contents · contents limits 3 deriv atives 14 integrals 33 appendix 42 ... dif fer ential equation...
TRANSCRIPT
Co
nte
nts
Lim
its
3
De
riva
tiv
es
14
Inte
gra
ls3
3
Ap
pe
nd
ix4
2
Th
isre
vie
wg
uid
ew
asw
ritt
enb
yD
ara
Ad
ib.
Po
rtio
ns
of
the
“Lim
its”
and
“Der
ivat
ives
”ch
apte
rsar
eb
ased
off
the
Cal
culu
sW
ikib
oo
kav
aila
ble
on
the
Inte
rnet
athttp://en.wikibooks.org/wiki/
Calculus
.C
HS
NR
evie
wP
roje
ctco
ntr
ibu
tors
Dar
aA
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and
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lS
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ico
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ibu
ted
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ibo
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isis
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evel
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men
tv
ersi
on
of
the
tex
tth
atsh
ou
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der
eda
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rog
ress
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Th
isre
vie
wg
uid
eis
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elo
ped
by
the
CH
SN
Rev
iew
Pro
ject
.T
od
ow
nlo
adth
isre
vie
wg
uid
ean
do
ther
rev
iew
gu
ides
,vis
itchsntech.org
.
Co
py
rig
ht
©20
08-2
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Dar
aA
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and
oth
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ntr
ibu
tors
toth
eC
alcu
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Wik
ibo
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lyli
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sed
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eD
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itio
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ltu
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Wo
rks
( freedomdefined.org
).E
x-
cep
tas
no
ted
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der
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ph
icC
red
its”
on
this
pag
e,it
isli
cen
sed
un
der
the
Cre
ativ
eC
om
mo
ns
Att
rib
uti
on
-Sh
are
Ali
ke
3.0
Un
po
rted
Lic
ense
.T
ov
iew
aco
py
of
this
lice
nse
,v
isit
http://creativecommons.org/licenses/by-sa/3.0/
or
sen
da
lett
erto
Cre
ativ
eC
om
mo
ns,
171
Sec
on
dS
tree
t,S
uit
e30
0,S
anF
ran
cisc
o,C
alif
orn
ia,9
4105
,US
A.
Th
isre
vie
wg
uid
eis
pro
vid
ed“a
sis
”w
ith
ou
tw
arra
nty
of
any
kin
d,
eith
erex
pre
ssed
or
imp
lied
.Y
ou
sho
uld
no
tas
sum
eth
atth
isre
vie
wg
uid
eis
erro
r-fr
eeo
rth
atit
wil
lb
esu
itab
lefo
rth
ep
arti
cula
rp
urp
ose
wh
ich
yo
uh
ave
inm
ind
wh
enu
sin
git
.In
no
even
tsh
all
the
CH
SN
Rev
iew
Pro
ject
be
liab
lefo
ran
ysp
ecia
l,in
cid
enta
l,in
dir
ect
or
con
seq
uen
tial
dam
ages
of
any
kin
d,
or
any
dam
ages
wh
atso
ever
,in
clu
din
g,w
ith
ou
tli
mit
atio
n,t
ho
sere
sult
ing
fro
mlo
sso
fu
se,d
ata
or
pro
fits
,wh
eth
ero
rn
ot
adv
ised
of
the
po
ssib
ilit
yo
fd
amag
e,an
do
nan
yth
eory
of
liab
ilit
y,ar
isin
go
ut
of
or
inco
nn
ecti
on
wit
hth
eu
seo
rp
erfo
rman
ceo
fth
isre
vie
wg
uid
eo
ro
ther
do
cum
ents
wh
ich
are
refe
ren
ced
by
or
lin
ked
toin
this
rev
iew
gu
ide.
Gra
ph
icC
red
its
•F
igu
re0.
1o
np
age
24is
ap
ub
lic
do
mai
ng
rap
hic
by
Ind
uct
ivel
oad
:http://commons.wikimedia.org/wiki/File:Maxima_and_Minima.svg
•F
igu
re0.
2o
np
age
26is
ap
ub
lic
do
mai
ng
rap
hic
by
Ind
uct
ivel
oad
:http://commons.wikimedia.org/wiki/File:X_cubed_(narrow).svg
2
Ap
pen
dix
Tri
go
no
me
tric
Ide
nti
tie
s
Py
thag
ore
an
Ide
nti
tie
s
1.si
n2θ
+co
s2θ
=1
2.1
+ta
n2θ
=se
c2θ
3.1
+co
t2θ
=cs
c2θ
Qu
oti
en
tId
en
titi
es
1.
tan
θ=
sin
θ
cosθ
2.
cotθ
=co
sθ
sin
θ
Su
mo
fTw
oA
ng
les
1.si
n(A
+B)
=si
nA
cosB
+co
sA
sin
B
2.co
s(A
+B)
=co
sA
cosB
−si
nA
sin
B
42
Lim
its
Th
isch
apte
rw
aso
rig
inal
lyd
esig
ned
for
ate
sto
nli
mit
sad
min
iste
red
by
Jean
ine
Len
no
nto
her
Mat
h12
H(4
H/
Pre
calc
ulu
s)cl
ass
on
Ap
ril
2,20
08.
Itw
asla
ter
up
dat
edw
ith
an“A
dd
end
um
”se
ctio
n(p
age
12)
for
ate
sto
nli
mit
sad
min
iste
red
by
Jon
ath
anC
her
nic
kto
his
AP
1C
alcu
lus
BC
clas
so
nS
epte
mb
er18
,200
8.
Intr
od
uc
tio
n
Ali
mit
loo
ks
atw
hat
hap
pen
sto
afu
nct
ion
wh
enth
ein
pu
tap
pro
ach
es,
bu
td
oes
no
tn
eces
sari
lyre
ach
,ace
rtai
nv
alu
e.T
he
gen
eral
no
tati
on
for
ali
mit
isb
elo
w.
lim
x→
cf(
x)
=L
Th
isis
read
as“t
he
lim
ito
ff(
x)
asx
app
roac
hes
cis
L.”
Info
rma
lD
efi
nit
ion
of
aL
imit
Lis
the
lim
ito
ff(
x)
asx
app
roac
hes
c.
Th
ev
alu
eo
ff(
x)
com
escl
ose
toL
wh
enx
iscl
ose
(bu
tn
ot
nec
essa
rily
equ
al)
toc.
Itca
nb
ere
pre
sen
ted
by
eith
ero
fth
efo
llo
win
gfo
rms,
wit
hth
efo
rmer
bei
ng
far
mo
reco
mm
on
.
•li
mx→
cf(
x)
=L
•f(
x)
→L
asx
→c
Ru
les
No
wth
ata
lim
ith
asb
een
info
rmal
lyd
efin
ed,
som
eru
les
that
are
use
ful
for
man
ipu
lati
ng
ali
mit
are
list
ed.
Ide
nti
tie
s
Th
efo
llo
win
gid
enti
ties
assu
me
lim
x→
cf(
x)
=L
and
lim
x→
cg(x
)=
M.
Usi
ng
thes
eid
enti
ties
,o
ther
rule
s
can
be
ded
uce
d.
1A
Pis
are
gis
tere
dtr
adem
ark
of
the
Co
lleg
eB
oar
d,
wh
ich
was
no
tin
vo
lved
inth
ep
rod
uct
ion
of,
and
do
esn
ot
end
ors
e,th
isp
rod
uct
.
3
Sc
ala
rM
ult
ipli
ca
tio
n
Asc
alar
isa
con
stan
t.W
hen
afu
nct
ion
ism
ult
ipli
edb
ya
con
stan
t,sc
alar
mu
ltip
lica
tio
nis
per
form
ed.
lim
x→
ckf(
x)
=k·l
imx→
cf(
x)
=kL
Ad
dit
ion
lim
x→
c[f
(x)+
g(x
)]=
lim
x→
cf(
x)+
lim
x→
cg(x
)=
L+
M
Su
btr
ac
tio
n
lim
x→
c[f
(x)−
g(x
)]=
lim
x→
cf(
x)−
lim
x→
cg(x
)=
L−
M
Mu
ltip
lic
ati
on
lim
x→
c[f
(x)·g
(x)]
=li
mx→
cf(
x)·l
imx→
cg(x
)=
L·M
Div
isio
n
lim
x→
c
f(x)
g(x
)=
lim
x→
cf(
x)
lim
x→
cg(x
)=
L M,w
her
eM
6=0
Co
ns
tan
tR
ule
Th
eco
nst
ant
rule
stat
esth
atif
f(x)
=k
isco
nst
ant
for
allx
,th
enth
eli
mit
asx
app
roac
hes
cm
ust
be
equ
alto
k.
lim
x→
ck
=k
Ide
nti
tyR
ule
Th
eid
enti
tyru
lest
ates
that
iff(
x)
=x
,th
enth
eli
mit
asx
app
roac
hes
cis
equ
alto
c.
lim
x→
cx
=c
4
Ex
am
ple
dy
dx
=yx
dy y
=xdx
ln|y
|=
1 2x2
eln
|y|
=e
1 2x
2
y=
e1 2x
2+
C
Inte
gra
tio
nB
yS
ub
sti
tuti
on
Inte
gra
tio
nb
ysu
bst
itu
tio
nis
am
eth
od
for
inte
gra
tin
ga
com
po
siti
on
of
fun
ctio
n,
wh
enth
een
tire
inte
gra
lca
nb
eex
pre
ssed
inte
rms
of
con
stan
ts,u
,an
ddu
.
Inte
gra
tio
nb
ysu
bst
itu
tio
nm
ayb
eu
sed
inco
mb
inat
ion
wit
hru
les
for
inv
erse
trig
on
om
etri
cfu
nc-
tio
ns.
Tri
go
no
me
tric
Ide
nti
tie
s
Tri
go
no
met
ric
iden
titi
es(p
age
42)
can
be
use
dto
sim
pli
fyex
pre
ssio
ns
bef
ore
or
afte
rin
teg
rati
ng
.
41
Slo
pe
Fie
lds
Slo
pe
fiel
ds
(als
ok
no
wn
asd
irec
tio
nfi
eld
s)ar
ea
log
ical
exte
nsi
on
toin
itia
lv
alu
ep
rob
lem
sas
they
pro
vid
ea
sket
cho
fth
ed
iffe
ren
tial
equ
atio
nfo
ran
yv
alu
eo
fC
.
Ata
ble
con
tain
ing
the
val
ue
of
dy
dx
(th
efu
nct
ion
’ssl
op
e)at
dif
fere
ntx
and
yv
alu
esis
use
dto
crea
tea
slo
pe
fiel
d.
Ap
pro
ach
es
Th
ese
app
roac
hes
red
uce
the
tim
ere
qu
ired
tom
ake
or
anal
yze
slo
pe
fiel
ds
and
the
po
ssib
ilit
yo
fm
akin
ger
rors
.
Pa
tte
rns
Ho
rizo
nta
lP
att
ern
Wh
enth
ed
iffe
ren
tial
equ
atio
no
nly
con
tain
sth
ele
tter
y(e
.g.
dy
dx
=y
),th
ere
isa
ho
rizo
nta
lp
atte
rn.
Ve
rtic
al
Pa
tte
rnW
hen
the
dif
fere
nti
aleq
uat
ion
on
lyco
nta
ins
the
lett
erx
(e.g
.dy
dx
=x
),th
ere
isa
ver
tica
lp
atte
rn.
Dir
ec
tio
no
fS
lop
e
Det
erm
inin
gw
het
her
the
slo
pes
of
po
ints
ina
cert
ain
vic
init
yar
ep
osi
tiv
eo
rn
egat
ive
isu
sefu
lfo
rco
mp
arin
gsl
op
efi
eld
s.
Ze
ro/N
oS
lop
e
Det
erm
inin
gw
her
eth
esl
op
eso
fp
oin
tsar
ein
fin
ity
(ver
tica
lan
du
nd
efin
ed)
and
wh
ere
they
are
zero
isu
sefu
lfo
rco
mp
arin
gsl
op
efi
eld
s.
Se
pa
rati
on
of
Va
ria
ble
s
Sep
arat
ion
of
var
iab
les
iso
ne
met
ho
dto
iso
late
var
iab
les
ina
dif
fere
nti
able
equ
atio
n.
Th
ese
par
ated
var
iab
les
can
then
be
inte
gra
ted
.
Ifdy
dx
=g(x
)h(y
),th
endy
h(y
)=
g(x
)dx
.
Bas
ical
ly,
dy
dx
isb
ein
gtr
eate
das
afr
acti
on
,wh
ich
can
be
can
be
sep
arat
ed.
40
Po
we
rR
ule
Th
eru
lefo
rp
rod
uct
sm
any
tim
esre
sult
sin
det
erm
inin
gth
ep
ow
erru
le.
lim
x→
cf(
x)n
=(
lim
x→
cf(
x))
n
Fin
din
gL
imit
s
Ifc
isin
the
do
mai
no
fth
efu
nct
ion
and
the
fun
ctio
nca
nb
eb
uil
to
ut
of
rati
on
al,
trig
on
om
etri
c,lo
gar
ith
mic
and
exp
on
enti
alfu
nct
ion
s,th
enth
eli
mit
issi
mp
lyth
ev
alu
eo
fth
efu
nct
ion
atc.
Ifc
isn
ot
inth
ed
om
ain
of
the
fun
ctio
n,
then
inm
any
case
s(a
sw
ith
rati
on
alfu
nct
ion
s)th
ed
om
ain
of
the
fun
ctio
nin
clu
des
all
of
the
po
ints
nea
rc,
bu
tn
otc.
An
exam
ple
wo
uld
be
ifo
ne
wan
ted
to
fin
dli
mx→
0
x x,
wh
ere
the
do
mai
nin
clu
des
all
real
nu
mb
ers
exce
pt0.
