C\ropf*"1
Fu^r*;',neonJ" Li- 't s
Funcfi'n
1'1 Four W^1sto RuP"'*"t &-
a
is ^ r,^le -fk^t Ase,;lh, tn ",.*h efet'^*-"tX i t ^ s * b b
A fu^rL;2n f
fCy),, i., ^ sut E '
Ca[.{
eXaCtll on€ *le**r.f,
r/
,{
J-n
ftn
Tf.,,an\rrF,b"e-r
J[rs rah8e 4
(
b
fl,e s*
reaL unbers.
tet'
("" urhieh *['*
We usi4a11.r
4e,ngiJ*-f funtti'^,
is aqn*J
E are s*ftt
b oJ
*lru d'*a'in '+ *l-
"-f *+ x."
is fl., valu. '+ f ^t X ad 'i r*A
,'s fu
S.l
.* all possill'g Value's4
Varie> thr.11ho,'k 4lnc-lor"ai"'
nr,rtb*.-o- i n 'tt- Jtrt',ain
-+
6.
fun"tt'r,.
fq1 as X
A s6mbd 'lh*t rtT-.c.unf' on arLifitTl
,^r."ti-,^ *
f*""t
;; calk l, an i"A:Iendent vadqae
^
h s)mL-l "n^t re-?rzs"',tt
Jepunde'.t
F.,Fll't
.{
^
hLryyrb*r
in *f,.
raDa{
e- n*
( l
+
i
is c.^lt.a( x
tlqria blq.
h"fyhl
t- 'tk^K ,+ A f**ti",,
tf',* gia main
A5 {h. let 'f
ag ^
rrtaahie'g
w+- e.an t'A;"k
-t1,,*ra?Je
ae -fhuE.t "f ell potsiLle in yafs an/.
.
attyo*i Lle
"u'fyuts
fr"1
fuufy"t)
%
* Anolhrr
"U
'to p{cture ^ ft*ncti.n is
b
An Ar-row Aia6ram.
X
d\
b
TA, noif
f
--F
f
E
IqL+A,J,A" vfsuali*;ryqa. fun"{;.n i s its
$
ato.Ph.
i s ^4 *not;*n wi*A dsnain b, -flrto,ffryra1,hls *Au 5st t4 orderqJ.
&Prrrn'ort
pairs
{(*,
k^))
lx€Dl
(d, fut
'^at
d or*,ain
R"Fre-.sentatione
^+ F*t gf;'n9
Thera Ar"- (un. Ftesitl€. ot,
e Nor,,eri
,^b
€ Yiso^l)
to
rePr*s*"t & firnttio, ", ;
( 6J ^ taLle-.+ vqlu*tJ
(bJ ^
6rnfi'h)
G hl'broicctl! t
^n .xpliaf"L {'r*.^1^)
b
%.,
Ex"(a zs ) rl /t4= zx\-x -r 2, f;^) f rz1,
f c-r),f@t, f {_o),
f ( o*t) , L f{q, f tznl, 1(*), [-fto,)", anlf'(6,.+,,')"
{t4= g(z)t-(z)-rL = tz
f 1 - r 1= 3 G r ' - ( - z ) 1 - L = t L - Y 2 1 L = 1 6
f Gl= 3(u)L- (a) +z
- 3*'-_o- a-L
f ( - o l - - 3 ( - o ) t - ( - ^ )t L
= 3^'-r
a'+ L
3(a+l)"-(a+t)tL=3(ct-r-La+t)
= 3a"+5a-+\
=3a>+6o+3-a+l
fbrt)=
-^
-l+L
z froy
f tLn)
ft^')
lf^,)
Lu!-'+ t''') - ^ - A+ L
=
L
z
(
o''*
(
(
a
+h)
r.+h
r
I
)'=
f G+h)
=3t+6o.l+3h'-a-h 1L
%",
Ex- (#26)
La h.or, wi*h rad i ns r i a ehes hns volur,,
e
A 5phe.r ical
V ( r ) = L r r 1 3 , FinA ^ funrti.^ Jht reprias*,.ts*h. 6.rnou.,t,+ air
3
f <1uir*d tn i"{hte-
tk* bu..\\6.,r\
f..'^
^ ro.Jiug '+ r in ch" s
t o ^ .oJios -t- rt \ i r.ch.s ,
Y(rrl) - 1r(') :
- rtJ
{3" ( r r r ) ' - - I n 13= ln It'-r.r)s
5
\
l
l
'
|
J
'n3
= Lu (/"3r'+ 3r+l /;
] v
-
Trr(3r'*3r
Ex. (+ z")
5ir^gli{; b.*
x -l
+r)
3
A
13a1L13"f,
stb + saf
3
*h.
