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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