Inth
atca
se,
on
ew
ou
ldw
ant
to
fin
da
sim
ilar
fun
ctio
n,
wit
hth
eh
ole
fill
edin
.T
he
lim
ito
fth
isfu
nct
ion
atc
wil
lb
eth
esa
me,
wh
ile
the
fun
ctio
nis
the
sam
eat
all
po
ints
no
teq
ual
toc.
Th
eli
mit
defi
nit
ion
dep
end
so
nf(
x)
on
lyat
the
po
ints
wh
ere
xis
clo
seto
cb
ut
no
teq
ual
toit
.A
nd
sin
ceth
ed
om
ain
of
the
new
fun
ctio
nin
clu
des
c,
on
eca
nn
ow
(ass
um
ing
it’s
stil
lb
uil
to
ut
of
rati
on
al,
trig
on
om
etri
c,lo
gar
ith
mic
and
exp
on
enti
alfu
nct
ion
s)ju
stev
alu
ate
the
fun
ctio
nat
cas
bef
ore
.
Inth
eab
ov
eex
amp
le,
this
isea
sy;
can
celi
ng
the
x’s
giv
es1,
wh
ich
equ
als
x /x
atal
lp
oin
tsex
cep
t0.
Th
us,
lim
x→
0
x x=
lim
x→
01
=1.
Ing
ener
al,
wh
enco
mp
uti
ng
lim
its
of
rati
on
alfu
nct
ion
s,it
’sa
go
od
idea
to
loo
kfo
rco
mm
on
fact
ors
inth
en
um
erat
or
and
den
om
inat
or.
Do
es
No
tE
xis
t
No
teth
atth
eli
mit
mig
ht
no
tex
ist
atal
l.T
her
ear
ea
nu
mb
ero
fw
ays
inw
hic
hth
isca
no
ccu
r.
No
tS
am
efr
om
Bo
thS
ide
s
Ale
ft-h
and
edli
mit
isd
iffe
ren
tfr
om
the
rig
ht-
han
ded
lim
ito
fth
esa
me
var
iab
le,v
alu
e,an
dfu
nct
ion
.S
ince
,th
ele
ft-h
and
edli
mit6=
rig
ht-
han
ded
lim
it,t
he
lim
itd
oes
no
tex
ist.
Th
isin
clu
des
case
sin
wh
ich
the
lim
ito
fa
cert
ain
sid
ed
oes
no
tex
ist
(e.g
.li
mx→
2
√x
−2,w
hic
hh
asn
ole
ft-h
and
edli
mit
).
Ga
p
Th
ere
isa
gap
(mo
reth
ana
po
int
wid
e)in
the
fun
ctio
nw
her
eth
efu
nct
ion
isn
ot
defi
ned
.A
san
exam
ple
,in
f(x)
=√
x2
−16,f(
x)
do
esn
ot
hav
ean
yli
mit
wh
en−
4≤
x≤
4.
Th
ere
isn
ow
ayto
“ap
pro
ach
”th
em
idd
leo
fth
eg
rap
h.
No
teal
soth
atth
efu
nct
ion
also
has
no
lim
itat
the
end
po
ints
of
the
two
curv
esg
ener
ated
(atx
=−
4an
dx
=4)
sin
celi
mit
sfr
om
bo
thsi
des
do
no
tex
ist.
5
Ju
mp
Ifth
eg
rap
hsu
dd
enly
jum
ps
toa
dif
fere
nt
lev
el,
ther
eis
no
lim
it.
Th
isis
illu
stra
ted
inth
efl
oo
rfu
nct
ion
(in
wh
ich
the
ou
tpu
tv
alu
eis
the
gre
ates
tin
teg
ern
ot
gre
ater
than
the
inp
ut
val
ue)
.T
he
lim
itd
oes
no
tex
ist
wh
enth
eg
reat
est
inte
ger
fun
ctio
nap
pro
ach
esan
inte
ger
(li
mx→
inte
ger⌊x⌋,
also
wri
tten
as
intx
).|x
| /x
and
x /|x
|ar
eo
ther
exam
ple
so
fg
rap
hs
that
con
tain
jum
ps.
Infi
nit
eO
sc
illa
tio
n
Th
isca
nb
etr
ick
yto
vis
ual
ize.
Ag
rap
hco
nti
nu
ally
rise
sab
ov
ean
db
elo
wa
ho
rizo
nta
lli
ne
asit
app
roac
hes
ace
rtai
nx
-val
ue,
for
inst
ance
infi
nit
y.T
his
oft
enm
ean
sth
atth
eli
mit
do
esn
ot
exis
t,as
the
gra
ph
nev
erap
pro
ach
esa
par
ticu
lar
val
ue.
Ho
wev
er,i
fth
eh
eig
ht
(an
dd
epth
)o
fea
cho
scil
lati
on
dim
inis
hes
asth
eg
rap
hap
pro
ach
esth
ex
-val
ue,
soth
atth
eo
scil
lati
on
sg
etar
bit
rari
lysm
alle
r,th
enth
ere
mig
ht
actu
ally
be
ali
mit
.
Th
eu
seo
fo
scil
lati
on
nat
ura
lly
call
sto
min
dth
etr
igo
no
met
ric
fun
ctio
ns.
An
exam
ple
of
atr
igo
no
-
met
ric
fun
ctio
nth
atd
oes
no
th
ave
ali
mit
asx
app
roac
hes
0is
f(x)
=si
n1 x
.A
sx
get
scl
ose
rto
0,
the
fun
ctio
nk
eep
so
scil
lati
ng
bet
wee
n−
1an
d1.
Inc
om
ple
teG
rap
h
Co
nsi
der
the
foll
ow
ing
exam
ple
.
g(x
)=
{2,
ifx
isra
tio
nal
0,
ifx
isir
rati
on
al
g(x
)d
oes
no
th
ave
ali
mit
.F
or
letx
be
are
aln
um
ber
,g(x
)ca
n’t
hav
ea
lim
itat
x.
No
mat
ter
ho
wcl
ose
on
eg
ets
tox
,th
ere
wil
lb
era
tio
nal
nu
mb
ers
(wh
eng(x
)w
ill
be
2)
and
irra
tio
nal
nu
mb
ers
(wh
eng
wil
lb
e0).
Th
us
g(x
)h
asn
oli
mit
atan
yre
aln
um
ber
.
On
e-S
ide
dL
imit
s
So
met
imes
,it
isn
eces
sary
toco
nsi
der
wh
ath
app
ens
wh
eno
ne
app
roac
hes
anx
val
ue
fro
mo
ne
par
-ti
cula
rd
irec
tio
n.
To
acco
mm
od
ate
for
this
,th
ere
are
on
e-si
ded
lim
its.
Ina
left
-han
ded
lim
it,x
ap-
pro
ach
esa
fro
mth
ele
fth
and
sid
e(n
egat
ive)
.L
ikew
ise,
ina
rig
ht-
han
ded
lim
it,x
app
roac
hes
afr
om
the
rig
ht
han
dsi
de
(po
siti
ve)
.
Fo
rex
amp
le,
lim
x→
2
√x
−2
do
esn
ot
exis
tb
ecau
seth
ere
isn
ole
ft-h
and
edli
mit
.
Th
ele
ft-h
and
edli
mit
,wh
ich
do
esn
ot
exis
t,is
exp
ress
edas
the
foll
ow
ing
.
lim
x→
2−
√x
−2
Th
eri
gh
t-h
and
edli
mit
,wh
ich
equ
als
0,i
sex
pre
ssed
asth
efo
llo
win
g.
lim
x→
2+
√x
−2
=0
6
Me
an
Va
lue
of
De
fin
ite
Inte
gra
ls
Me
an
Va
lue
Th
eav
erag
e(a
rith
met
icm
ean
)y
-val
ue
of
afu
nct
ion
ov
eran
inte
rval
isth
ein
teg
ral
ov
erth
ein
terv
ald
ivid
edb
yth
ele
ng
tho
fth
ein
terv
al.
f avg
=
∫b a[f
(x)d
x]
b−
a
Me
an
Va
lue
Th
eo
rem
Iff
isco
nti
nu
ou
so
nth
ecl
ose
din
terv
al[a
,b],
then
atso
me
po
intc
in[a
,b]
ther
eex
ists
the
foll
ow
ing
:
f(c)
=
∫b a[f
(x)d
x]
b−
a
Init
ial
Va
lue
Pro
ble
ms
Intr
od
uc
tio
n
An
equ
atio
nth
atco
nta
ins
ad
eriv
ativ
eis
call
eda
dif
fere
nti
aleq
uat
ion
.F
or
exam
ple
,dy
dx
=2x
isa
dif
fere
nti
aleq
uat
ion
.E
ver
yd
iffe
ren
tial
equ
atio
no
fa
fun
ctio
nco
rres
po
nd
sto
asp
ecifi
ceq
uat
ion
ata
par
ticu
lar
po
int
(ref
erre
dto
asa
par
ticu
lar
solu
tio
n),
assu
min
gth
ep
oin
tis
inth
efu
nct
ion
’sd
om
ain
.
An
init
ial
val
ue
pro
ble
mp
rov
ides
ad
iffe
ren
tial
equ
atio
nan
da
par
ticu
lar
po
int
thro
ug
hw
hic
hth
efu
nct
ion
pas
ses
thro
ug
h.
Th
esp
ecifi
ceq
uat
ion
isd
eter
min
edb
yca
lcu
lati
ng
the
val
ue
ofC
.
Ex
am
ple
dy
dx
=2x
,y(1
)=
6
∫[
dy
dx
]
=
∫
[2xdx]
y=
x2
+C
6=
(1)2
+C
6=
1+
C
C=
5
y=
x2
+5
39
ax
Ru
le
∫
[ axdx]=
ax
lna
+C
Tri
go
no
me
try
•in
teg
rati
ng
the
der
ivat
ives
of
the
six
trig
on
om
etri
cfu
nct
ion
s
•in
teg
rati
ng
the
der
ivat
ives
of
the
inv
erse
trig
on
om
etri
cfu
nct
ion
s
See
the
the
trig
on
om
etri
cse
ctio
no
fth
ed
eriv
ativ
esch
apte
ro
np
age
20fo
rm
ore
info
rmat
ion
.
Co
ns
tan
t
Ifth
eco
nst
ant
iso
uts
ide
the
trig
on
om
etri
cfu
nct
ion
,u
seth
eco
nst
ant
mu
ltip
lier
rule
(Sec
tio
n).
Ifth
eco
nst
ant
isin
sid
eth
etr
igo
no
met
ric
fun
ctio
n,u
seth
efo
llo
win
gru
le.
∫
[(tr
igkx)d
x]=
(∫[t
rig]k
x)
k+
C
wh
ere
kis
aco
nst
ant.
De
fin
ite
Inte
gra
ls
Ad
dit
ivit
yR
ule
Th
ear
eau
nd
erth
eg
rap
ho
ff(
x)
bet
wee
na
and
bis
the
area
bet
wee
na
and
cp
lus
the
area
bet
wee
nc
and
b.
∫b a[f
(x)d
x]=
∫c a[f
(x)d
x]+
∫b c[f
(x)d
x]
Ze
roR
ule
∫a a[f
(x)d
x]=
0
Ord
er
of
Inte
gra
tio
nR
ule
∫a b[f
(x)d
x]=
−
∫b a[f
(x)d
x]
38
Infi
nit
eL
imit
s
Lim
its
can
also
inv
olv
elo
ok
ing
atw
hat
hap
pen
sto
f(x)
asx
get
sv
ery
big
.F
or
exam
ple
,co
nsi
der
the
fun
ctio
nf(
x)
=1 x
.A
sx
bec
om
esv
ery
big
,1 x
bec
om
escl
ose
rto
zero
.W
ith
ou
tli
mit
sit
isv
ery
dif
ficu
ltto
talk
abo
ut
this
fact
,b
ecau
se1 x
nev
erac
tual
lyb
eco
mes
zero
.B
ut
the
lan
gu
age
of
lim
its
exis
tsp
reci
sely
tole
to
ne
talk
abo
ut
the
beh
avio
ro
fa
fun
ctio
nas
itap
pro
ach
esso
met
hin
g,
wit
ho
ut
cari
ng
abo
ut
the
fact
that
itw
ill
nev
erg
etth
ere.
Inth
isca
se,
ho
wev
er,
the
sam
ep
rob
lem
asb
efo
reex
ists
;ho
wb
igd
oes
xh
ave
tob
eto
be
sure
that
f(x)
isre
ally
go
ing
tow
ard
s0?
Inth
isca
se,
the
big
ger
xg
ets,
the
clo
ser
f(x)
sho
uld
get
to0.
Rea
lly,
this
mea
ns
that
ho
wev
ercl
ose
on
ew
ants
f(x)
tog
etto
0,o
ne
can
fin
dan
xb
igen
ou
gh
sof(
x)
isth
atcl
ose
.T
his
isw
ritt
enin
asi
mil
arw
ayto
fin
ite
lim
its
and
isre
adas
“th
eli
mit
,as
xap
pro
ach
esin
fin
ity,
equ
als
0,”
or
“as
xap
pro
ach
esin
fin
ity,
the
fun
ctio
nap
pro
ach
es0.”
lim
x→
∞
1 x=
0
Ru
les
Th
eea
sies
tw
ayto
det
erm
ine
lim
its
asx
app
roac
hes
±∞
isb
yu
sin
gth
eg
rap
hin
gca
lcu
lato
rto
mak
eo
bse
rvat
ion
s,o
rb
yp
lug
gin
gin
hig
hv
alu
eso
fp
osi
tiv
ean
dn
egat
ive
nu
mb
ers
ina
calc
ula
tor.
Ho
wev
er,
ther
ear
eth
ree
rule
sfo
rd
eter
min
ing
ali
mit
of
afr
acti
on
anal
yti
call
yas
av
aria
ble
ap-
pro
ach
esin
fin
ity.
