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Mathematics SL guide 17
Sylla
bu
s
Sylla
bu
s co
nte
nt
Top
ic 1
—A
lge
bra
9
ho
urs
T
he
aim
of
this
to
pic
is
to i
ntr
oduce
stu
den
ts t
o s
om
e bas
ic a
lgeb
raic
conce
pts
and a
ppli
cati
ons.
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
1.1
A
rith
met
ic s
equen
ces
and s
erie
s; s
um
of
finit
e
arit
hm
etic
ser
ies;
geo
met
ric
sequen
ces
and s
erie
s;
sum
of
finit
e an
d infi
nit
e geo
met
ric
seri
es.
Sig
ma
nota
tion.
Tec
hnolo
gy m
ay b
e use
d t
o g
ener
ate
and
dis
pla
y s
equen
ces
in s
ever
al w
ays.
Lin
k t
o 2
.6, ex
ponen
tial
funct
ions.
In
t: T
he
ches
s le
gen
d (
Sis
sa i
bn D
ahir
).
In
t: A
ryab
hat
ta i
s so
met
imes
consi
der
ed t
he
“fat
her
of
algeb
ra”.
Com
par
e w
ith
al-K
haw
ariz
mi.
TO
K:
How
did
Gau
ss a
dd u
p i
nte
ger
s fr
om
1 t
o 1
00?
Dis
cuss
the
idea
of
mat
hem
atic
al
intu
itio
n a
s th
e bas
is f
or
form
al p
roof.
TO
K:
Deb
ate
over
the
val
idit
y o
f th
e noti
on o
f
“infi
nit
y”:
fin
itis
ts s
uch
as
L. K
ronec
ker
consi
der
that
“a
mat
hem
atic
al o
bje
ct d
oes
not
exis
t unle
ss i
t ca
n b
e co
nst
ruct
ed f
rom
nat
ura
l
num
ber
s in
a f
init
e num
ber
of
step
s”.
TO
K:
What
is
Zen
o’s
dic
hoto
my p
arad
ox?
How
far
can
mat
hem
atic
al f
acts
be
from
intu
itio
n?
A
ppli
cati
ons.
E
xam
ple
s in
clude
com
pound i
nte
rest
and
po
pula
tion g
row
th.
Mathematics SL guide18
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
1.2
E
lem
enta
ry t
reat
men
t o
f ex
po
nen
ts a
nd
logar
ith
ms.
Examples:
3 4
16
8;
16
3lo
g8
4;
log
32
5lo
g2
; 4
31
2(2
)2
.
Ap
pl:
Ch
emis
try 1
8.1
(C
alcu
lati
on
of
pH
).
TO
K:
Are
lo
gar
ith
ms
an i
nven
tio
n o
r
dis
cover
y?
(Th
is t
opic
is
an o
ppo
rtu
nit
y f
or
teac
her
s to
gen
erat
e re
flec
tio
n o
n “
the
nat
ure
of
mat
hem
atic
s”.)
L
aws
of
expo
nen
ts;
law
s of
logar
ith
ms.
Ch
ange
of
bas
e.
Examples:
4
ln7
log
ln4
7,
25
5 5
log
log
log
125
3125
25
2.
Lin
k t
o 2
.6, lo
gar
ith
mic
funct
ion
s.
1.3
T
he
bin
om
ial
theo
rem
:
expan
sio
n o
f (
),n
ab
n.
Co
un
tin
g p
rin
ciple
s m
ay b
e u
sed
in
th
e
dev
elo
pm
ent
of
the
theo
rem
.
Aim
8:
Pas
cal’
s tr
ian
gle
. A
ttri
buti
ng t
he
ori
gin
of
a m
ath
emat
ical
dis
cover
y t
o t
he
wro
ng
mat
hem
atic
ian
.
Int:
Th
e so
-cal
led
“P
asca
l’s
tria
ngle
” w
as
kn
ow
n i
n C
hin
a m
uch
ear
lier
th
an P
asca
l.
Cal
cula
tio
n o
f bin
om
ial
coef
fici
ents
usi
ng
Pas
cal’
s tr
ian
gle
an
dn r
.
n r s
ho
uld
be
fou
nd
usi
ng b
oth
th
e fo
rmu
la
and
tec
hn
olo
gy.
Example
: fi
ndin
g
6 r f
rom
in
pu
ttin
g
6n
ry
CX
an
d t
hen
rea
din
g c
oef
fici
ents
fro
m
the
table
.
Lin
k t
o 5
.8,
bin
om
ial
dis
trib
uti
on
. N
ot
req
uir
ed
:
form
al t
reat
men
t o
f per
mu
tati
on
s an
d f
orm
ula
for n
rP.
Mathematics SL guide 19
Syllabus content
Top
ic 2
—Fu
nc
tio
ns
an
d e
qu
atio
ns
24
ho
urs
T
he
aim
s of
this
to
pic
are
to e
xplo
re t
he
noti
on o
f a
funct
ion a
s a
unif
yin
g t
hem
e in
mat
hem
atic
s, a
nd t
o a
pply
funct
ional
met
hods
to a
var
iety
of
mat
hem
atic
al s
ituat
ions.