x + J -''* ,
-F-F_*
X-t3-2x-L
X+l
x-l
r-r 3
X-+l
5*3 - z(x+t)
2
X+l
X-1
x*l
*X+l
x+l
x-l
I
- ( 4/i';
/ r'
(Xr-l
1)
) (X.=
;'1
*t' I
5i're'nfun"tio".
fut- ft,'
x-l
=
&
f
Evoluotr-*h. A;({etenca.lqsti<-"rb f"
rtnsi^,tr,
ft*y= *o> ,
x+l
$q)- {r,y
=
r
+ t)3=
l(e
i ( "- L)'=
x+l
@7,
,1
-
*xr-l
x+l
x*l
EX, ( *
Finr\ *he ds'air, ,+ *h* f*r,efion.
3L)
z x t. . 5
ftxt =
X2+X- 6
eN(. pt wh*n X) X- 6 = 0
frr) ,s d.(;nQ f,r a,ll x
K'+x-6=0 +
,o.
(n-t X(+3)=o
D f = ( - o 4 ,* 3 ) u ( - 3 ) ? : ) U ( z ' . c )
E x .( + 3 5 ) A t x l = 'xz
+ *
xz-fx)o
x(,(-5)
X=0t X
x ( x - 5 ) = Q t+
X
x
5
0
-o{
e
{
+
JE-E
5
od
t
-f
{*5
x (r-5)
fx
+
*t
\-#
Y(O
Dh=
or
( - o a ,o ) U
w
O/,,
T 1," V*.'fie^l L;r,*T.sT
A eune" ir-t tl.,*
tS
Xr{- phne-
*h*
( ) f
^not"t
.+a- f*."tion
.( x ;f
6raph
i.f ho vcr'tiealli.,* inters".ts
*he.
CUf rlV lno fe
-{han o n ce
"
.)
X
Tlris r,u(ve regfeser,tg ^ f*n"t,bn
Tl.,is cclrvv doe. not
reff€-S€\t a. f*r."tior.
?t".*w ise D.f in.J F*r,.ti,'.s
T 1",* 4*r, ctio,.g
diff"rentp"'tr
J.{i .J
6{e
A*F;'*A b
.+{t c-ir
oliff.r*",t"f-rrnulas
i'"
Ar * atins &(r. callto[
p ie.e,wi:s
{*n"tiong .
Ex. ,SKetcl. *l*
Ixl
*k,'t
lx
]-,*
6raph
o{ *h-.- uLsol,^tcValue {tn"tt'",
\l xlo
J
;f x(o
fcr,= lxl
lxl
a=
x
fu.,
*"
Fva,loqt. f(-3);
(*qy)
{**tion,
f@l ,
T1.." s K"{ch *he
4nJt ft4
-+
6raph
(or *h-
piecewfse ,1"4;n."t
*h- f*,.*tion.
x!')
3- \x
{r*)=
2x- 5
{
; t x<L ilt 0;.)
;{
x > 2 #-
x
3 l l
q
l e ) = 3 * t ( - t ) =3 - rL = z
Sqol=
3 - + ( o=
)3
{ { 2 1= z ( z )- 5 : * l
EK,
(* qs) 5 K"t. h fl.,* 6ray\ *+
*h. $u"rtion.
I , f t:t l +I r I t - " r l
t = D , t + l = oa { = I
il +<*t;>
r f * ' ( f < o + A ( 1=1- t n ( t + r ) =
/4,
r{
-7t-l
t tt,=
{
I
zt+l
',f 't1- |
i{ -l(t<o
"t$ t>o
= l
t ) o 2 A ( + ) t=+ ( t r t ) = z t + l
ti3
-TlTfior")
-L'3
+ t!