Fo
rea
chru
le,o
ne
mu
stlo
ok
atth
ev
aria
ble
so
nb
oth
the
nu
mer
ato
ran
dd
eno
min
a-to
ro
fth
efu
nct
ion
.
Lo
ok
for
the
hig
hes
tte
rm(w
ith
the
hig
hes
tex
po
nen
t)in
the
nu
mer
ato
r.L
oo
kfo
rth
esa
me
inth
ed
eno
min
ato
r.T
hes
eru
les
are
bas
edo
nth
atin
form
atio
n.
Fo
rli
mit
sas
the
var
iab
leap
pro
ach
esin
fin
ity
:
•If
the
exp
on
ent
of
the
hig
hes
tte
rmin
the
nu
mer
ato
rm
atch
esth
eex
po
nen
to
fth
eh
igh
est
term
inth
ed
eno
min
ato
r,th
eli
mit
isth
efr
acti
on
alra
tio
of
the
coef
fici
ents
of
the
hig
hes
tte
rms.
•If
the
nu
mer
ator
has
the
hig
hes
tte
rm,
then
the
frac
tio
nis
call
ed“t
op
hea
vy
”an
dth
eli
mit
isin
fin
ity.
•If
the
den
omin
ator
has
the
hig
hes
tte
rm,
then
the
frac
tio
nis
call
ed“b
ott
om
hea
vy
”an
dth
eli
mit
isze
ro.
Ifth
ere
isn
od
eno
min
ato
rst
ated
,it
isu
nd
erst
oo
dth
atth
ed
eno
min
ato
ris
1o
r1n
0,a
nd
the
lim
itw
ill
be
infi
nit
y.
As
ym
pto
tes
Ali
nea
ras
ym
pto
teis
ast
raig
ht
lin
eth
ata
gra
ph
app
roac
hes
,b
ut
do
esn
ot
bec
om
eid
enti
cal
to.
Asy
mp
tote
sar
efo
rmal
lyd
efin
edu
sin
gli
mit
s.
7
Ve
rtic
al
As
ym
pto
tes
Th
eli
ne
x=
ais
av
erti
cal
asy
mp
tote
for
the
fun
ctio
nf(
x)
ifat
leas
to
ne
of
the
foll
ow
ing
stat
emen
tsis
tru
e.
1.li
mx→
af(
x)
=±
∞
2.li
mx→
a−
f(x)
=±
∞
3.li
mx→
a+
f(x)
=±
∞
Th
eli
mit
sfr
om
bo
thd
irec
tio
ns
do
no
th
ave
tob
eeq
ual
toh
ave
anas
ym
pto
te,b
ut
they
may
be
equ
al.
Ess
enti
ally
,a
ver
tica
las
ym
pto
teo
ccu
rsw
her
eth
eth
ev
alu
eo
fa
lim
itis
po
siti
ve
or
neg
ativ
ein
fin
ity
fro
man
yd
irec
tio
n.
Rec
all
that
this
occ
urs
wh
ere
the
frac
tio
no
fa
fun
ctio
nis
un
defi
ned
(den
om
inat
or
equ
als
zero
).
Re
mo
va
ble
Dis
co
nti
nu
itie
s
Th
efu
nct
ion
f(x)
=x
2−
9x−
3is
con
sid
ered
toh
ave
are
mo
vab
led
isco
nti
nu
ity
atx
=3.
Itis
dis
con
tin
uo
us
atth
atp
oin
tb
ecau
seth
efr
acti
on
then
bec
om
es0 0
wh
ich
isu
nd
efin
ed.
Sta
nd
ard
alg
ebra
icte
chn
iqu
esfo
rsi
mp
lify
ing
frac
tio
ns
and
alg
ebra
icex
pre
ssio
ns
(i.e
.fa
cto
rin
g,m
ul-
tip
lyin
gb
yco
nju
gat
es)
can
be
use
dto
elim
inat
eth
ed
isco
nti
nu
ity.
f(x)
=x2
−9
x−
3=
(x+
3)(
x−
3)
(x−
3)
=x
+3
1·x
−3
x−
3=
x+
3
1·1
=x
+3
Ho
wev
er,
the
fun
ctio
nis
no
tre
ally
con
tin
uo
us,
and
ano
pen
circ
lem
ust
be
left
inth
eg
rap
hat
the
rem
ov
able
dis
con
tin
uit
y.
Ho
rizo
nta
lA
sy
mp
tote
s
Th
eli
ne
y=
ais
ah
ori
zon
tal
asy
mp
tote
for
the
fun
ctio
nf(
x)
ifli
mx→
∞f(
x)
=a
or
lim
x→
−∞
f(x)
=a
.
Ifli
mx→
∞f(
x)
=a
and
lim
x→
−∞
f(x)
=b
,th
enth
efu
nct
ion
f(x)
has
two
asy
mp
tote
sat
y=
aan
dy
=b
.
No
teth
atin
som
efu
nct
ion
s,th
eg
rap
hm
ayp
ass
thro
ug
hth
eh
ori
zon
tal
asy
mp
tote
atan
xv
alu
eo
fze
ro.
Ess
enti
ally
,a
ho
rizo
nta
las
ym
pto
teo
ccu
rsat
the
val
ue
of
ali
mit
wh
ere
xap
pro
ach
esp
osi
tiv
eo
rn
egat
ive
infi
nit
y.
Rec
all
that
rule
sex
ist
for
calc
ula
tin
gth
eth
ev
alu
eo
fa
lim
itw
her
ex
app
roac
hes
po
siti
ve
or
neg
ativ
ein
fin
ity.
Ru
les
Th
eea
sies
tw
ayto
det
erm
ine
lim
its
asx
app
roac
hes
±∞
isb
yu
sin
gth
eg
rap
hin
gca
lcu
lato
rto
mak
eo
bse
rvat
ion
s,o
rb
yp
lug
gin
gin
hig
hv
alu
eso
fp
osi
tiv
ean
dn
egat
ive
nu
mb
ers
ina
calc
ula
tor.
8
Co
ns
tan
tM
ult
ipli
er
Ru
le
∫
[c×
f(x)d
x]=
c
∫
[f(x
)dx]
∫b a[c×
f(x)d
x]=
c
∫b a[f
(x)d
x]
Po
we
rR
ule
∫
[xndx]=
xn
+1
n+
1+
C
∫b a[x
ndx]=
bn
+1
−a
n+
1
n+
1
wh
ere
nis
aco
nst
ant
exp
on
ent
no
teq
ual
to−
1an
dx6=
0.
Ex
pre
ssio
ns
con
tain
ing
roo
ts(i
.e.
squ
are
roo
ts)
can
be
intr
egra
ted
by
usi
ng
afr
acti
on
alv
alu
efo
rn
(b√
xa
=x
a/
b).
Ex
pre
ssio
ns
con
tain
ing
alg
ebra
icm
on
om
ials
inth
ed
eno
min
ato
ro
fa
frac
tio
nca
nb
e
inte
gra
ted
by
inv
erti
ng
the
sig
no
fn
(1 xn
=x
−n
).
Lo
ga
rith
ms
1 xR
ule
∫[
dx x
]
=ln
|x|+
C
∫b a
[
dx x
]
=ln
|b|−
ln|a
|
wh
ere
x6=
0.
ex
Ru
le
∫[
ekxdx]
=ekx
k+
C
∫b a
[
ekxdx]
=ekb
k−
eka
k
wh
ere
kis
aco
nst
ant.
37
Co
roll
ary
Inte
gra
tio
nan
dd
iffe
ren
tiat
ion
are
inv
erse
so
fea
cho
ther
.
Iff
isco
nti
nu
ou
so
nth
ecl
ose
din
terv
al[a
,b]
then
:
d dx
[∫
x a[f
(t)d
t]
]
=f(
x)
d du
[∫
u a[f
(t)dt]
]
=f(
u)d
u
Inte
gra
lR
ule
s
Ru
les
for
calc
ula
tin
gth
ein
teg
rals
of
gen
eral
fun
ctio
ns
hav
eb
een
dev
elo
ped
.A
sa
resu
lt,i
tis
po
ssib
leto
calc
ula
teth
ein
teg
rals
of
aw
ide
var
iety
of
fun
ctio
ns.
Inm
any
case
sth
eu
seo
fm
ult
iple
rule
sar
ere
qu
ired
.
Inth
efo
llo
win
gru
les,
Cre
pre
sen
tsth
eco
nst
ant
of
inte
gra
tio
n.
Co
ns
tan
tF
un
cti
on
Th
ed
efin
ite
inte
gra
lo
fa
con
stan
tfu
nct
ion
isa
rect
ang
lew
ith
the
hei
gh
tb
ein
gth
eco
nst
ant
and
the
wid
thb
ein
gth
ein
terv
alo
fin
teg
rati
on
.
∫
[cdx]=
cx
+C
∫b a[c
dx]=
c(b
−a)
wh
ere
cis
aco
nst
ant.
Ad
dit
ion
/Su
btr
ac
tio
nR
ule
Iff(
x)
and
g(x
)ar
eco
nti
nu
ou
so
nth
ecl
ose
din
terv
al[a
,b],
then
:
∫
[(f(
x)±
g(x
))dx]=
∫
[f(x
)dx]±
∫
[g(x
)dx]+
C
∫b a[(
f(x)±
g(x
))dx]=
∫b a[f
(x)d
x]±
∫b a[g
(x)d
x]
As
are
sult
,o
ne
can
tak
ean
equ
atio
n,
bre
akit
up
into
term
s,fi
gu
reo
ut
the
defi
nit
ein
teg
rals
ind
ivid
-u
ally
,an
db
uil
dth
ean
swer
bac
ku
p.
36
Ho
wev
er,
ther
ear
eth
ree
rule
sfo
rd
eter
min
ing
ali
mit
of
afr
acti
on
anal
yti
call
yas
av
aria
ble
ap-
pro
ach
esin
fin
ity.
Fo
rea
chru
le,o
ne
mu
stlo
ok
atth
ev
aria
ble
so
nb
oth
the
nu
mer
ato
ran
dd
eno
min
a-to
ro
fth
efu
nct
ion
.
Lo
ok
for
the
hig
hes
tte
rm(w
ith
the
hig
hes
tex
po
nen
t)in
the
nu
mer
ato
r.L
oo
kfo
rth
esa
me
inth
ed
eno
min
ato
r.T
hes
eru
les
are
bas
edo
nth
atin
form
atio
n.
•If
the
exp
on
ent
of
the
hig
hes
tte
rmin
the
nu
mer
ato
rm
atch
esth
eex
po
nen
to
fth
eh
igh
est
term
inth
ed
eno
min
ato
r,th
eli
mit
isth
efr
acti
on
alra
tio
of
the
coef
fici
ents
of
the
hig
hes
tte
rms.
•If
the
nu
mer
ator
has
the
hig
hes
tte
rm,
then
the
frac
tio
nis
call
ed“t
op
hea
vy
”an
dth
eli
mit
isin
fin
ity.
•If
the
den
omin
ator
has
the
hig
hes
tte
rm,
then
the
frac
tio
nis
call
ed“b
ott
om
hea
vy
”an
dth
eli
mit
isze
ro.
Ifth
ere
isn
od
eno
min
ato
rst
ated
,it
isu
nd
erst
oo
dth
atth
ed
eno
min
ato
ris
1o
r1n
0,a
nd
the
lim
itw
ill
be
infi
nit
y.
Sk
etc
hin
gw
ith
As
ym
pto
tes
Ase
ries
of
step
sca
nb
eta
ken
tosk
etch
wit
has
ym
pto
tes.
As
are
sult
,cu
rves
may
be
sket
ched
wit
ho
ut
ag
rap
hin
gca
lcu
lato
r.
1.F
ind
the
x-i
nte
rcep
tb
yse
ttin
gy
equ
alto
zero
.
2.F
ind
the
y-i
nte
rcep
tb
yse
ttin
gx
equ
alto
zero
.
3.F
ind
the
ho
rizo
nta
las
ym
pto
te(s
).
4.F
ind
the
ver
tica
las
ym
pto
tes(
s).
5.P
lot
the
x-i
nte
rcep
tan
dy
-in
terc
ept.
6.S
ket
chth
eas
ym
pto
te(s
).
7.F
ind
the
lim
its
of
bo
thsi
des
of
the
ver
tica
las
ym
pto
teb
yu
sin
gte
stp
oin
ts.
8.S
ket
chth
ecu
rve
usi
ng
the
det
erm
ined
info
rmat
ion
and
the
sket
ched
asy
mp
tote
s.
Inso
me
pro
ble
ms
on
lyli
mit
sw
ill
be
pro
vid
ed.
Fro
mth
ese
lim
its
ho
rizo
nta
lan
dv
erti
cal
asy
mp
-to
tes
can
be
det
erm
ined
.W
hil
eth
ex
-in
terc
ept
and
y-i
nte
rcep
tar
en
ot
pro
vid
ed,
itis
stil
lp
oss
ible
tosk
etch
the
gra
ph
.T
he
sket
chw
ill
be
less
accu
rate
,b
ut
that
isac
cep
tab
lew
hen
pro
vid
edw
ith
lim
ited
info
rmat
ion
.
Co
nti
nu
ity
De
fin
itio
n
Th
efo
rmal
defi
nit
ion
of
con
tin
uit
yis
sim
ple
.
Iff(
x)
isd
efin
edo
nan
op
enin
terv
alco
nta
inin
gc,t
hen
f(x)
issa
idto
be
con
tin
uo
us
atc
ifan
do
nly
ifth
eli
mit
asx
app
roac
hes
ceq
ual
sf(
c).
lim
x→
cf(
x)
=f(
c)
9
No
teth
atfo
rf(
x)
tob
eco
nti
nu
ou
sat
c,t
he
defi
nit
ion
req
uir
esth
ree
con
dit
ion
s.