It
is e
xpec
ted t
hat
exte
nsi
ve
use
wil
l be
mad
e of
tech
nolo
gy i
n b
oth
the
dev
elo
pm
ent
and t
he
appli
cati
on o
f th
is t
opic
, ra
ther
than
elab
ora
te a
nal
yti
cal
tech
niq
ues
. O
n e
xam
inat
ion p
aper
s, q
ues
tions
may
be
set
requir
ing t
he
gra
phin
g o
f fu
nct
ions
that
do n
ot
expli
citl
y a
ppea
r on t
he
syll
abus,
and s
tuden
ts m
ay n
eed t
o c
hoose
the
appro
pri
ate
vie
win
g w
indow
. F
or
those
funct
ions
expli
citl
y m
enti
oned
, ques
tions
may a
lso b
e se
t on
com
posi
tion o
f th
ese
funct
ions
wit
h t
he
linea
r fu
nct
ion yaxb
.
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
2.1
C
once
pt
of
funct
ion
:(
)fx
fx
.
Dom
ain, ra
nge;
im
age
(val
ue)
.
Example
: fo
r 2
xx
, dom
ain i
s 2
x,
range
is
0y
.
A g
raph i
s hel
pfu
l in
vis
ual
izin
g t
he
range.
Int:
The
dev
elo
pm
ent
of
funct
ions,
Ren
e
Des
cart
es (
Fra
nce
), G
ott
frie
d W
ilhel
m L
eibniz
(Ger
man
y)
and L
eonhar
d E
ule
r (S
wit
zerl
and).
Com
posi
te f
unct
ions.
(
)(
())
fgx
fgx
. T
OK
: Is
zer
o t
he
sam
e as
“noth
ing”?
TO
K:
Is m
athem
atic
s a
form
al l
anguag
e?
Iden
tity
funct
ion. In
ver
se f
unct
ion
1f
. 1
1(
)()
()(
)ff
xf
fx
x.
On e
xam
inat
ion p
aper
s, s
tuden
ts w
ill only
be
asked
to f
ind the
inver
se o
f a one-to-one
funct
ion.
Not
req
uir
ed
:
dom
ain r
estr
icti
on.
2.2
T
he
gra
ph o
f a
funct
ion;
its
equat
ion
()
yfx
.
Ap
pl:
Chem
istr
y 1
1.3
.1 (
sket
chin
g a
nd
inte
rpre
ting g
raphs)
; geo
gra
phic
skil
ls.
TO
K:
How
acc
ura
te i
s a
vis
ual
rep
rese
nta
tion
of
a m
athem
atic
al c
once
pt?
(L
imit
s of
gra
phs
in d
eliv
erin
g i
nfo
rmat
ion a
bout
funct
ions
and
phen
om
ena
in g
ener
al, re
levan
ce o
f m
odes
of
repre
senta
tion.)
Funct
ion g
raphin
g s
kil
ls.
Inves
tigat
ion o
f key
fea
ture
s of
gra
phs,
such
as
max
imu
m a
nd m
inim
um
valu
es, in
terc
epts
,
hori
zonta
l an
d v
erti
cal
asym
pto
tes,
sym
met
ry,
and c
onsi
der
atio
n o
f dom
ain a
nd r
ange.
Note
the
dif
fere
nce
in t
he
com
man
d t
erm
s
“dra
w”
and “
sket
ch”.
Use
of
tech
nolo
gy t
o g
raph a
var
iety
of
funct
ions,
incl
udin
g o
nes
not
spec
ific
ally
men
tioned
.
An a
nal
yti
c ap
pro
ach i
s al
so e
xpec
ted f
or
sim
ple
funct
ions,
incl
udin
g a
ll t
hose
lis
ted
under
to
pic
2.
The
gra
ph o
f 1(
)y
fx
as
the
refl
ecti
on i
n
the
line yx
of
the
gra
ph o
f (
)y
fx
.
Lin
k t
o 6
.3, lo
cal
max
imum
and m
inim
um
poin
ts.
Mathematics SL guide20
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
2.3
T
ran
sfo
rmat
ion
s of
gra
ph
s.
Tec
hn
olo
gy s
ho
uld
be
use
d t
o i
nves
tigat
e th
ese
tran
sform
atio
ns.
Ap
pl:
Eco
no
mic
s 1
.1 (
shif
tin
g o
f su
pply
an
d
dem
and
cu
rves
).
Tra
nsl
atio
ns:
(
)y
fx
b;
()
yfxa
.
Ref
lect
ion
s (i
n b
oth
axes
):
()
yfx
;
()
yfx
.
Ver
tica
l st
retc
h w
ith
sca
le f
acto
r p:
()
ypfx
.
