie

SGU$800*SGU$800*


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N%!


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REALD86 e6π, ....

Rationa

86 F6+46 .....

?#pe o' N%!


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REAL LINE & Inter"as

0 +

+D8

D8

a6


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roperties o' Rea N%!


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So"ing IneQ%ait#

5 Linear IneQ%ait#

5 %adractic IneQ%ait#

5 2raction IneQ%ait#5 In"o"ing A


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So"ing Linear IneQ%ait#

&ro'lem. So"e the to$sided ineQ%ait# / N + $, +0.

Solution.

?he a


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So"ing %adratic IneQ%ait#

&ro'lem. So"e the Q%adratic ineQ%ait# , 8 , $ / O 0

Solution.=rite the ineQ%ait# so that the rhs 0=rite the eQ%ation as !%tipication o' 'actors

(,)(,$8)O01ra a rea ine

;ar7 the roots ocations (,$ and ,8)?his i di"ide the rea ine into se"era seg!ents (e.g )Ana#se the sign o' (,)(,$8) in each seg!ent3


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So"ing IneQ%ait# o' the 'or! a<

ro


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So"ing IneQ%ait#In"o"ing A


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Lines

5 Sope (gradient)> specia cases "ertica6 horionta6

positi"e6 negati"e

5 EQ%ations o' straight ine

5 Specia ines5 erpendic%ar

5 arae

5 An# ange


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Cartesian aneCartesian ane A "ersatie too 'or e,poring reationship


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SL3E 1e'inition

1 2

E2

E1

E

α

! tan (α) sope gradientr%n


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SL3E !eaning

Slopem

'

'

x

y

x

y

∆

∆=

∆

∆


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Specia Cases o' Sope

Spe(ial "ases:

+) ertica ine6 ! %nde'ined

8) 9orionta ine6 !0

) !O06 santed to the right4) !06 santed to the e't

E

m+

m+

m@


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EQ%ation o' Straight Line A point & Sope

Gi"en a point (,06#0) and a ine passing thro%ght this point ith

sope !. 2ind the eQ%ation o' the ine.

S%ppose a point (,6#) is in the ine6 then the sope !%st


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EQ%ation o' Straight Line 8 points

12

12

1

1

x x

y y

x x

y y

−−

=−−

Gi"en to points A(,+6#+) and B(,86#8). 2ind the eQ%ation o' the

ine passing thro%gh AB.2irst6 e cac%ate the sope6 !

12

12

x x

y ym

−

−

=

)( 112

121 x x

x x

y y y y −

−−=−

And %se one o' the point(etVs sa# A) and thepre"io%s 'or!%a to get

the ine

O!


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Specia Reationship Beteen ?o Lines

5 arae ines sa!e sopes


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Specia Reationship Beteen ?o Lines

5erpendic%ar ines !+ W !8 $+


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Appication E,trapoation

5 See te,t


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2%nctions

# 9 # 9 # 9 # 9

?1A ?2A ?)A ?*A

-hi(h of ?12)*A des(ri'e a fun(tion/f?3A7 5here 3H# and /H9


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Graph o' 2%nctions


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?esting A Graph Is it a '%nction

Jerti(al Line Test


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So!e Ee!entar# 2%nctions

5 o#no!ias5 Rationa

5 SQ%are root

5 ?rigono!etric

5 E,ponentia

5 Logarith!ic

1572)( 23 +−+= x x x x f

1

157)(

2−

+=

x

x x f

162)( 2 −= x x f

)3cos(4)sin(15)( x x x f +=

x x f )3(10)( =

)4log(2)( +=

x x f

3amples:


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o#no!ias


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Rationa 2%nctions

Graph 5ith 4erti(al

as/mptot

Graph 5ith

horiontal as/mptot


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1o!ain o' a Rationa 2%nction

$8 8

1o!ain

1e'inition & 1o!ain o'


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1e'inition & 1o!ain o'SQ%are Root 2%nctions

-hen 5e 5rite /K37 5e mean that:

/ is a positi4e num'er ?B+A7 su(h that

/2 3.