1.f(
x)
isd
efin
edat
c
a)f(
c)
exis
ts
2.T
he
lim
itas
xap
pro
ach
esc
exis
ts.
a)li
mx→
cf(
x)
exis
ts
3.T
he
lim
itan
df(
c)
are
equ
al.
a)f(
c)
=li
mx→
cf(
x)
Ifan
yo
fth
ese
do
no
th
old
then
f(x)
isn
ot
con
tin
uo
us
atc.
No
tice
ho
wth
isre
late
sto
the
idea
of
con
tin
uit
y.T
ob
eco
nti
nu
ou
s,th
efu
nct
ion
mu
stb
eu
nif
orm
ly“s
mo
oth
”(e
.g.
no
“gap
s,”
bre
aks,
or
shar
ptu
rns/
corn
ers)
wit
hin
anin
terv
al.
Afu
nct
ion
issa
idto
be
con
tin
uo
us
ifit
isco
nti
nu
ou
sat
ever
yp
oin
tc
init
sd
om
ain
.
Afu
nct
ion
may
be
con
tin
uo
us
ata
cert
ain
po
int,
bu
tn
ot
aco
nti
nu
ou
sfu
nct
ion
(th
rou
gh
ou
t).
Lik
e-w
ise,
ad
isco
nti
nu
ou
sfu
nct
ion
may
be
con
tin
uo
us
ata
cert
ain
po
int.
Re
mo
va
ble
Dis
co
nti
nu
itie
s
dis
co
nti
nu
ity
po
int
wh
ere
afu
nct
ion
isn
ot
con
tin
uo
us
Ifth
ere
isa
“gap
”o
ne
po
int
wid
eo
na
gra
ph
(f(c
)d
oes
no
tex
ist)
or
ifth
ere
isa
“ju
mp
”o
ne
po
int
wid
eo
na
gra
ph
(f(c
)6=
lim
x→
cf(
x))
,th
ed
isco
nti
nu
ity
isre
mo
vab
le.
Gap
dis
con
tin
uit
ies
(lim
x→
cf(
x)
do
esn
ot
ex-
ist)
,ju
mp
dis
con
tin
uit
ies
(f(c
)6=
lim
x→
cf(
x))
,an
din
fin
ite
osc
illa
tio
nd
isco
nti
nu
itie
sar
en
on
-rem
ov
able
.
Th
efu
nct
ion
f(x)
=x
2−
9x−
3is
con
sid
ered
toh
ave
are
mo
vab
led
isco
nti
nu
ity
atx
=3.
Itis
dis
con
tin
uo
us
atth
atp
oin
tb
ecau
seth
efr
acti
on
then
bec
om
es0 0
wh
ich
isu
nd
efin
ed.
Th
eref
ore
the
fun
ctio
nfa
ils
the
ver
yfi
rst
con
dit
ion
of
con
tin
uit
y.
Ifth
efu
nct
ion
issl
igh
tly
mo
difi
ed,
the
dis
con
tin
uit
yca
nb
ere
mo
ved
and
the
fun
ctio
nb
eco
mes
con
-ti
nu
ou
s.S
tan
dar
dal
geb
raic
tech
niq
ues
for
sim
pli
fyin
gfr
acti
on
san
dal
geb
raic
exp
ress
ion
s(e
.g.
fac-
tori
ng
,mu
ltip
lyin
gb
yco
nju
gat
es)
can
be
use
d.
To
mak
eth
efu
nct
ion
f(x)
con
tin
uo
us,
f(x)
mu
stb
esi
mp
lifi
ed.
f(x)
=x2
−9
x−
3=
(x+
3)(
x−
3)
(x−
3)
=x
+3
1×
x−
3
x−
3=
x+
3
1×
1=
x+
3
As
lon
gas
x6=
3,t
he
fun
ctio
nf(
x)
can
be
sim
pli
fied
tog
eta
new
fun
ctio
ng(x
).
g(x
)=
{x
+3,
ifx6=
3
6,
ifx
=3
No
teth
atth
efu
nct
ion
g(x
)is
no
tth
esa
me
asth
eo
rig
inal
fun
ctio
nf(
x),
bec
ause
g(x
)h
asth
eex
tra
po
int(3
,6).
g(x
)is
no
wd
efin
edfo
rx
=3,a
nd
ther
efo
reco
nti
nu
ou
s.
10
Gra
ph
ing
Ca
lcu
lato
r
Th
ese
inst
ruct
ion
sar
ed
esig
ned
for
aT
I-84
Plu
sca
lcu
lato
r,b
ut
they
may
use
do
no
ther
Tex
asIn
stru
-m
ents
gra
ph
ing
calc
ula
tors
,th
ou
gh
slig
ht
mo
difi
cati
on
may
be
nec
essa
ry.
Un
less
oth
erw
ise
spec
ified
,th
eg
rap
hin
gca
lcu
lato
rsh
ou
ldb
ein
rad
ian
mo
de.
De
fin
ite
Inte
gra
lR
ec
tan
gu
lar
Ap
pro
xim
ati
on
s
Inso
me
case
sit
may
be
easi
ero
rre
qu
ired
toca
lcu
late
rect
ang
ula
rap
pro
xim
atio
ns
of
defi
nit
ein
teg
rals
usi
ng
the
gra
ph
ing
calc
ula
tor,
esp
ecia
lly
wh
enu
sin
ga
larg
en
um
ber
of
rect
ang
les.
Th
ep
rog
ramRAM
mu
stb
ead
ded
toth
eca
lcu
lato
r’s
mem
ory
.O
nce
inst
alle
d,
set
the
y1
of
the
calc
u-
lato
r’s
gra
ph
toth
efu
nct
ion
bei
ng
inte
gra
ted
and
run
the
pro
gra
mw
ithPRGM−→
RAM
.
De
fin
ite
Inte
gra
lC
alc
ula
tio
ns
Inso
me
case
sit
may
be
easi
ero
rre
qu
ired
toca
lcu
late
defi
nit
ein
teg
rals
usi
ng
the
gra
ph
ing
calc
ula
tor,
esp
ecia
lly
wh
enth
efu
nct
ion
isto
oco
mp
lex
.It
can
also
be
use
dto
chec
ko
ne’
san
swer
.
Fu
nd
am
en
tal
Th
eo
rem
of
Ca
lcu
lus
Ev
ery
con
tin
uo
us
fun
ctio
nh
asan
anti
der
ivat
ive.
Pa
rtI
Iff
isco
nti
nu
ou
so
nth
ecl
ose
din
terv
al[a
,b]
and
F(x
)=
∫x a[f
(t)d
t]o
nth
ecl
ose
din
terv
al[a
,b],
then
Fis
dif
fere
nti
able
on
the
op
enin
terv
al(a
,b)
and
F′ (
x)
=f(
x)
for
allx
inth
eo
pen
inte
rval
(a,b
).
By
defi
nit
ion
F(x
)is
the
anti
der
ivat
ive
off(
x)
inth
eo
pen
inte
rval
(a,b
).
Pa
rtII
Iff
isco
nti
nu
ou
so
nth
ecl
ose
din
terv
al[a
,b]
and
Fis
anan
tid
eriv
ativ
eo
ff,
then
:
∫b a[f
(x)d
x]=
F(b
)−
F(a
)
Itis
ther
efo
rep
oss
ible
toca
lcu
late
ad
efin
ite
inte
gra
lu
sin
gru
les
for
anti
der
ivat
ives
(in
defi
nit
ein
te-
gra
ls).
35
Re
cta
ng
ula
rA
pp
rox
ima
tio
nM
eth
od
Rec
tan
gu
lar
Ap
pro
xim
atio
nM
eth
od
(RA
M)
isa
met
ho
do
fes
tim
atin
gd
efin
ite
inte
gra
lsb
yca
lcu
lati
ng
the
area
of
ace
rtai
nn
um
ber
of
rect
ang
les.
Ala
rger
nu
mb
ero
fre
ctan
gle
sw
ill
giv
ea
mo
reac
cura
tees
tim
ate.
Le
ftR
ec
tan
gu
lar
Ap
pro
xim
ati
on
Me
tho
d(L
RA
M)
∫b a[f
(x)d
x]≈
∆x(f
(a)+
f(a
+∆
x)+···+
f(b
−2∆
x)+
f(b
−∆
x))
wh
ere
∆x
isth
ew
idth
of
the
rect
ang
les
(b−
an
)an
dn
isth
en
um
ber
of
rect
ang
les.
Rig
ht
Re
cta
ng
ula
rA
pp
rox
ima
tio
nM
eth
od
(RR
AM
)
∫b a[f
(x)d
x]≈
∆x(f
(a+
∆x)+
f(a
+2∆
x)+···+
f(b
−∆
x)+
f(b))
wh
ere
∆x
isth
ew
idth
of
the
rect
ang
les
(b−
an
)an
dn
isth
en
um
ber
of
rect
ang
les.
Mid
po
int
Re
cta
ng
ula
rA
pp
rox
ima
tio
nM
eth
od
(MR
AM
)
∫b a[f
(x)d
x]≈
∆x(f
(a+
∆x 2)+
f(a
+∆
x)+···+
f(b
−∆
x)+
f(b
−∆
x 2))
wh
ere
∆x
isth
ew
idth
of
the
rect
ang
les
(b−
an
)an
dn
isth
en
um
ber
of
rect
ang
les.
Tra
pe
zo
ida
lA
pp
rox
ima
tio
nM
eth
od
∫b a[f
(x)d
x]≈(
1 2
)
(∆x)( f
(a)+
2f(
a+
∆x)+···+
2f(
b−
∆x)+
f(b))
wh
ere
∆x
isth
ew
idth
of
the
trap
ezo
ids
(b−
an
)an
dn
isth
en
um
ber
of
trap
ezo
ids.
An
inte
gra
lap
pro
xim
ated
wit
hth
isru
leo
na
con
cav
e-u
pfu
nct
ion
wil
lb
ean
ov
eres
tim
ate
bec
ause
the
trap
ezo
ids
incl
ud
eal
lo
fth
ear
eau
nd
erth
ecu
rve
and
exte
nd
ov
erit
.U
sin
gth
ism
eth
od
on
aco
nca
ve-
do
wn
fun
ctio
ny
ield
san
un
der
esti
mat
eb
ecau
sear
eais
un
acco
un
ted
for
un
der
the
curv
e,b
ut
no
ne
isco
un
ted
abo
ve.
34
Pro
pe
rtie
s
Iff(
x)
and
g(x
)ar
eco
nti
nu
ou
s,th
enth
efo
llo
win
gar
eal
soco
nti
nu
ou
s:
•f(
x)+
g(x
)
•f(
x)·g
(x)
•f(
x)−
g(x
)
•f(
x)
g(x
),g
6=0
•k×
f(x),
wh
ere
kis
aco
nst
ant
Inte
rme
dia
teV
alu
eT
he
ore
m
Ag
rap
ho
fa
con
tin
uo
us
fun
ctio
nh
asn
ob
reak
s,so
ap
oin
tb
etw
een
two
x-v
alu
esh
asa
y-v
alu
eb
etw
een
the
y-v
alu
eso
fth
ere
spec
tiv
ex
-val
ues
.
Ifa
fun
ctio
nis
con
tin
uo
us
on
the
clo
sed
inte
rval
[a,b
],th
enfo
rev
ery
val
ue
kb
etw
een
f(a)
and
f(b)
ther
eis
av
alu
ec
on
[a,b
]su
chth
atf(
c)
=k
.
Th
isca
nb
eu
sed
toap
pro
xim
ate
wh
enth
ey
-val
ue
of
afu
nct
ion
wil
lb
ea
cert
ain
val
ue
(e.g
.th
ex
-val
ue
wh
eny
=4).
Ca
lcu
lati
ng
Co
nti
nu
itie
s
On
esh
ou
ldb
eab
leto
det
erm
ine
wh
ere
afu
nct
ion
isd
isco
nti
nu
ou
s.In
som
eca
ses,
on
em
ayb
ere
qu
ired
tod
eter
min
eth
ev
alu
e(s)
of
var
iab
le(s
)in
rule
(s)
of
afu
nct
ion
soth
atth
efu
nct
ion
wil
lb
eco
nti
nu
ou
s.A
syst
emo
feq
uat
ion
sis
req
uir
edw
hen
ther
ear
em
ult
iple
var
iab
les.
Tri
go
no
me
tric
Fu
nc
tio
ns
Inm
ost
case
s,li
mit
sw
ith
trig
on
om
etri
cfu
nct
ion
sca
nb
etr
eate
dth
esa
me
way
aso
ther
lim
its.
On
eca
nsu
bst
itu
tein
toth
eex
pre
ssio
nif
po
ssib
le,o
ru
seth
eg
rap
hin
gca
lcu
lato
r.
Ifd
ivid
eb
yze
roo
ccu
rs,
on
em
ayel
imin
ate
rem
ov
able
dis
con
tin
uit
ies
ifth
eyex
ist
or
use
the
gra
ph
-in
gca
lcu
lato
r.In
som
eca
ses,
fact
ori
ng
toel
imin
ate
rem
ov
able
dis
con
tin
uit
ies
can
on
lyb
ed
on
eif
trig
on
om
etri
cid
enti
ties
are
use
dfi
rst.
No
teW
hen
gra
ph
ing
,st
ayin
rad
ian
mo
de
asth
eli
mit
sar
ep
rov
ided
inra
dia
nm
od
eu
nle
ssst
ated
oth
erw
ise.
Tri
go
no
me
tric
Ide
nti
tie
s
Tri
go
no
met
ric
iden
titi
es(p
age
42)
can
be
use
dto
sim
pli
fyex
pre
ssio
ns
bef
ore
or
afte
rfi
nd
ing
ali
mit
.