Str
etch
in
th
e x-
dir
ecti
on
wit
h s
cale
fac
tor
1 q:
yfqx
.
Tra
nsl
atio
n b
y t
he
vec
tor
3 2 d
eno
tes
ho
rizo
nta
l sh
ift
of
3 u
nit
s to
th
e ri
gh
t, a
nd
ver
tica
l sh
ift
of
2 d
ow
n.
Co
mpo
site
tra
nsf
orm
atio
ns.
Example
: 2
yx
use
d t
o o
bta
in
23
2y
x b
y
a st
retc
h o
f sc
ale
fact
or
3 i
n t
he y-
dir
ecti
on
foll
ow
ed b
y a
tra
nsl
atio
n o
f 0 2
.
2.4
T
he
qu
adra
tic
funct
ion
2
xax
bxc
: it
s
gra
ph
, y-
inte
rcep
t (0
,)c
. A
xis
of
sym
met
ry.
Th
e fo
rm
()(
)x
axpxq
,
x-in
terc
epts
(,
0)
p a
nd
(,
0)
q.
Th
e fo
rm
2(
)x
axh
k,
ver
tex (
,)
hk
.
Can
did
ates
are
expec
ted
to
be
able
to
chan
ge
fro
m o
ne
form
to
an
oth
er.
Lin
ks
to 2
.3,
tran
sform
atio
ns;
2.7
, q
uad
rati
c
equ
atio
ns.
Ap
pl:
Ch
emis
try 1
7.2
(eq
uil
ibri
um
law
).
Ap
pl:
Ph
ysi
cs 2
.1 (
kin
emat
ics)
.
Ap
pl:
Ph
ysi
cs 4
.2 (
sim
ple
har
mo
nic
mo
tio
n).
Ap
pl:
Ph
ysi
cs 9
.1 (
HL
only
) (p
roje
ctil
e
mo
tio
n).
Mathematics SL guide 21
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
2.5
T
he
reci
pro
cal
fun
ctio
n
1x
x,
0x
: it
s
gra
ph
an
d s
elf-
inver
se n
ature
.
Th
e ra
tio
nal
fu
nct
ion
axb
xcx
d a
nd
its
gra
ph
.
Examples:
4
2(
),
32
3hx
xx
;
75
, 2
52
xy
xx
.
Ver
tica
l an
d h
ori
zon
tal
asym
pto
tes.
D
iagra
ms
sho
uld
in
clu
de
all
asym
pto
tes
and
inte
rcep
ts.
2.6
E
xpo
nen
tial
fu
nct
ion
s an
d t
hei
r gra
ph
s:
xx
a,
0a
, ex
x.
In
t: T
he
Bab
ylo
nia
n m
eth
od
of
mu
ltip
lica
tion
: 2
22
()
2
ab
ab
ab
. S
ulb
a S
utr
as i
n a
nci
ent
Ind
ia a
nd
th
e B
akh
shal
i M
anu
scri
pt
conta
ined
an a
lgeb
raic
form
ula
for
solv
ing q
uad
rati
c
equ
atio
ns.
Lo
gar
ith
mic
fu
nct
ions
and t
hei
r gra
ph
s:
loga
xx
, 0
x,
lnx
x,
0x
.
Rel
atio
nsh
ips
bet
wee
n t
hes
e fu
nct
ion
s:
lne
xxa
a;
log
x
aa
x;
logax
ax
, 0
x.
Lin
ks
to 1
.1,
geo
met
ric
sequ
ence
s; 1
.2,
law
s o
f
expo
nen
ts a
nd
lo
gar
ith
ms;
2.1
, in
ver
se
fun
ctio
ns;
2.2
, gra
phs
of
inver
ses;
an
d 6
.1,
lim
its.
Mathematics SL guide22
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
2.7
S
olv
ing e
qu
atio
ns,
bo
th g
raph
ical
ly a
nd
anal
yti
call
y.
Use
of
tech
no
logy t
o s
olv
e a
var
iety
of
equ
atio
ns,
in
clu
din
g t
hose
wh
ere
ther
e is
no
appro
pri
ate
anal
yti
c ap
pro
ach
.
So
luti
ons
may
be
refe
rred
to
as
roo
ts o
f
equ
atio
ns
or
zero
s of
fun
ctio
ns.
Lin
ks
to 2
.2,
funct
ion
gra
ph
ing s
kil
ls;
and
2.3
–
2.6
, eq
uat
ion
s in
vo
lvin
g s
pec
ific
fu
nct
ion
s.
Examples:
4
56
0e
sin
,x
xx
x.
So
lvin
g
20
ax
bxc
, 0
a.
Th
e q
uad
rati
c fo
rmu
la.
Th
e d
iscr
imin
ant
24
bac
an
d t
he
nat
ure
of
the
roo
ts,
that
is,
tw
o d
isti
nct
rea
l ro
ots
, tw
o
equ
al r
eal
roots
, n
o r
eal
roots
.