-llr
J
3
2
(oill
o I r ftiotc)
t tJ
0lr
l l J
Q,.t
I
I
I
*L't o
Ex, SK"tch*h"
eqq)
lxl
{r*=
1|
I
4t*t=
{(^t =
lxI
fun"fion.
i+ l x l < l
i+ l x l > l
{
*--1
T*"n;1 i,t";r)
-9upposeA)0.
i{ *tr(r (o
*x
I
!-x
I
I
i{
o(X.( |
if
if
x<*l
x>t
;$ K<- |
(-r,l)
i$-t.((<o
x, ,( o(x( I
I
il
x>l
@/,-,
Then
/=!^
fXl=a
lxl(a €> *q(x1a
lxl >n <.+ X(-q or
* , < -| o r X > l
i{
I
i
L-
l( -l .(X _<I
(
f,r*r
,+ *[.
6.aph
I
I
*>"I
Ex. (*tS)
| Lox wi*[ an or"n *.p is t' Le aonstrrtt.l fro* ^
re"ton6ular yi..- o( cardtoarJ wi{h di-"nsions 12,r, L7 20 in l7
cornertti
,^+t!U 'wf nual eltl'res"f side x at "a"l'
sio\.sas i'. *h' {i6uce" Exy"ss*ks v'l*-.
oF h.
V "{:: *l.- to,,t
qs n fu'."ti'n # x.
T,,
fr
()
+o
'{)
-u
f
V(x)= t 1r/X = (zo-LxXIz*zx)x
= 2 ( l o* x ) ( z )( e * x ) x= 4 ( r o* x ) ( 6 * x ) x
= 4( 6o*lox-6xad")x
= 4 ( e u- t 6x - +K- " ) X : . ( z + o 6 1 * + 1 x ' ) X
= 4 x t - 6 7 x " - *z 4 o x
L > o+
W > o+
x
o
Zo-Lx)O+
zx 1 Z s + X < f o
lz-zx )o +
2x 1 l z > x < 6 @
0 @
O,@r@ +
o<x 16 in
4^3- 61xt* zlox
o(x < 6Ln
%,
?'"t',
EX. @tQ An ele&ci.ifJ a.r"paiy el,a'6esits attsfo-ers 4. basera-te 'f
c.,ibs per Kilowatt-Lo*c (KWh) 1"r *k- firtt
6
-o,.tL,
ylus
o
ffO
KWL #,. all Vsa6zovtr l2oo KW\' Exyre's
l2oo kWh anl 7 ce"ts per
*h. ..r'*t',! e.st E as o'fun"ti.n 4 *he ar,,o,^"tx 4
us*[. Th.",,
th- f,,r,.ti" n E f..
6ra7l',.
Fori[,. (;,*
c,;t E is:
tZookWh,th. no"fll,J
"l"etricit.;
0 -{ X-( 2ooo.
EU)= t0+ o.o6x, o(x<l2oo
F-,ra"llusryeovLr IZoo Khth, fr,n ^""th!1 c'st E is ,
x>rm
s> F$)=10+
6(tLoo)-vo.o7(x-l2oo)
= l o + V Z + O . o VU * l L o o )
= 8Z+ o.ovu-l ?f,o)
Thvef o, e
,
; { o r( {"..( lLao
( fo + o - o 5 x
E6)- |
v{ x /> l2oo
+o-o7fr-t:-)
ltt
E ("'t(+)
JEU)
tts"l 8z (hacS
2ool i36
t80
(6e
llo
l)D
loo
8o
6o
1o
f(rzoor
LO 6'tot
&
lt lt
tte sW l6a l&o 2ooo
9,,
5t*p {u^cti.p,e ^(e f *nrtlong *h^t j"*f
th- n*fi.
4.o*.. one va lue t,
Th<"f, tt,*,?) vx^^ple i s 6r t t.f
f *n cti,r, .
Ex.@)SktcV"t[,u-6'( , leh .+ th* f*n.ti on "
(b) WhdtarL
+l* dIo bat^o*{ rary 4 {?
*2
i+-s<x<-t (q
?
T ( x )=
+ - l ( x< 1
?
I
,
J
f
3
:
b
(L) Df=[-;, 6)
)
R f= { - 2 , ' r 3 }
SJrnrn.ttJ
O tf n {u^"t;"nf
sa.t;tf;r, ft-rl= ft^) f.. .urg nunt*r in ifs
6lomain,+hu f i s c".ll.J an evenf uncti"n
-fA"
fh.
1rayl.,
.f
an e(en f,n"ti"r, is Jlr^".,ctric wi*h fespect f"
3-axis.