11
Ad
de
nd
um
Th
isse
ctio
nw
asd
esig
ned
for
ate
sto
nli
mit
sad
min
iste
red
by
Jon
ath
anC
her
nic
kto
his
AP
2C
alcu
lus
BC
clas
so
nS
epte
mb
er18
,200
8.It
isn
ot
cov
ered
inM
ath
12H
/4H
.
Fu
rth
er
Tri
go
no
me
tric
Ide
nti
tie
s
Th
ese
iden
titi
esca
nb
eu
sed
for
the
sam
ep
urp
ose
asth
eo
ther
trig
on
om
etri
cid
enti
ties
.T
ou
seth
ese
iden
titi
es,
the
lim
its
may
nee
dto
be
mu
ltip
lied
by
ace
rtai
nfa
cto
ro
rse
par
ated
bas
edo
nth
eru
les
on
pag
e3.
Sin
e
lim
x→
0
sin
x
x=
1
Co
sin
e
lim
x→
0
1−
cosx
x=
0
Sq
ue
eze
(Sa
nd
wic
h)
Th
eo
rem
Th
esq
uee
zeth
eore
m,
also
kn
ow
nas
the
san
dw
ich
theo
rem
,is
use
dto
fin
dth
eli
mit
of
afu
nct
ion
by
com
par
iso
nw
ith
two
oth
erfu
nct
ion
sw
ho
seli
mit
sar
ek
no
wn
or
easi
lyco
mp
ute
d.
Itre
fers
toa
fun
ctio
nf(
x)
wh
ose
val
ues
are
squ
eeze
db
etw
een
the
val
ues
of
two
oth
erfu
nct
ion
sg(x
)an
dh(x
),b
oth
of
wh
ich
hav
eth
esa
me
lim
itL
.If
the
val
ue
off(
x)
istr
app
edb
etw
een
the
val
ues
of
the
two
fun
ctio
ns
g(x
)an
dh(x
),th
ev
alu
eso
ff(
x)
mu
stal
soap
pro
ach
L.
Ifth
efo
llo
win
gar
etr
ue:
1.g(x
)≤
f(x)≤
h(x
)fo
ral
lx
no
teq
ual
toc
2.li
mx→
cg(x
)=
lim
x→
ch(x
)=
L
Th
enli
mx→
cf(
x)
=L
.
Ex
am
ple
:
lim
x→
0x
sin
1 x
2A
Pis
are
gis
tere
dtr
adem
ark
of
the
Co
lleg
eB
oar
d,
wh
ich
was
no
tin
vo
lved
inth
ep
rod
uct
ion
of,
and
do
esn
ot
end
ors
e,th
isp
rod
uct
.
12
Inte
gra
ls
Th
isch
apte
rw
asd
esig
ned
for
ate
sto
nin
teg
rals
adm
inis
tere
db
yJo
nat
han
Ch
ern
ick
toh
isA
P5
Cal
-cu
lus
BC
clas
so
nN
ov
emb
er26
,200
8.It
isn
ot
cov
ered
inM
ath
12H
/4H
.
De
fin
ite
Inte
gra
ls
De
fin
itio
n
de
fin
ite
inte
gra
lar
eab
etw
een
acu
rve
and
the
x-a
xis
(are
au
nd
ern
eath
the
x-a
xis
isn
egat
ive)
Afi
nit
en
um
ber
of
rect
ang
les
can
be
use
dto
esti
mat
eth
isar
ea.
Ala
rger
nu
mb
ero
fre
ctan
gle
sw
ill
giv
ea
mo
reac
cura
tees
tim
ate,
and
anin
fin
ite
nu
mb
ero
fre
ctan
gle
sca
ng
ive
anex
act
answ
er.
∫b a[f
(x)d
x]≈
Ak
=
n ∑ k=
1
ak
=a
1+
a2
+···+
an
−1
+a
n
Rie
ma
nn
Su
ms
Th
isar
eaca
nb
eex
pre
ssed
asth
ein
fin
ite
lim
ito
fR
iem
ann
sum
s.A
sn
get
sla
rger
the
wid
tho
fth
ere
ctan
gle
sg
ets
smal
ler
and
wh
enn
app
roac
hes
infi
nit
y,th
eex
act
area
isca
lcu
late
d.
Iff(
x)
isa
con
tin
uo
us
on
the
clo
sed
inte
rval
[a,b
],th
ed
efin
ite
inte
gra
lo
ff(
x)
bet
wee
na
and
bis
:
∫b a[f
(x)]
dx
=li
mn
→∞
(
n ∑ k=
1
f(ck))
(
b−
a
n
)
wh
ere
ck
are
sam
ple
po
ints
inth
ein
terv
al.
No
tati
on
Wh
enco
nsi
der
ing
the
exp
ress
ion
∫b a[f
(x)]
dx
,th
efu
nct
ion
f(x)
isca
lled
the
inte
gra
nd
and
the
inte
rval
[a,b
]is
the
inte
rval
of
inte
gra
tio
n.a
and
bar
eth
elo
wer
and
up
per
lim
its
of
inte
gra
tio
n,r
esp
ecti
vel
y.
5A
Pis
are
gis
tere
dtr
adem
ark
of
the
Co
lleg
eB
oar
d,
wh
ich
was
no
tin
vo
lved
inth
ep
rod
uct
ion
of,
and
do
esn
ot
end
ors
e,th
isp
rod
uct
.
33
Ex
tre
me
Va
lue
Th
eo
rem
Iff
isco
nti
nu
ou
so
nth
ein
terv
al[a
,b],
fh
asb
oth
am
axim
um
and
am
inim
um
val
ue
inth
ein
terv
al.
No
teth
atb
rack
ets
[]re
fer
toa
clo
sed
inte
rval
incl
ud
ing
the
end
po
ints
wh
ile
par
enth
eses
()
refe
rto
anin
terv
aln
ot
incl
ud
ing
the
end
po
ints
.
Me
an
Va
lue
Th
eo
rem
Iff
isco
nti
nu
ou
so
nth
ein
terv
al[a
,b]
and
dif
fere
nti
able
on
the
inte
rval
(a,b
),th
ere
exis
tsa
po
intc
on
(a,b
)su
chth
atf′
(c)
=f(
b)−
f(a)
b−
a.
Ino
ther
wo
rds,
som
ewh
ere
on
the
inte
rval
the
slo
pe
of
the
tan
gen
tli
ne
equ
als
(at
leas
to
nce
)th
esl
op
eo
fth
ese
can
tli
ne
con
nec
tin
gth
etw
oen
dp
oin
ts.
Ro
lle
’sT
he
ore
m
Ro
lle’
sT
heo
rem
isa
spec
ial
case
of
the
Mea
nV
alu
eT
heo
rem
.
Iff
isco
nti
nu
ou
so
nth
ein
terv
al[a
,b],
dif
fere
nti
able
on
the
inte
rval
(a,b
),an
df(
a)
=f(
b),
then
ther
eex
ists
ap
oin
tc
on
(a,b
)su
chth
atf′
(c)
=0.
32
No
teth
atth
esi
ne
of
any
thin
gis
inth
ein
terv
al[−
1,1
].In
oth
erw
ord
s,−
1≤
sin
x≤
1fo
ral
lx
).A
sa
resu
lt,f
or
all
no
nze
rox
,−1×
| x|≤
xsi
n1 x≤
1×
| x| .
Sim
pli
fied
,th
ism
ean
s−
| x|≤
xsi
n1 x≤
| x| .
Sin
ce
lim
x→
0−
| x|=
lim
x→
0| x
|=
0,
lim
x→
0x
sin
1 x=
0.
En
dB
eh
av
ior
Th
een
db
ehav
ior
of
ag
rap
hd
escr
ibes
ho
wit
app
ears
asx
app
roac
hes
infi
nit
yto
the
rig
ht
(xin
crea
ses)
or
toth
ele
ft(x
dec
reas
es).
En
db
ehav
ior
isex
pre
ssed
asa
beh
avio
rm
od
el.
Th
eb
ehav
ior
mo
del
of
ag
rap
hd
epen
ds
on
the
hig
hes
to
rder
term
inth
eeq
uat
ion
.In
rati
on
alex
pre
ssio
ns
(fra
ctio
ns)
,th
isw
ou
ldb
eth
ed
ivis
ion
of
the
hig
hes
to
rder
term
inth
en
um
erat
or
by
the
hig
hes
to
rder
term
inth
ed
eno
min
ato
r.
Fo
rex
amp
le,t
he
beh
avio
rm
od
elo
f2x5
+x4
−x2
+1
3x2
−5x
+7
is2x5
3x2
.T
he
lim
itas
xap
pro
ach
esb
oth
po
siti
ve
and
neg
ativ
ein
fin
ity
wo
uld
be
po
siti
ve
infi
nit
y.
Dif
feri
ng
Be
hav
ior
So
met
imes
,rig
ht-
han
dan
dle
ft-h
and
beh
avio
rd
iffe
r.
Ifth
efu
nct
ion
isf(
x)
and
its
left
-han
db
ehav
ior
mo
del
isg(x
),li
mx→
∞−
f(x)
g(x
)=
1.
Lik
ewis
e,if
the
fun
c-
tio
nis
f(x)
and
its
rig
ht-
han
db
ehav
ior
mo
del
ish(x
),li
mx→
∞+
f(x)
h(x
)=
1.
Ex
am
ple
:f(
x)
=x
+e−
x
lim
x→
∞−
x+
e−
x
e−
x=
lim
x→
∞−
x
e−
x+
lim
x→
∞−
e−
x
e−
x=
0+
1=
1.
Th
eref
ore
,y=
e−
xis
the
left
-han
db
ehav
ior
mo
del
.
lim
x→
∞+
x+
e−
x
x=
lim
x→
∞+
x x=
lim
x→
∞+
e−
x
x=
1+
0=
1.
Th
eref
ore
,y=
xis
the
rig
ht-
han
db
ehav
ior
mo
del
.
13
Deri
vati
ves
Th
isch
apte
rw
aso
rig
inal
lyd
esig
ned
for
ate
sto
nd
eriv
ativ
esad
min
iste
red
by
Jean
ine
Len
no
nto
her
Mat
h12
H(4
H/
Pre
calc
ulu
s)cl
ass
on
Ap
ril
18,
2008
.It
was
up
dat
edfo
ra
qu
izo
nth
ed
eriv
ativ
eso
ftr
igo
no
met
ric
fun
ctio
ns
on
Ap
ril
29,
2008
,an
dla
ter
up
dat
edw
ith
an“A
dd
end
um
”se
ctio
n(p
age
27)
for
ate
sto
nd
eriv
ativ
esad
min
iste
red
by
Jon
ath
anC
her
nic
kto
his
AP
3C
alcu
lus
BC
clas
so
nO
cto
ber
14,2
008.
Intr
od
uc
tio
n
Th
esl
op
eo
fa
curv
eca
nn
ot
be
det
erm
ined
by
usi
ng
the
form
ula
m=
y2−
y1
x2−
x1
,bu
tth
esl
op
eso
fta
ng
ent
lin
esd
raw
nto
acu
rve
can
be
det
erm
ined
.T
ocr
eate
anin
fin
ite
nu
mb
ero
fta
ng
ent
lin
es,
two
po
ints
on
the
curv
em
ust
be
“pu
shed
”to
get
her
soth
atth
eir
dis
tan
ce,h
,ap
pro
ach
esze
ro.
Th
eco
nce
pt
of
ali
mit
isu
sed
tofi
nd
ad
eriv
ativ
e.T
he
der
ivat
ive
isth
em
tan
(slo
pe
of
tan
gen
tli
ne)
on
acu
rve
ata
spec
ific
po
int.
de
riva
tiv
esl
op
eo
fa
curv
eat
ag
iven
po
int
on
the
curv
e
no
rma
lli
ne
lin
ep
erp
end
icu
lar
toa
tan
gen
tli
ne
atth
ep
oin
to
fta
ng
ency
De
fin
itio
n
f′(x
)=
lim
h→
0
f(x
+h)−
f(x)
h
Ta
ng
en
tL
ine
s
Th
ed
eriv
ativ
eca
nb
eu
sed
toca
lcu
late
the
equ
atio
no
fa
lin
eta
ng
ent
toa
curv
eat
ace
rtai
np
oin
t.T
he
der
ivat
ive
isth
esl
op
eo
fth
eta
ng
ent
lin
e,an
dw
hen
the
coo
rdin
ates
of
the
cert
ain
po
int
on
the
curv
ear
ek
no
wn
,th
eca
lcu
late
dsl
op
ean
dth
eco
ord
inat
eso
fth
ece
rtai
np
oin
to
nth
ecu
rve
(val
ues
can
be
calc
ula
ted
by
plu
gg
ing
into
equ
atio
no
fcu
rve)
can
be
plu
gg
edin
toy
=m
x+
bo
rth
ep
oin
t-sl
op
efo
rmu
lato
det
erm
ine
the
equ
atio
no
fth
eta
ng
ent
lin
e.
Ifth
esl
op
eo
fa
curv
eat
ag
iven
po
int
(der
ivat
ive)
iseq
ual
toth
esl
op
eo
fan
oth
ercu
rve
ata
giv
enp
oin
t,th
enth
etw
ocu
rves
hav
ep
aral
lel
tan
gen
tli
nes
atth
ein
dic
ated
po
ints
.
No
tati
on
Th
ed
eriv
ativ
en
ota
tio
nis
spec
ial
and
un
iqu
ein
mat
hem
atic
s.T
her
ear
etw
ok
ind
so
fn
ota
tio
ns:
Lei
bn
izn
ota
tio
nan
dN
ewto
nia
nn
ota
tio
n.
3A
Pis
are
gis
tere
dtr
adem
ark
of
the
Co
lleg
eB
oar
d,
wh
ich
was
no
tin
vo
lved
inth
ep
rod
uct
ion
of,
and
do
esn
ot
end
ors
e,th
isp
rod
uct
.