Example
: F
ind
k g
iven
that
th
e eq
uat
ion
23
20
kxxk
has
tw
o e
qu
al r
eal
roo
ts.
So
lvin
g e
xpo
nen
tial
eq
uat
ion
s.
Examples:
1
210
x,
11
93
x
x.
Lin
k t
o 1
.2, ex
po
nen
ts a
nd
lo
gar
ith
ms.
2.8
A
ppli
cati
ons
of
gra
ph
ing s
kil
ls a
nd
solv
ing
equ
atio
ns
that
rel
ate
to r
eal-
life
sit
uat
ion
s.
Lin
k t
o 1
.1,
geo
met
ric
seri
es.
Ap
pl:
Co
mpo
un
d i
nte
rest
, gro
wth
an
d d
ecay
;
pro
ject
ile
mo
tio
n;
bra
kin
g d
ista
nce
; el
ectr
ical
circ
uit
s.
Ap
pl:
Ph
ysi
cs 7
.2.7
–7
.2.9
, 1
3.2
.5,
13
.2.6
,
13
.2.8
(ra
dio
acti
ve
dec
ay a
nd
hal
f-li
fe)
Mathematics SL guide 23
Syllabus content
Top
ic 3
—C
irc
ula
r fu
nc
tio
ns
an
d t
rig
on
om
etr
y
16
ho
urs
T
he
aim
s o
f th
is t
opic
are
to
ex
plo
re t
he
circ
ula
r fu
nct
ion
s an
d t
o s
olv
e pro
ble
ms
usi
ng t
rigo
no
met
ry.
On
ex
amin
atio
n p
aper
s, r
adia
n m
easu
re s
ho
uld
be
assu
med
un
less
oth
erw
ise
ind
icat
ed.
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
3.1
T
he
circ
le:
rad
ian
mea
sure
of
angle
s; l
ength
of
an a
rc;
area
of
a se
ctor.
Rad
ian
mea
sure
may
be
expre
ssed
as
exac
t
mu
ltip
les
of
, o
r d
ecim
als.
In
t: S
eki
Tak
akaz
u c
alcu
lati
ng
to
ten
dec
imal
pla
ces.
In
t: H
ippar
chus,
Men
elau
s an
d P
tole
my.
In
t: W
hy a
re t
her
e 36
0 d
egre
es i
n a
co
mple
te
turn
? L
inks
to B
abylo
nia
n m
ath
emat
ics.
TO
K:
Whic
h i
s a
bet
ter
mea
sure
of
angle
:
rad
ian
or
deg
ree?
Wh
at a
re t
he
“bes
t” c
rite
ria
by w
hic
h t
o d
ecid
e?
TO
K:
Eu
clid
’s a
xio
ms
as t
he
bu
ildin
g b
lock
s
of
Eu
clid
ean
geo
met
ry.
Lin
k t
o n
on
-Eu
clid
ean
geo
met
ry.
3.2
D
efin
itio
n o
f co
s a
nd
sin
in
ter
ms
of
the
un
it c
ircl
e.
A
im 8
: W
ho
rea
lly i
nven
ted
“P
yth
ago
ras’
theo
rem
”?
In
t: T
he
firs
t w
ork
to
ref
er e
xpli
citl
y t
o t
he
sin
e as
a f
unct
ion
of
an a
ngle
is
the
Ary
abh
atiy
a of
Ary
abh
ata
(ca.
51
0).
TO
K:
Tri
go
no
met
ry w
as d
evel
oped
by
succ
essi
ve
civil
izat
ion
s an
d c
ult
ure
s. H
ow
is
mat
hem
atic
al k
no
wle
dge
con
sider
ed f
rom
a
soci
ocu
ltura
l per
spec
tive?
Def
init
ion
of
tan
as
sin
cos
. T
he
equat
ion
of
a st
raig
ht
lin
e th
rou
gh
th
e
ori
gin
is
tan
yx
.
Ex
act
val
ues
of
trig
on
om
etri
c ra
tio
s of
0,
,,
,6
43
2 a
nd
thei
r m
ult
iple
s.
Exa
mple
s:
sin
,co
s,
tan
21
03
24
32
.
Mathematics SL guide24
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
3.3
T
he
Pyth
ago
rean
id
enti
ty
22
cos
sin
1.
Do
uble
an
gle
iden
titi
es f
or
sin
e an
d c
osi
ne.
Sim
ple
geo
met
rica
l d
iagra
ms
and
/or
tech
no
logy m
ay b
e u
sed
to
ill
ust
rate
th
e d
ou
ble
angle
form
ula
e (a
nd o
ther
tri
gono
met
ric
iden
titi
es).
Rel
atio
nsh
ip b
etw
een
tri
go
no
met
ric
rati
os.
Examples:
Giv
en s
in,
find
ing p
oss
ible
val
ues
of
tan
wit
ho
ut
fin
din
g
.