F", ax^-It-,
f CU _ X
L
r
t9
eVt n
ta aorrs€ :
P
ft-x)= (-x)"=K'= ru)
w,,
@ Tf f s^tisf ;., fkx) = - ftxl f", ev(rJ hLrrnber
a
',
ts do rnain, -fA*n
in
f is c,*ll"-J A"n , Jrl fn^.ti'q ,
- TA.
aL",^t {k' oriin'
Ara,pl o* an ool) funat;"n is Tnne*ric
F"r ?xd.^Fl. tie fun.ti*,
,
fG x): (- ^)t =
x
? A f unrt\-n can te
!:F
ft^) = x? i s .JA
3= G
fr*l
evln,
b ecaus,.
J
1d4, o( nei*her even hsr .41 .
The zero fun,tion ({ul=
o) \s *le ""b {unut;on *ha,t is
b*h eve^
an/', oo(rt.
(ae;fA*o
Int"a:i
n ^nl'
u1
F,on.t,oh s
be.r.asir.
v-'L - j
u
;f
A funot; on is calteJ, increa,sin16n arl interv6.l I
ftx,) 1 ftrS
It
is caltJ
dec(?ati.)
whene{e(
oh
fu,)
rro
r oXamyle., *h*
t-dt,0]
?
xr(x,
T
in
T 1+
Wlrens{z(
f*",rti.n
e(<n ne{
,
'
ftxl=X"
Xl
i s Aecrectsin
anl inc-t.,"sttJ
6n *1.. i^brv* I
9,
1*z in T
d
[0, *) "
o n +h. i"rtt-rva I
a
EX , D eternainew hetllus "f is * s ? n t c > d J * (
,
fcr)= ,x
fV3
x L +|
{Gr)=
*x
=
\r)
(-x) -r-|
5ince.&-*S -- - f ,x)
r
*71
n ei*htr"
rt+ I
{Lr- I
f is &n "lJ
f un"tion.
fr*t
L
fut
x7+l
- L
P
f (-x)=
=
x
f*l
C41r
Since ft-I
# 75.
= fc*;
= frrl , f is an ?Ven frntti,^
o
fks=
X+l
?
t(-x)=
-{
Cx,t =
-X -+ |
(-x) i.l
{(-r)+ft*1
=
?
f ( - ^ ' ) t ' - f i r tI +
-x+ |
e
rf r gnei*h*,
€.Vtn hor*
+ 7 6 {ut = x
lrl
f ( - { ) =( x ; l - " 1= - x l r l = - f r * )
1inc" f Gr) = -fu1 , f
is 8n
"dA
fun"ti-n.
"A
A
Ex"(* vt)pllt f anrl,6 6re
rbtIf f a""t6a(e
t dfh & Y e n
f" " tti
ons,
IS
fu
even
1
"eA f^ nc'ti ,ne, i s f,NJ o*le 7
b dth
(e) rwhmf i {- f is Etl&na"Jr1 is o dA ?
J*"'b
our- Ang UcrS"
fl
( q) l* f o^J ars L*41, €ven fun.t;,ns fA^
t
d
Now :
f(*r)= f,rl anJ
6Fx)Vu)
V*6)G4 = fGx) +d?x) = ft^;+fiu)=V*6) t^;
TAeref"c,
@ ff
f ^J
Now :
is 6n etl (^ functi,n.
f -J
-ll*^
areL"/A nddf,uoct;^nrt
fr-x)= -fu. o*l
6
AFx)=-du)
Vra) t-K)= {C^t +g( x) = -frrl
= - ffv)kl
*f,r)-6JU1
=- fftr, *gort)
" La,^,J=
TAer.f-r9 f
*rJ is en .,tA frn&;-n
(c) ff
f is €vtn o^l
d
dA' fA*nf(-n ^ frn ^J
*7J(x)
flFx) =
is
NoW :
V*O)Gx) = fG4+d? x) = fu; rf- Aul = frrl - u)
A
V+5; (-n * (fra)txt
(f*,il(-r) f -(fr6)k)
Ex
".f
l + / *J
)
i s neitA.r even
hor odJ
t;oa : If / is fA" zerof un.t;^ tA.n
,
f *,, will6e od{ .
0
is tl"
If
o,I
Z.ro frn"t;^, ,-f6un
f-t
:
@t,
o
witl be Bvel .
^(t, Erone'
C A L L U L U SStew
,
h n rn<-worK
5..'tion l-|
Exetc ises #
4, 3, 7, 1
2V* 33
"dA
3 V * 1 7 .,riJ
5 t - 6 l *,JJ
6 5 , 6 r , vl, v7