14
Bas
edo
nth
ech
ain
rule
(pag
e17
),w
her
eu
isan
yd
iffe
ren
tiab
leex
pre
ssio
n,
d dx[e
u]=
eu×
du
dx
cx
cre
pre
sen
tsa
con
stan
t.T
he
der
ivat
ive
ofcx
iscx×
lnc,c
>0
and
c6=
1.
Bas
edo
nth
ech
ain
rule
(pag
e17
),w
her
ec
isa
con
stan
t,d dx[c
u]=
lnc×
cu×
du
dx
,c>
0an
dc6=
1
lnx
Th
ed
eriv
ativ
eo
fln
xis
1 x,x
>0.
Bas
edo
nth
ech
ain
rule
(pag
e17
),w
her
eu
isan
yd
iffe
ren
tiab
leex
pre
ssio
n,
d dx[l
nu]=
1 u×
du
dx
,u>
0
Lo
ga
rith
ms
Pro
pe
rtie
sT
hes
ep
rop
erti
esh
old
tru
efo
rb
oth
log
and
ln.
•lo
g(x
y)
=lo
gx
+lo
gy
•lo
g(x
/y)
=lo
gx
−lo
gy
•lo
gxa
=a
lnx
Ch
an
ge
of
Ba
se
log
ax
=lo
gx
log
a=
lnx
lna
log
bx
Th
ed
eriv
ativ
eo
flo
gb
xis
1
xln
(b)
.
Bas
edo
nth
ech
ain
rule
(pag
e17
),w
her
eu
isan
yd
iffe
ren
tiab
leex
pre
ssio
n,
d dx[l
og
bu]=
1
uln
(b)×
du
dx
;b>
0,b
6=1,a
nd
u>
0
Lo
ga
rith
mic
Dif
fere
nti
ati
on
Lo
gar
ith
mic
dif
fere
nti
atio
nis
ad
iffe
ren
tiat
ion
pro
cess
use
dto
tak
eth
ed
eriv
ativ
eo
fa
var
iab
lera
ised
toa
var
iab
leo
ro
ther
com
ple
xsi
tuat
ion
s.T
he
nat
ura
llo
g(l
n)
of
bo
thsi
des
of
aneq
uat
ion
are
tak
en,a
nd
the
resu
ltis
imp
lici
tly
dif
fere
nti
ated
.
31
Tri
go
no
met
ric
Fu
nct
ion
Inv
erse
(arc
no
tati
on
)In
ver
se(−
1n
ota
tio
n)
sin
arcs
insi
n−
1
cos
arcc
os
cos−
1
tan
arct
anta
n−
1
cot
arcc
ot
cot−
1
sec
arcs
ecse
c−1
csc
arcc
sccs
c−1
Inth
eta
ble
bel
ow
,uca
nre
pre
sen
tan
yd
iffe
ren
tiab
leex
pre
ssio
n,u
sin
gth
ech
ain
rule
(pag
e17
).
Inv
erse
Tri
go
no
met
ric
Fu
nct
ion
Der
ivat
ive
arcs
inu
1√
1−
u2×
du
dx
,|u|<
1
arcc
osu
−1
√1
−u
2×
du
dx
,|u|<
1
arct
anu
1
1+
u2×
du
dx
arcc
otu
−1
1+
u2×
du
dx
arcs
ecu
1
|u|√
u2
−1×
du
dx
,|u|>
1
arcc
scu
−1
|u|√
u2
−1×
du
dx
,|u|>
1
Str
ate
gie
sfo
rS
imp
lify
ing
Inm
any
dif
ficu
ltp
rob
lem
s(e
.g.
mu
ltip
lech
oic
e)w
her
esi
mp
lify
ing
isn
eces
sary
,th
ere
are
som
est
rate
gie
sfo
rd
oin
gso
.If
sim
pli
fyin
gis
no
tre
qu
ired
,th
ese
stra
teg
ies
are
no
tn
eces
sary
.
•If
anex
pre
ssio
nu
nd
eran
abso
lute
val
ue
isal
way
sp
osi
tiv
e,th
eab
solu
tev
alu
esy
mb
ols
can
be
rem
ov
ed.
•C
om
bin
ete
rms
into
term
sw
ith
aco
mm
on
den
om
inat
or.
•F
acto
ro
ut
var
iab
les
fro
msq
uar
ero
ots
.
Mo
reR
ule
s
Ifth
eo
rig
inal
exp
ress
ion
isa
con
stan
tra
ised
toa
var
iab
lep
ow
er,
use
the
cx
rule
().
Ifth
eo
rig
inal
exp
ress
ion
con
tain
sa
var
iab
lein
the
bas
ean
dex
po
nen
t,lo
gar
ith
mic
dif
fere
nti
atio
n(p
age
31)
mu
stb
eu
sed
.
ex Th
ed
eriv
ativ
eo
fex
isit
self
.
30
Le
ibn
izN
ota
tio
n
Th
eL
eib
niz
no
tati
on
isex
pre
ssed
asdy
dx
,m
ean
ing
“rat
eo
fch
ang
ein
yw
ith
resp
ect
tox
”o
ras
d dx
,w
hic
hli
tera
lly
mea
ns
“der
ivat
ive
wit
hre
spec
tto
x.”
Bec
ause
the
der
ivat
ive
of
fun
ctio
ny
isd
efin
edas
afu
nct
ion
rep
rese
nti
ng
the
slo
pe
of
fun
ctio
ny
,th
ese
con
d(o
rd
ou
ble
)d
eriv
ativ
eis
the
fun
ctio
nre
pre
sen
tin
gth
esl
op
eo
fth
efi
rst
der
ivat
ive
fun
ctio
n.
InL
eib
niz
no
tati
on
,th
isis
wri
tten
as:
d dx
(
dy
dx
)
=d
2y
dx2
New
ton
ian
No
tati
on
Wit
hth
eN
ewto
nia
nn
ota
tio
n,
the
der
ivat
ive
of
the
fun
ctio
nf(
x)
isd
eno
ted
by
f′(x
),an
dit
sse
con
d(o
rd
ou
ble
)d
eriv
ativ
eis
den
ote
db
yf′′ (
x).
Th
isis
read
as“f
do
ub
lep
rim
eo
fx
,”o
r“t
he
seco
nd
der
ivat
ive
off(
x).
”
Hig
he
rO
rde
rD
eri
va
tiv
es
Th
ese
con
dd
eriv
ativ
eis
the
der
ivat
ive
of
the
der
ivat
ive
of
afu
nct
ion
.S
ub
seq
uen
td
eriv
ativ
esca
nb
eca
lcu
late
db
yca
lcu
lati
ng
the
der
ivat
ive
of
the
pre
vio
us
der
ivat
ive.
Th
efo
llo
win
gar
en
ota
tio
ns
for
der
ivat
ives
of
dif
fere
nt
ord
ers.
Ord
erN
ewto
nia
nN
ota
tio
nL
eib
niz
No
tati
on
Lei
bn
izN
ota
tio
n
Fir
stD
eriv
ativ
ef′
(x)
dy
dx
d dx
[ f(x
)]
Sec
on
dD
eriv
ativ
ef′′ (
x)
d2y
dx2
d2
dx2
[ f(x
)]
Th
ird
Der
ivat
ive
f′′′ (
x)
d3y
dx3
d3
dx3
[f(x
)]
Fo
urt
hD
eriv
ativ
ef(
4)(x
)d
4y
dx4
d4
dx4
[ f(x
)]
Nth
Der
ivat
ive
f(n
)(x
)d
ny
dxn
dn
dxn
[ f(x
)]
On
esh
ou
ldn
ot
wri
tefn
(x)
toin
dic
ate
the
nth
der
ivat
ive,
asth
isis
easi
lyco
nfu
sed
wit
hth
eq
uan
tity
f(x)
all
rais
edto
the
nth
po
wer
.
Ru
les
Ru
les
for
calc
ula
tin
gth
ed
eriv
ativ
eso
fg
ener
alfu
nct
ion
sh
ave
bee
nd
evel
op
ed.
As
are
sult
,it
isp
oss
ible
toca
lcu
late
the
der
ivat
ive
of
aw
ide
var
iety
of
fun
ctio
ns.
Inm
any
case
sth
eu
seo
fm
ult
iple
rule
sar
ere
qu
ired
.
15
Co
ns
tan
tF
un
cti
on
Fo
ran
yco
nst
antc,
d dx[c
]=
0
Th
efu
nct
ion
f(x)
=c
isa
ho
rizo
nta
lli
ne,
wh
ich
has
aco
nst
ant
slo
pe
of
zero
.T
her
efo
re,
itsh
ou
ldb
eex
pec
ted
that
the
der
ivat
ive
of
this
fun
ctio
nis
zero
,re
gar
dle
sso
fth
ev
alu
eo
fx
.It
isim
po
rtan
tto
un
der
stan
dth
ate
and
πar
eco
nst
ants
,an
dth
atth
eir
der
ivat
ive
isth
eref
ore
zero
.
Lin
ea
rF
un
cti
on
Fo
ran
yco
nst
ants
man
dc,
d dx[m
x+
c]=
m
Th
efu
nct
ion
f(x)
=m
x+
cis
ali
ne
wit
ha
slo
pe
ofm
.
Co
ns
tan
tM
ult
ipli
er
Ru
le
Fo
ran
yco
nst
antc,
d dx[c
f(x)]
=c
d dx[f
(x)]
Inth
ed
efin
itio
no
fa
der
ivat
ive,
on
eca
nfa
cto
rc
ou
to
fth
en
um
erat
or
and
then
ou
to
fth
een
tire
lim
it.
Ad
dit
ion
/Su
btr
ac
tio
nR
ule
Fo
rth
eg
iven
fun
ctio
ns
f(x)
and
g(x
),
d dx[f
(x)±
g(x
)]=
d dx[f
(x)]±
d dx[g
(x)]
As
are
sult
,o
ne
can
tak
ean
equ
atio
n,
bre
akit
up
into
term
s,fi
gu
reo
ut
the
der
ivat
ive
ind
ivid
ual
ly,
and
bu
ild
the
answ
erb
ack
up
.
Po
we
rR
ule
Fo
ran
yco
nst
ant
exp
on
entn
,
d dx
[ xn]=
nxn
−1,x
6=0
16
Wit
hu
seo
fth
ech
ain
rule
(pag
e17
),th
ere
lati
on
ship
bet
wee
nth
ed
eriv
ativ
eo
fa
fun
ctio
nan
dth
ed
eriv
ativ
eo
fit
sin
ver
seca
nb
ed
eter
min
ed. f(
f−1(x
))
=x
f′[
f−1(a
)]
×[
f−1]
′ (a)
=1
[
f−1]
′ (a)
=1
f′[
f−1(a
)]
Afu
nct
ion
and
its
inv
erse
hav
ere
cip
roca
lsl
op
esw
ith
rev
erse
d(x
,y)
val
ues
.
[
f−1]
′ (a)
=1
f′[
f−1(a
)]
Ex
am
ple
:f(
x)
=x3
+x
−2,fi
nd[
f−1]
′ (0)
0=
x3
+x
−2
x=
1
f′(x
)=
3x2
+1
f′(1
)=
4
[
f−1]
′ (a)
=1
f′[
f−1(a
)]
[
f−1]
′ (0)
=1
f′[
f−1(0
)]
[
f−1]
′ (0)
=1
f′(1
)[
f−1]
′ (0)
=1 4
Inv
ers
eTri
go
no
me
tric
Fu
nc
tio
ns
Th
ein
ver
setr
igo
no
met
ric
fun
ctio
ns
are
the
inv
erse
fun
ctio
ns
of
the
trig
on
om
etri
cfu
nct
ion
s.T
he
inv
erse
of
the
trig
on
om
etri
cfu
nct
ion
ssi
n,
cos,
tan
,co
t,se
c,an
dcs
cis
arcs
in,
arcc
os,
arct
an,
arcc
ot,
arcs
ec,a
nd
arcc
sc,r
esp
ecti
vel
y.
Th
en
ota
tio
ns
sin
−1,c
os−
1,e
tc.
are
oft
enu
sed
for
arcs
in,a
rcco
s,et
c.,r
esp
ecti
vel
y,b
ut
this
con
ven
tio
nm
ayre
sult
inco
nfu
sio
nb
etw
een
mu
ltip
lica
tiv
ein
ver
sean
dco
mp
osi
tio
nal
inv
erse
sin
ceth
islo
gic
ally
con
flic
tsw
ith
the
stru
ctu
reo
fex
pre
ssio
ns
lik
esi
n2x
,w
hic
hd
on
ot
refe
rto
fun
ctio
nco
mp
osi
tio
nb
ut
rath
erm
ult
ipli
cati
on
.
Eac
hin
ver
setr
igo
no
met
ric
fun
ctio
nh
asa
der
ivat
ive.
29
Imp
lic
itD
iffe
ren
tia
tio
n
ex
pli
cit
rela
tio
ns
hip
fun
ctio
nin
wh
ich
f(x)
isg
iven
inte
rms
ofx
and
con
stan
ts;
for
ever
yx
-val
ue
ther
eis
on
ey
-val
ue
imp
lic
itre
lati
on
sh
ipre
lati
on
ship
bet
wee
ntw
oo
rm
ore
var
iab
les;
two
or
mo
refu
nct
ion
sp
ut
to-
get
her
Ord
inar
yd
iffe
ren
tiat
ion
isex
pli
cit
dif
fere
nti
atio
n.
Imp
lici
td
iffe
ren
tiat
ion
isu
sefu
lw
hen
dif
fere
nti
-at
ing
aneq
uat
ion
that
can
no
tb
eex
pli
citl
yd
iffe
ren
tiat
edb
ecau
seit
isim
po
ssib
leo
rh
ard
tois
ola
tev
aria
ble
s(e
.g.x2
+xy
+y
2=
16).