Giv
en
3co
s4
x,
and
x i
s ac
ute
, fi
nd
sin
2x
wit
ho
ut
fin
din
g x
.
3.4
T
he
circ
ula
r fu
nct
ions
sinx
, co
sx
an
d t
anx
:
thei
r d
om
ain
s an
d r
anges
; am
pli
tud
e, t
hei
r
per
iod
ic n
atu
re;
and
th
eir
gra
ph
s.
Appl:
Ph
ysi
cs 4
.2 (
sim
ple
har
mo
nic
mo
tio
n).
Co
mpo
site
fun
ctio
ns
of
the
form
()
sin
()
fx
abxc
d.
Examples:
()
tan
4fx
x,
()
2co
s3
(4)
1fx
x.
Tra
nsf
orm
atio
ns.
Example
: si
ny
x u
sed
to
obta
in
3si
n2
yx
by a
str
etch
of
scal
e fa
cto
r 3
in
th
e y-
dir
ecti
on
and
a s
tret
ch o
f sc
ale
fact
or
1 2 i
n t
he
x-d
irec
tio
n.
Lin
k t
o 2
.3, tr
ansf
orm
atio
n o
f gra
ph
s.
Ap
pli
cati
ons.
E
xam
ple
s in
clu
de
hei
gh
t o
f ti
de,
mo
tio
n o
f a
Fer
ris
wh
eel.
Mathematics SL guide 25
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
3.5
S
olv
ing t
rigo
no
met
ric
equ
atio
ns
in a
fin
ite
inte
rval
, both
gra
ph
ical
ly a
nd
an
alyti
call
y.
Exa
mple
s: 2
sin
1x
, 0
2x
,
2si
n2
3co
sx
x,
oo
0180
x,
2ta
n3
(4)
1x
, x
.
E
qu
atio
ns
lead
ing t
o q
uad
rati
c eq
uat
ion
s in
sin
,co
so
rta
nx
xx
.
No
t req
uir
ed
:
the
gen
eral
so
luti
on
of
trig
on
om
etri
c eq
uat
ion
s.
Exa
mple
s:
22
sin
5co
s1
0x
x f
or
04
x,
2si
nco
s2
xx
, x
.
3.6
S
olu
tio
n o
f tr
ian
gle
s.
Pyth
ago
ras’
th
eore
m i
s a
spec
ial
case
of
the
cosi
ne
rule
.
Aim
8:
Att
ribu
tin
g t
he
ori
gin
of
a
mat
hem
atic
al d
isco
ver
y t
o t
he
wro
ng
mat
hem
atic
ian
.
Int:
Cosi
ne
rule
: A
l-K
ashi
and
Pyth
ago
ras.
T
he
cosi
ne
rule
.
Th
e si
ne
rule
, in
clu
din
g t
he
ambig
uo
us
case
.
Are
a of
a tr
ian
gle
, 1
sin
2a
bC
.
Lin
k w
ith
4.2
, sc
alar
pro
duct
, n
oti
ng t
hat
: 2
22
2cab
ca
bab
.
Ap
pli
cati
ons.
E
xam
ple
s in
clu
de
nav
igat
ion
, pro
ble
ms
in t
wo
and
thre
e d
imen
sion
s, i
ncl
ud
ing a
ngle
s of
elev
atio
n a
nd
dep
ress
ion
.
TO
K:
No
n-E
ucl
idea
n g
eom
etry
: an
gle
su
m o
n
a glo
be
gre
ater
th
an 1
80
°.
Mathematics SL guide26
Syllabus content
Top
ic 4
—V
ec
tors
1
6 h
ou
rs
The
aim
of
this
topic
is
to p
rovid
e an
ele
men
tary
intr
oduct
ion t
o v
ecto
rs,
incl
udin
g b
oth
alg
ebra
ic a
nd g
eom
etri
c ap
pro
aches
. T
he
use
of
dynam
ic g
eom
etry
soft
war
e is
extr
emel
y h
elpfu
l to
vis
ual
ize
situ
atio
ns
in t
hre
e dim
ensi
ons.
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
4.1
V
ecto
rs a
s dis
pla
cem
ents
in t
he
pla
ne
and i
n
thre
e dim
ensi
ons.
Lin
k t
o t
hre
e-dim
ensi
onal
geo
met
ry, x,
y a
nd z
-
axes
.
Appl:
Physi
cs 1
.3.2
(vec
tor
sum
s an
d
dif
fere
nce
s) P
hysi
cs 2
.2.2
, 2.2
.3 (
vec
tor
resu
ltan
ts).
TOK:
How
do w
e re
late
a t
heo
ry t
o t
he
auth
or?
Who d
evel
oped
vec
tor
anal
ysi
s:
JW G
ibbs
or
O H
eavis
ide?
Com
ponen
ts o
f a
vec
tor;
colu
mn
repre
senta
tion;
1 21
23
3v vv
vv
v
vi
jk
.