Thus K* 27 NOT K* 2.

If 32 * then 3 K*7 or 3 2

NOT onl/ 3 K* ?5e need 3 =K* alsoA


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Graphing Using Co!p%ters

Co!p%ter ith appropriate so'tares can hep a ot in

"is%aiing a '%nction


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So"ing EQ%ations

# solution of the e8uation f ?3A + is an/ num'er 3+

that satisfies the e8uation Ci.e.7for 5hi(h f ?3+A +M.In this (ase7 3+ is (alled a !O of the fun(tion f?sin(e 3+ is a 4alue that maes f eroA or a !OOT ofthe e8uation f ?3A +.

Spe(ial (ase: 8uadrati( e8uation the a'( formula


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3ther !ethods to 'ind Roots

Fa(toring : not al5a/s eas/

Guessing : /ou ma/ 'e lu(/

If /ou su'stitute 317 then /ouPll find f?1A+. Thus 31 is oneof the roots "an /ou find the remaining roots


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3ther !ethods to 'ind RootsGraphing: it helps a lot

&ossi'le roots:a'out =17

'et5een + and 2

oom

"loser inspe(tion re4eals t5oroots 'et5een + and 2: at31 and a'out 1.*

In this case6 e can 'actor o%t the

eQ%ation (,$+)(,8

$8)0


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3ther !ethods to 'ind Roots

=hen e need a "er# acc%rate res%ts6 then e cane!po# n%!erica a#s ith the hep o' co!p%ter

progra!s. ;an# di''erent agorith!s e,ist6 each hasits strengths and ea7nesses. ?hose are Neton$Rhapson6 Bisection6 Secant etc.

=e =i Learn this !ethod ater

N%!erica =a#s


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#ngle : Unit of measurement

#ngle is measured in radian and degree ?QA. In (al(ulus7

most fre8uentl/ 5e use radian ?radA. !ad is a real num'er.

+

!θ

#

9

R

ABrad =)(θ

DFINITION of !#DI#N "ONJ!SION !#D Degree

2π (rad) = 360°

0X 0 rad

0X π/6 rad

4*X π/4 rad

/0X π/3 rad

0X π/2 rad

Some spe(ial angles


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?rigono!etric 2%nctions

Sin ?θA /r"os ?RA 3r

Tan ?RA sin?3A(os?3A

"tan ?RA 1tan?3A

Se( ?RA 1(os?3A"ose(?RA 1sin?3A

E

(,6#)

,

#

θ

r

Sin ?θ) and "os ?RA are periodi( fun(tions5ith period 2π. -hi(h means:

Sin ?R2A Sin ?RA"os ?R2A "os ?RA

Do /ou no5 the period of othertrigonometri( fun(tions


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?rigono!etric 2%nctions graphs


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?rigono!etric Identities

See e3amples in te3t'oo


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E,ponentia 2%nction

F?3A ?'A37 '>+

'ase e3ponent


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?#pica shapes o' E,ponentia 2%nctions

The e3ponent ma/ 'e : an integer7 a rationalnum'er or irrational num'er7 thus a real num'er.


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Con"erting to E,ponentia 2or!


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Appication E,ponentia Groth

Initia# there are +00


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Logarith!

0,log >= b xb xb

3ample.og+0 +0 + ( since +0+ +0)6og+0 +00 8 ( since +08 +00)6og+0 +000 ( since +0 +000)

#s a (onse8uen(e:

Important 'ases:

log1+?3A

loge ?3A ln ?3A7 e 8.-+8+84*.....


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roperties o' Log

"hange of 'ase7 from ' a:

Log' 3 Loga3Loga'


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;ore E,a!pes


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Sti ;ore E,a!pes

"hange of 'ase


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Appication 1eci


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?rans'or!ation o' '%nctions


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Co!position o' 2%nctions


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Shi'ting & Stretching A Graph

Stret(hing4erti(all/

Shiftdo5n5ard


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Introduction Calculus

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