Inm
any
dif
ficu
ltp
rob
lem
sin
vo
lvin
gim
pli
cit
dif
fere
nti
atio
n(e
.g.
mu
ltip
lech
oic
e),
itis
nec
essa
ryto
sub
stit
ute
the
dep
end
ent
var
iab
le(e
.g.
y)
and
its
der
ivat
ives
(e.g
.dy
dx
,d
2y
dx
2)
bas
edo
nth
eo
rig
inal
equ
atio
no
rp
rev
iou
sd
eter
min
edd
eriv
ativ
eex
pre
ssio
ns.
Ex
am
ple
:x2
+y
2=
1
Ex
pli
cit
Dif
fere
nti
ati
on
x2
+y
2=
1
y2
=1
−x2
y=
±√
1−
x2
y=
±(1
−x2)
1 2
dy
dx
=−
x y
Imp
lic
itD
iffe
ren
tia
tio
n
x2
+y
2=
1
2x
+2y
dy
dx
=0
2y
dy
dx
=−
2x
dy
dx
=−
2x
2y
dy
dx
=−
x y
Inv
ers
eF
un
cti
on
s
inv
ers
efu
nc
tio
n“o
pp
osi
te”
of
afu
nct
ion
;iff(
x)
=a
,f−
1(a
)=
f(x);
refl
ecte
do
ver
lin
ey
=x
Th
eco
mp
osi
tio
no
fa
fun
ctio
nan
dit
sin
ver
seis
xb
ecau
seth
etw
ofu
nct
ion
s“u
nd
o”
each
oth
er.
f(
f−1(x
))
=x
28
Th
isru
leis
actu
ally
inef
fect
inli
nea
req
uat
ion
sto
o,
sin
cexn
−1
=x0
wh
enn
=1,
and
any
real
nu
mb
ero
rv
aria
ble
toth
eze
rop
ow
eris
on
e.
Th
isru
leal
soap
pli
esto
frac
tio
nal
and
neg
ativ
ep
ow
ers.
Th
eref
ore
,
d dx
[√
x]
=d dx
[
x1
/2]
=1 2x
−1
/2
=1
2√
x
Sin
cep
oly
no
mia
lsar
esu
ms
of
mo
no
mia
ls,
usi
ng
this
rule
and
the
add
itio
n/
sub
trac
tio
nru
le(p
age
16)
lets
on
eca
lcu
late
the
der
ivat
ive
of
any
po
lyn
om
ial.
Sim
ple
Fra
cti
on
s
Wh
enta
kin
gth
ed
eriv
ativ
eo
fsi
mp
lefr
acti
on
s,o
ne
can
use
the
foll
ow
ing
sho
rtcu
tto
qu
ick
lyd
oso
.T
he
calc
ula
tio
ns
of
der
ivat
ives
of
mo
reco
mp
lex
frac
tio
ns
req
uir
eu
seo
fth
eq
uo
tien
tru
le.
d dx
[
c xb
]
=d dx
[
cx
−b]
=−
cbx
−b
−1
=−
cbx
−(b
+1)
=−
cb
xb
+1
,wh
ere
cis
aco
nst
ant
Ch
ain
Ru
le
Th
ech
ain
rule
allo
ws
on
eto
calc
ula
teth
ed
eriv
ativ
eo
fan
un
exp
and
edex
pre
ssio
nw
ith
ou
tex
pan
din
gth
eex
pre
ssio
n.
Th
isis
do
ne
by
calc
ula
tin
gth
ed
eriv
ativ
eo
fth
eco
mp
osi
teo
ftw
ofu
nct
ion
s.
Fo
rex
amp
le,
see
the
fun
ctio
nf(
x)
=(a
+b)c
.T
om
ake
this
the
com
po
site
of
two
fun
ctio
ns,
g(x
)=
a+
ban
df(
x)
=g(x
)c.
Th
isfu
nct
ion
can
be
rew
ritt
enas
the
com
po
site
fun
ctio
nf(
g(x
)),
wh
ere
g(x
)
isth
ep
oly
no
mia
l(a
+b
)an
df(
x)
isg(x
)to
the
cth
po
wer
.
Acc
ord
ing
toth
ech
ain
rule
,
d dx[f
(g(x
))]=
f′(g
(x))×
g′ (
x)
An
exam
ple
of
this
situ
atio
nis
f(x)
=(3
x+
4)3
.In
this
case
,g(x
)=
3x+
4an
df(
x)
=g(x
)3.
Acc
ord
ing
toth
ech
ain
rule
,
d dx
[
(3x
+4)3]
=3(3
x+
4)2
×d dx
[ 3x
+4]=
3(3
x+
4)2
×(3
+0)
=9(3
x+
4)2
Pro
du
ct
Ru
le
Th
ed
eriv
ativ
eo
fth
efu
nct
ion
f(x)
=A×
Bw
ou
ldn
otb
ef′
(a)×
f′(b
).T
he
pro
du
ctru
leal
low
so
ne
toco
rrec
tly
calc
ula
teth
ed
eriv
ativ
eo
fth
ep
rod
uct
of
two
fun
ctio
ns.
Acc
ord
ing
toth
ep
rod
uct
rule
, d dx[f
(x)×
g(x
)]=
f(x)×
g′ (
x)+
g(x
)×
f′(x
)
Th
ed
eriv
ativ
eo
fth
ep
rod
uct
of
two
fun
ctio
ns
isth
efi
rst
fun
ctio
nm
ult
ipli
edb
yth
ed
eriv
ativ
eo
fth
eo
ther
fun
ctio
n,a
dd
edto
the
firs
tfu
nct
ion
mu
ltip
lied
by
the
der
ivat
ive
of
the
seco
nd
fun
ctio
n.
Th
em
nem
on
icd
evic
e“o
ne-
D-t
wo
plu
stw
o-D
-on
e”ca
nb
eu
sed
tore
mem
ber
this
rule
.
17
Qu
oti
en
tR
ule
As
wit
hm
ult
iply
ing
,th
ed
eriv
ativ
eo
fa
qu
oti
ent
isn
ot
the
qu
oti
ent
of
the
der
ivat
ives
.T
he
qu
oti
ent
rule
allo
ws
on
eto
corr
ectl
yca
lcu
late
the
der
ivat
ive
of
the
qu
oti
ent
of
two
fun
ctio
ns.
Acc
ord
ing
toth
eq
uo
tien
tru
le, d d
x
[
f(x)
g(x
)
]
=g(x
)×
f′(x
)−
f(x)×
g′ (
x)
g(x
)2
Th
em
nem
on
icd
evic
e“l
ow
-D-h
igh
min
us
hig
h-D
-lo
wo
ver
the
squ
are
of
wh
at’s
bel
ow
”ca
nb
eu
sed
tore
mem
ber
this
rule
.
Ba
sic
Po
lyn
om
ials
Wit
hth
ese
rule
s,th
ed
eriv
ativ
eo
fan
yp
oly
no
mia
lca
nb
ed
eter
min
ed.
Her
eis
ast
ep-b
y-s
tep
exam
ple
of
the
pro
cess
of
calc
ula
tin
gth
ed
eriv
ativ
eo
fa
fair
lysi
mp
lep
oly
no
mia
l.T
he
chai
n,
pro
du
ct,
and
qu
oti
ent
rule
sar
en
ot
cov
ered
.
d dx
[
6x5
+3x2
+3x
+1]
Th
ead
dit
ion
/su
btr
acti
on
rule
(pag
e16
)sp
lits
the
equ
atio
nin
tose
ver
alte
rms.
d dx
[
6x5]
+d dx
[
3x2]
+d dx[3
x]+
d dx[1
]
Th
eco
nst
ant
(pag
e16
)an
dli
nea
r(p
age
16)
rule
sg
etri
do
fso
me
term
s.
d dx
[
6x5]
+d dx
[
3x2]
+3
+0
Th
eco
nst
ant
mu
ltip
lier
rule
(pag
e16
)m
ov
esth
eco
nst
ants
ou
tsid
eo
fth
ed
eriv
ativ
es.
6d dx
[
x5]
+3
d dx[x
]+
3
Th
ep
ow
erru
le(p
age
16)
wo
rks
on
the
ind
ivid
ual
mo
no
mia
ls.
6(
5x4)
+3(2
x)+
3
Sim
pli
fyin
go
bta
ins
the
fin
alan
swer
.
30x4
+6x
+3
Gra
ph
ing
Ca
lcu
lato
r
Inso
me
case
sit
may
be
easi
ero
rre
qu
ired
toca
lcu
late
der
ivat
ives
usi
ng
the
gra
ph
ing
calc
ula
tor.
Itca
nal
sob
eu
sed
toch
eck
on
e’s
answ
er.
Th
ere
are
two
met
ho
ds
of
calc
ula
tin
ga
der
ivat
ive
of
ag
rap
hw
ith
aT
exas
Inst
rum
ents
gra
ph
ing
calc
ula
tor.
Th
ese
inst
ruct
ion
sar
ed
esig
ned
for
aT
I-84
Plu
sca
lcu
lato
r,b
ut
they
may
use
do
no
ther
Tex
asIn
stru
men
tsg
rap
hin
gca
lcu
lato
rs,t
ho
ug
hsl
igh
tm
od
ifica
tio
nm
ayb
en
eces
sary
.
Un
less
oth
erw
ise
spec
ified
,th
eg
rap
hin
gca
lcu
lato
rsh
ou
ldb
ein
rad
ian
mo
de.
18
rec
tan
gu
lar
pri
sm
V=
abc,w
her
ea
,b,a
nd
car
eth
ele
ng
ths
of
the
3si
des
of
the
pri
sm
cy
lin
de
rV
=πr2
h,w
her
er
isth
era
diu
san
dh
isth
eh
eig
ht
of
the
cyli
nd
er
sp
he
reV
=4 3πr3
,wh
ere
rre
pre
sen
tsth
era
diu
so
fth
esp
her
e
Su
rfa
ce
Are
a
cu
be
A=
6a
2,w
her
ea
isth
ele
ng
tho
fth
esi
de
of
each
edg
eo
fth
ecu
be
rec
tan
gu
lar
pri
sm
A=
2ab
+2bc+
2ac,w
her
ea
,b,a
nd
car
eth
ele
ng
ths
of
the
3si
des
of
the
pri
sm
sp
he
reA
=4πr2
,wh
ere
ris
rad
ius
of
the
sph
ere
cy
lin
de
rA
=2πr2
+2πrh
,wh
ere
ris
the
rad
ius
and
his
the
hei
gh
to
fth
ecy
lin
der
Ad
de
nd
um
Th
isse
ctio
nw
asd
esig
ned
for
ate
sto
nd
eriv
ativ
esad
min
iste
red
by
Jon
ath
anC
her
nic
kto
his
AP
4
Cal
culu
sB
Ccl
ass
on
Oct
ob
er14
,200
8.It
isn
ot
cov
ered
inM
ath
12H
/4H
.
Alt
ern
ati
ve
De
fin
itio
no
fD
eri
va
tiv
e
f′(x
)=
lim
x→
a
f(x)−
f(a)
x−
a
Pa
ram
etr
icE
qu
ati
on
s
Par
amet
ric
equ
atio
ns
are
typ
ical
lyd
efin
edb
ytw
oeq
uat
ion
sth
atsp
ecif
yb
oth
the
xan
dy
coo
rdin
ates
of
ag
rap
hu
sin
ga
par
amet
er.
Th
eyar
eg
rap
hed
usi
ng
the
par
amet
er(u
sual
lyt)
tofi
gu
reo
ut
bo
thth
ex
and
yco
ord
inat
es.
Th
ed
eriv
ativ
eo
fth
ep
aram
etri
zed
curv
ex(t
),y(t
)is
:
dy
dx
=
dy
dt
dx
dt
,dx
dt6=
0
Ex
am
ple
:
x=
t,y
=t2
dy
dx
=
dy
dt
dx
dt
=2t 1
=2t
4A
Pis
are
gis
tere
dtr
adem
ark
of
the
Co
lleg
eB
oar
d,
wh
ich
was
no
tin
vo
lved
inth
ep
rod
uct
ion
of,
and
do
esn
ot
end
ors
e,th
isp
rod
uct
.
27
1.M
ath−→
8(8
.n
Der
iv)−→
ente
rw
ith
form
fun
ctio
n,x
,xva
lue−→
En
ter
a)re
pla
cefu
nct
ion
wit
hth
eap
pro
pri
ate
fun
ctio
n
b)
rep
lace
xva
lue
wit
hth
eap
pro
pri
ate
val
ue
2.G
rap
hfu
nct
ion−→
2nd−→
Tra
ce(C
alc)
−→en
terx
val
ue−→
En
ter
a)u
seth
eap
pro
pri
ate
xv
alu
e
Do
es
No
tE
xis
t
Th
eg
rap
hin
gca
lcu
lato
rm
ayd
isp
lay
anin
corr
ect
answ
erw
hen
calc
ula
tin
gd
eriv
ativ
esth
atd
on
ot
exis
t(e
.g.
ata
corn
er).
Gra
ph
ing
calc
ula
tors
lik
eth
eT
I-84
Plu
sca
lcu
late
der
ivat
ives
by
usi
ng
the
sym
met
ric
dif
fere
nce
qu
oti
ent.
f′(x
)=
lim
h→
0
f(x
+h)−
f(x
−h)
2h
Th
ep
rob
lem
wit
hth
ism
eth
od
isth
atth
eca
lcu
lato
rw
ill
actu
ally
calc
ula
teth
eav
erag
esl
op
eo
ver
av
ery
smal
lar
eain
stea
do
fth
etr
ue
der
ivat
ive
(in
stan
tan
eou
ssl
op
e).