Com
ponen
ts a
re w
ith r
espec
t to
the
unit
vec
tors
i, j
and k
(st
andar
d b
asis
).
Alg
ebra
ic a
nd g
eom
etri
c ap
pro
aches
to t
he
foll
ow
ing:
Appli
cati
ons
to s
imple
geo
met
ric
figure
s ar
e
esse
nti
al.
th
e su
m a
nd d
iffe
rence
of
two v
ecto
rs;
the
zero
vec
tor,
the
vec
tor v
;
The
dif
fere
nce
of v
and
w i
s
()
vw
vw
. V
ecto
r su
ms
and d
iffe
rence
s
can b
e re
pre
sente
d b
y t
he
dia
gonal
s of
a
par
alle
logra
m.
m
ult
ipli
cati
on b
y a
sca
lar,
kv
; par
alle
l
vec
tors
;
Mult
ipli
cati
on b
y a
sca
lar
can b
e il
lust
rate
d b
y
enla
rgem
ent.
m
agnit
ude
of
a vec
tor,
v
;
unit
vec
tors
; bas
e vec
tors
; i,
j a
nd k
;
posi
tion v
ecto
rs O
Aa
;
A
BO
BO
Aba
. D
ista
nce
bet
wee
n p
oin
ts A
and B
is
the
magnit
ude
of
AB
.
Mathematics SL guide 27
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
4.2
T
he
scal
ar p
rod
uct
of
two
vec
tors
. T
he
scal
ar p
rod
uct
is
also
kn
ow
n a
s th
e “d
ot
pro
du
ct”.
Lin
k t
o 3
.6, co
sin
e ru
le.
Per
pen
dic
ula
r vec
tors
; par
alle
l vec
tors
. F
or
no
n-z
ero
vec
tors
, 0
vw
is
equ
ival
ent
to
the
vec
tors
bei
ng p
erpen
dic
ula
r.
Fo
r par
alle
l vec
tors
, k
wv
, vw
vw
.
Th
e an
gle
bet
wee
n t
wo
vec
tors
.
4.3
V
ecto
r eq
uat
ion
of
a li
ne
in t
wo
an
d t
hre
e
dim
ensi
on
s:
tra
b.
Rel
evan
ce o
f a
(po
siti
on)
and
b (
dir
ecti
on
).
Inte
rpre
tati
on
of t a
s ti
me
and
b a
s vel
oci
ty,
wit
h b
rep
rese
nti
ng s
pee
d.
Aim
8:
Vec
tor
theo
ry i
s use
d f
or
trac
kin
g
dis
pla
cem
ent
of
obje
cts,
incl
ud
ing f
or
pea
cefu
l
and
har
mfu
l pu
rpo
ses.
TO
K:
Are
alg
ebra
an
d g
eom
etry
tw
o s
epar
ate
do
mai
ns
of
kn
ow
led
ge?
(V
ecto
r al
geb
ra i
s a
go
od
op
po
rtu
nit
y t
o d
iscu
ss h
ow
geo
met
rica
l
pro
per
ties
are
des
crib
ed a
nd
gen
eral
ized
by
algeb
raic
met
ho
ds.
)
Th
e an
gle
bet
wee
n t
wo
lin
es.
4.4
D
isti
ngu
ishin
g b
etw
een
coin
cid
ent
and
par
alle
l
lines
.
Fin
din
g t
he
poin
t o
f in
ters
ecti
on
of
two
lin
es.
Det
erm
inin
g w
het
her
tw
o l
ines
in
ters
ect.
Mathematics SL guide28
Syllabus content
Top
ic 5
—Sta
tist
ics
an
d p
rob
ab
ility
3
5 h
ou
rs
Th
e ai
m o
f th
is t
opic
is
to i
ntr
odu
ce b
asic
co
nce
pts
. It
is
expec
ted
that
mo
st o
f th
e ca
lcu
lati
on
s re
quir
ed w
ill
be
do
ne
usi
ng t
ech
nolo
gy,
bu
t ex
pla
nat
ion
s of
calc
ula
tio
ns
by h
and
may
en
han
ce u
nd
erst
andin
g.
Th
e em
ph
asis
is
on
un
der
stan
din
g a
nd
in
terp
reti
ng t
he
resu
lts
obta
ined
, in
con
tex
t. S
tati
stic
al t
able
s w
ill
no
lon
ger
be
allo
wed
in
ex
amin
atio
ns.
Wh
ile
man
y o
f th
e ca
lcu
lati
on
s re
quir
ed i
n e
xam
inat
ions
are
esti
mat
es,
it i
s li
kel
y t
hat
th
e co
mm
and
ter
ms
“wri
te d
ow
n”,
“fin
d”
and
“ca
lcu
late
” w
ill
be
use
d.
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
5.1
C
on
cepts
of
po
pula
tio
n,
sam
ple
, ra
nd
om
sam
ple
, d
iscr
ete
and
co
nti
nu
ou
s d
ata.
Pre
senta
tion o
f dat
a: f
requen
cy d
istr
ibuti
ons
(tab
les)
; fr
equen
cy h
isto
gra
ms
wit
h e
qual
cla
ss
inte
rval
s;
Co
nti
nu
ou
s an
d d
iscr
ete
dat
a.
Ap
pl:
Psy
cholo
gy:
des
crip
tive
stat
isti
cs,
ran
do
m s
ample
(var
iou
s pla
ces
in t
he
gu
ide)
.
Aim
8:
Mis
lead
ing s
tati
stic
s.
Int:
Th
e S
t P
eter
sbu
rg p
arad
ox
, C
heb
ych
ev,
Pav
lovsk
y.
bo
x-a
nd
-wh
isker
plo
ts;
ou
tlie
rs.
Ou
tlie
r is
def
ined
as
mo
re t
han
1.5
IQR
fro
m
the
nea
rest
qu
arti
le.
Tec
hn
olo
gy m
ay b
e u
sed
to p
rod
uce
his
togra
ms
and
bo
x-a
nd-w
his
ker
plo
ts.
Gro
uped
dat
a: u
se o
f m
id-i
nte
rval
val
ues
for
calc
ula
tio
ns;
inte
rval
wid
th;
up
per
an
d l
ow
er
inte
rval
bou
nd
arie
s; m
od
al c
lass
.
No
t req
uir
ed
:
freq
uen
cy d
ensi
ty h
isto
gra
ms.
Mathematics SL guide 29
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
5.2
S
tati
stic
al m
easu
res
and t
hei
r in
terp
reta
tions.
Cen
tral
ten
den
cy:
mea
n, m
edia
n, m
ode.
Quar
tile
s, p
erce
nti
les.
On e
xam
inat
ion p
aper
s, d
ata
wil
l be
trea
ted a
s
the
popula
tion.
Cal
cula
tion o
f m
ean u
sing f
orm
ula
and
tech
nolo
gy. S
tuden
ts s
hould
use
mid
-inte
rval
val
ues
to e
stim
ate
the
mea
n o
f gro
uped
dat
a.
Ap
pl:
Psy
cholo
gy:
des
crip
tive
stat
isti
cs
(var
ious
pla
ces
in t
he
guid
e).
Ap
pl:
Sta
tist
ical
cal
cula
tions
to s
how
pat
tern
s
and c
han
ges
; geo
gra
phic
skil
ls;
stat
isti
cal
gra
phs.
Ap
pl:
Bio
logy 1
.1.2
(ca
lcula
ting m
ean a
nd
stan
dar
d d
evia
tion )
; B
iolo
gy 1
.1.4
(co
mpar
ing
mea
ns
and s
pre
ads
bet
wee
n t
wo o
r m
ore
sam
ple
s).
Int:
Dis
cuss
ion o
f th
e dif
fere
nt
form
ula
e fo
r
var
iance
.
TO
K:
Do d
iffe
rent
mea
sure
s of
centr
al
tenden
cy e
xpre
ss d
iffe
rent
pro
per
ties
of
the
dat
a? A
re t
hes
e m
easu
res
inven
ted o
r
dis
cover
ed?
Could
mat
hem
atic
s m
ake
alte
rnat
ive,
equal
ly t
rue,
form
ula
e? W
hat
does
this
tel
l us
about
mat
hem
atic
al t
ruth
s?
TO
K:
How
eas
y i
s it
to l
ie w
ith s
tati
stic
s?
Dis
per
sion:
range,
inte
rquar
tile
ran
ge,
var
iance
, st
andar
d d
evia
tion.
Eff
ect
of
const
ant
chan
ges
to t
he
ori
gin
al d
ata.
Cal
cula
tion o
f st
andar
d d
evia
tion/v
aria
nce
usi
ng o
nly
tec
hnolo
gy.
Lin
k t
o 2
.3, tr
ansf
orm
atio
ns.
Examples:
If 5
is
subtr
acte
d f
rom
all
the
dat
a it
ems,
then
the
mea
n i
s dec
reas
ed b
y 5
, but
the
stan
dar
d
dev
iati
on i
s unch
anged
.
If a
ll t
he
dat
a it
ems
are
double
d, th
e m
edia
n i
s
dou
ble
d,
but
the
var
iance
is
incr
ease
d b
y a
fact
or
of
4.
Appli
cati
ons.
5.3
C
um
ula
tive
freq
uen
cy;
cum
ula
tive
freq
uen
cy
gra
phs;
use
to f
ind m
edia
n, quar
tile
s,
per
centi
les.
Val
ues
of
the
med
ian a
nd q
uar
tile
s pro
duce
d
by t
echnolo
gy m
ay b
e dif
fere
nt
from
those
obta
ined
fro
m a
cum
ula
tive
freq
uen
cy g
raph.
Mathematics SL guide30
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
5.4
L
inea
r co
rrel
atio
n o
f biv
aria
te d
ata.