At
aco
rner
,th
eav
erag
esl
op
eo
ver
av
ery
smal
lar
eaw
ill
be
zero
,bu
tth
eco
rrec
tan
swer
isth
atth
ed
eriv
ativ
ed
oes
no
tex
ist.
Dif
fere
nti
ab
ilit
y
De
fin
itio
n
Fo
rf(
x)
tob
ed
iffe
ren
tiab
leat
po
intc,t
he
foll
ow
ing
mu
stb
etr
ue:
1.f(
x)
mu
stb
eco
nti
nu
ou
sat
po
intc
a)f(
x)
isd
efin
edat
c
i.f(
c)
exis
ts
b)
Th
eli
mit
asx
app
roac
hes
cex
ists
.
i.li
mx→
cf(
x)
exis
ts
c)T
he
lim
itan
df(
c)
are
equ
al.
i.f(
c)
=li
mx→
cf(
x)
2.T
he
der
ivat
ive
fro
mb
oth
sid
esm
ust
be
equ
al
a)li
mx→
c−
f′(x
)=
lim
x→
c+
f′(x
)
Ifan
yo
fth
ese
do
no
th
old
then
f(x)
isn
ot
dif
fere
nti
able
atc.
No
tice
ho
wth
isre
late
sto
the
idea
of
dif
fere
nti
abil
ity.
To
be
dif
fere
nti
able
,th
efu
nct
ion
mu
sth
ave
au
nif
orm
rate
of
chan
ge
(e.g
.n
oco
rner
s,cu
sps,
or
ver
tica
lta
ng
ents
)w
ith
inan
inte
rval
.
Afu
nct
ion
issa
idto
be
dif
fere
nti
able
ifit
isd
iffe
ren
tiab
leat
ever
yp
oin
tc
init
sd
om
ain
.
Afu
nct
ion
may
be
dif
fere
nti
able
ata
cert
ain
po
int,
bu
tn
ot
ad
iffe
ren
tiab
lefu
nct
ion
(th
rou
gh
ou
t).
Lik
ewis
e,a
no
n-d
iffe
ren
tiab
lefu
nct
ion
may
be
dif
fere
nti
able
ata
cert
ain
po
int.
19
No
tD
iffe
ren
tia
ble
Co
rne
r
Afu
nct
ion
do
esn
ot
hav
ea
der
ivat
ive
ata
corn
er.
lim
x→
a−
f′(x
)6=
lim
x→
a+
f′(x
)
Cu
sp
Acu
spo
ccu
rsw
hen
the
lim
ito
fth
esl
op
efr
om
on
esi
de
of
acu
rve
go
esto
−∞
and
the
oth
ersi
de
of
the
curv
eg
oes
to+
∞.
As
are
sult
,afu
nct
ion
do
esn
ot
hav
ea
der
ivat
ive
ata
cusp
.
lim
x→
a−
f′(x
)6=
lim
x→
a+
f′(x
)
Ve
rtic
al
Ta
ng
en
t
Afu
nct
ion
do
esn
ot
hav
ea
der
ivat
ive
ata
ver
tica
lta
ng
ent.
lim
x→
af′
(x)
=∞
En
dp
oin
t
Afu
nct
ion
isn
ot
dif
fere
nti
able
atan
end
po
int
bec
ause
the
der
ivat
ive
can
on
lyb
eca
lcu
late
dfr
om
on
esi
de.
Ho
wev
er,
sin
cean
end
po
int
has
ao
ne-
sid
edd
eriv
ativ
e,th
een
dp
oin
tso
nth
eg
rap
ho
fth
ed
eriv
ativ
eo
fa
fun
ctio
nar
efi
lled
in.
En
dp
oin
tsar
ea
sou
rce
of
alo
to
fse
emin
gin
con
sist
ency
inca
lcu
lus.
Tri
go
no
me
tric
Fu
nc
tio
ns
Tri
go
no
me
tric
Ide
nti
tie
s
Tri
go
no
met
ric
iden
titi
es(p
age
42)
can
be
use
dto
sim
pli
fyex
pre
ssio
ns
bef
ore
or
afte
rfi
nd
ing
ad
eriv
a-ti
ve.
De
riva
tio
n
Sin
e,co
sin
e,ta
ng
ent,
cota
ng
ent,
seca
nt,
and
cose
can
tar
etr
igo
no
met
ric
fun
ctio
ns.
Eac
htr
igo
no
met
ric
fun
ctio
nh
asa
der
ivat
ive.
20
the
do
mai
nis
rest
rict
ed,
the
end
po
ints
of
the
do
mai
nm
ust
also
be
chec
ked
tose
eif
they
are
glo
bal
extr
ema.
cri
tic
al
po
int
po
int
ind
om
ain
off
wh
ere
f′=
0o
rf′
do
esn
ot
exis
t
Ex
trem
aca
no
nly
occ
ur
atcr
itic
alp
oin
tsan
den
dp
oin
ts.
Tru
eex
trem
are
qu
ire
asi
gn
chan
ge
inth
efi
rst
der
ivat
ive.
Th
ism
akes
sen
se—
the
gra
ph
mu
stri
se(p
osi
tiv
efi
rst
der
ivat
ive)
and
fall
(neg
ativ
efi
rst
der
ivat
ive)
tofo
rma
max
imu
m.
Inb
etw
een
risi
ng
and
fall
ing
,o
na
smo
oth
curv
e,th
ere
wil
lid
eall
yb
ea
po
int
of
zero
slo
pe
—th
em
axim
um
.A
min
i-m
um
wo
uld
exh
ibit
sim
ilar
pro
per
ties
,bu
tin
rev
erse
.
Fir
st
De
riva
tiv
eTe
st
Th
isle
ads
toa
sim
ple
met
ho
dto
clas
sify
ast
atio
nar
yp
oin
t—
plu
gx
val
ues
(tes
tp
oin
ts)
slig
htl
yle
ftan
dri
gh
tin
toth
ed
eriv
ativ
eo
fth
efu
nct
ion
.If
the
resu
lts
hav
eo
pp
osi
tesi
gn
sth
enit
isa
tru
eex
trem
um
.T
oca
lcu
late
the
coo
rdin
ates
of
the
min
imu
mo
rm
axim
um
po
int,
on
ew
ou
ldp
lug
the
det
erm
ined
xv
alu
ein
toth
eo
rig
inal
fun
ctio
nto
fin
dit
sy
val
ue.
•If
f′(x
)<
0fo
rx
<c
and
f′(x
)>
0fo
rx
>c,t
hen
f(c)
isa
loca
lm
inim
um
.
•If
f′(x
)>
0fo
rx
<c
and
f′(x
)<
0fo
rx
>c,t
hen
f(c)
isa
loca
lm
axim
um
.
Cau
tio
nm
ust
be
exer
cise
dw
ith
this
met
ho
d,
as,
ifa
po
int
too
far
fro
mth
eex
trem
um
isp
ick
ed,
on
eco
uld
tak
eit
on
the
far
sid
eo
fan
oth
erex
trem
um
and
inco
rrec
tly
clas
sify
the
po
int.
Am
ore
rig
oro
us
met
ho
dto
clas
sify
ast
atio
nar
yp
oin
tis
call
edth
eex
trem
um
test
that
use
sth
ese
con
dd
eriv
ativ
e,b
ut
this
sim
ple
met
ho
dis
acce
pta
ble
.
Se
co
nd
De
riva
tiv
eTe
st
•If
f′(c
)=
0an
df′′ (
c)
>0,t
hen
cis
alo
cal
min
imu
m.
•If
f′(c
)=
0an
df′′ (
c)
<0,t
hen
cis
alo
cal
max
imu
m.
No
teth
atth
ese
con
dd
eriv
ativ
ete
stca
nn
ot
be
use
dto
ver
ify
anex
trem
aif
the
firs
to
rse
con
dd
eriv
a-ti
ve
do
esn
ot
exis
t.
Info
rma
tio
n
Sta
tio
nar
yP
oin
tF
irst
Der
ivat
ive
Sec
on
dD
eriv
ativ
e
Min
imu
mP
oin
tze
roo
ru
nd
efin
edp
osi
tiv
eo
ru
nd
efin
ed
Max
imu
mP
oin
tze
roo
ru
nd
efin
edn
egat
ive
or
un
defi
ned
“Fla
tpo
int”
zero
zero
25
Tri
go
no
met
ric
Fu
nct
ion
Der
ivat
ive
sin
xco
sx
cosx
−si
nx
tan
xse
c2x
cotx
−cs
c2x
secx
secx×
tan
x
cscx
−cs
cx×
cotx
Sin
e
Th
ed
eriv
ativ
eo
fsi
ne
isco
sin
e.
d dx[s
in(x
)]=
cos(
x)
Co
sin
e
Th
ed
eriv
ativ
eo
fco
sin
eis
neg
ativ
esi
ne.
d dx[c
os(
x)]
=−
sin(x
)
Ta
ng
en
t
Usi
ng
the
qu
oti
ent
rule
(pag
e18
)an
dth
eP
yth
ago
rean
iden
tity
cos2
(x)+
sin
2(x
)=
1,
the
der
ivat
ive
of
tan
gen
tca
nb
ed
eriv
ed.
tan(x
)=
sin(x
)
cos(
x)
d dx[t
an(x
)]=
cos2
(x)+
sin
2(x
)
cos2
(x)
d dx[t
an(x
)]=
1
cos2
(x)
d dx[t
an(x
)]=
sec2
(x)
Th
eref
ore
,th
ed
eriv
ativ
eo
fta
ng
ent
isth
esq
uar
eo
fse
can
t.
d dx
tan(x
)=
sec2
(x)
21
Co
tan
ge
nt
Usi
ng
the
qu
oti
ent
rule
(pag
e18
)an
dth
eP
yth
ago
rean
iden
tity
cos2
(x)+
sin
2(x
)=
1,
the
der
ivat
ive
of
cota
ng
ent
can
be
der
ived
.
cot(
x)
=co
s(x)
sin(x
)
d dx[c
ot(
x)]
=−
sin
2(x
)−
cos2
(x)
sin
2(x
)
d dx[c
ot(
x)]
=−
1
sin
2(x
)
d dx[c
ot(
x)]
=−
csc2
(x)
Th
eref
ore
,th
ed
eriv
ativ
eo
fco
tan
gen
tis
the
neg
ativ
eo
fth
esq
uar
eo
fco
seca
nt.
d dx[c
ot(
x)]
=−
csc2
(x)
Se
ca
nt
Usi
ng
the
qu
oti
ent
rule
(pag
e18
),th
ed
eriv
ativ
eo
fse
can
tca
nb
ed
eriv
ed.
sec(
x)
=1
cos(
x)
d dx[s
ec(x
)]=
sin(x
)
cos2
(x)
d dx[s
ec(x
)]=
1
cos(
x)×
sin(x
)
cos(
x)
d dx[s
ec(x
)]=
sec(
x)×
tan(x
)
Th
eref
ore
,th
ed
eriv
ativ
eo
fse
can
tis
seca
nt
mu
ltip
lied
by
tan
gen
t.
d dx[s
ec(x
)]=
sec(
x)×
tan(x
)
Co
se
ca
nt
Usi
ng
the
qu
oti
ent
rule
(pag
e18
),th
ed
eriv
ativ
eo
fco
seca
nt
can
be
der
ived
.
22
csc(
x)
=1
−si
n(x
)
d dx[c
sc(x
)]=
−co
s(x)
sin
2(x
)
d dx[c
sc(x
)]=
−1
sin(x
)×
cos(
x)
sin(x
)
d dx[c
sc(x
)]=
−cs
c(x)×
cot(
x)
Th
eref
ore
,th
ed
eriv
ativ
eo
fco
seca
nt
isth
en
egat
ive
of
cose
can
tm
ult
ipli
edb
yco
tan
gen
t.
d dx[c
sc(x
)]=
−cs
c(x)×
cot(
x)
Co
mb
inin
gw
ith
De
riva
tiv
eR
ule
s
Inm
ost
case
s,o
ne
mu
std
eter
min
eth
ed
eriv
ativ
eo
fan
anex
amp
leth
atre
qu
ires
the
use
of
der
ivat
ive
rule
sin
add
itio
nto
the
kn
ow
led
ge
of
the
der
ivat
ives
of
trig
on
om
etri
cfu
nct
ion
.O
ne
may
app
lyth
efo
rmtr
ig(a
)to
man
yex
amp
les,
wh
ere
trig
isth
etr
igo
no
met
ric
fun
ctio
nan
da
isth
ean
gle
.
Bas
edo
nth
ech
ain
rule
(pag
e17
),th
ed
eriv
ativ
eo
ftr
ig(a
)w
ou
ldb
e(
d dx[t
rig])
(a)×
d dx[a
].
d dx[t
rig
(a)]
=
(
d dx[t
rig](
a))
×d dx[a
],
wh
ere
d dx[t
rig]
isth
ed
eriv
ativ
eo
fth
etr
igo
no
met
ric
fun
ctio
n,a
nd
d dx[a
]is
the
der
ivat
ive
of
the
ang
le.
Ex
am
ple
:si
n(2
x)
d dx[s
in(2
x)]
=
(
d dx
[ sin
] (2x))
×d dx[2
x]
d dx[s
in(2
x)]
=(c
os(
2x))×
2
d dx[s
in(2
x)]
=2
cos(
2x)
As
ym
pto
tes
Ali
nea
ras
ym
pto
teis
ast
raig
ht
lin
eth
ata
gra
ph
app
roac
hes
,b
ut
do
esn
ot
bec
om
eid
enti
cal
to.
Asy
mp
tote
sar
efo
rmal
lyd
efin
edu
sin
gli
mit
s.S
eeth
eth
eas
ym
pto
tes
sect
ion
of
the
lim
its
chap
ter
on
pag
e7
for
mo
rein
form
atio
n.
23