In
dep
end
ent
var
iable
x, d
epen
den
t var
iable
y.
Ap
pl:
Ch
emis
try 1
1.3
.3 (
curv
es o
f bes
t fi
t).
Ap
pl:
Geo
gra
ph
y (
geo
gra
ph
ic s
kil
ls).
Mea
sure
s o
f co
rrel
atio
n;
geo
gra
ph
ic s
kil
ls.
Ap
pl:
Bio
logy 1
.1.6
(co
rrel
atio
n d
oes
not
imply
cau
sati
on).
TO
K:
Can
we
pre
dic
t th
e val
ue
of
x fr
om
y,
usi
ng t
his
eq
uat
ion?
TO
K:
Can
all
dat
a be
mo
del
led
by a
(kn
ow
n)
mat
hem
atic
al f
un
ctio
n?
Con
sid
er t
he
reli
abil
ity
and
val
idit
y o
f m
ath
emat
ical
mo
del
s in
des
crib
ing r
eal-
life
ph
eno
men
a.
Pea
rso
n’s
pro
du
ct–
mo
men
t co
rrel
atio
n
coef
fici
ent
r.
Tec
hn
olo
gy s
ho
uld
be
use
d t
o c
alcu
late
r.
Ho
wev
er,
han
d c
alcu
lati
on
s o
f r
may
en
han
ce
un
der
stan
din
g.
Po
siti
ve,
zer
o,
neg
ativ
e; s
tro
ng,
wea
k,
no
corr
elat
ion
.
Sca
tter
dia
gra
ms;
lin
es o
f bes
t fi
t.
Th
e li
ne
of
bes
t fi
t pas
ses
thro
ugh
th
e m
ean
po
int.
Eq
uat
ion
of
the
regre
ssio
n l
ine
of
y on
x.
Use
of
the
equ
atio
n f
or
pre
dic
tio
n p
urp
ose
s.
Mat
hem
atic
al a
nd
co
nte
xtu
al i
nte
rpre
tati
on.
No
t req
uir
ed
:
the
coef
fici
ent
of
det
erm
inat
ion
R2.
Tec
hn
olo
gy s
ho
uld
be
use
d f
ind
th
e eq
uat
ion
.
Inte
rpo
lati
on
, ex
trap
ola
tio
n.
5.5
C
on
cepts
of
tria
l, o
utc
om
e, e
qu
ally
lik
ely
ou
tco
mes
, sa
mple
spac
e (U
) an
d e
ven
t.
Th
e sa
mple
spac
e ca
n b
e re
pre
sen
ted
dia
gra
mm
atic
ally
in
man
y w
ays.
TO
K:
To
wh
at e
xte
nt
does
mat
hem
atic
s o
ffer
mo
del
s o
f re
al l
ife?
Is
ther
e al
way
s a
fun
ctio
n
to m
od
el d
ata
beh
avio
ur?
Th
e pro
bab
ilit
y o
f an
even
t A
is
()
P(
)(
)
nA
An
U.
Th
e co
mple
men
tary
even
ts A
an
d A
(n
ot
A).
Use
of
Ven
n d
iagra
ms,
tre
e d
iagra
ms
and
table
s of
outc
om
es.
Ex
per
imen
ts u
sin
g c
oin
s, d
ice,
car
ds
and
so
on
,
can
enh
ance
un
der
stan
din
g o
f th
e d
isti
nct
ion
bet
wee
n (
exper
imen
tal)
rel
ativ
e fr
equ
ency
an
d
(th
eore
tica
l) p
robab
ilit
y.
Sim
ula
tions
may
be
use
d t
o e
nhan
ce t
his
topic
.
Lin
ks
to 5
.1,
freq
uen
cy;
5.3
, cu
mu
lati
ve
freq
uen
cy.
Mathematics SL guide 31
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
5.6
C
om
bin
ed e
ven
ts,
P(
)A
B.
Mutu
ally
excl
usi
ve
even
ts:
P(
)0
AB
.
Condit
ional
pro
bab
ilit
y;
the
def
init
ion
P(
)P
|P
()
AB
AB
B.
Indep
enden
t ev
ents
; th
e def
init
ion
P|
P(
)P
|AB
AAB
.
Pro
bab
ilit
ies
wit
h a
nd w
ithout
repla
cem
ent.
The
non-e
xcl
usi
vit
y o
f “o
r”.
Pro
ble
ms
are
oft
en b
est
solv
ed w
ith t
he
aid o
f a
Ven
n d
iagra
m o
r tr
ee d
iagra
m, w
ithout
expli
cit
use
of
form
ula
e.
Aim
8:
The
gam
bli
ng i
ssue:
use
of
pro
bab
ilit
y
in c
asin
os.
Could
or
should
mat
hem
atic
s hel
p
incr
ease
inco
mes
in g
ambli
ng?
TO
K:
Is m
athem
atic
s use
ful
to m
easu
re r
isks?
TO
K:
Can
gam
bli
ng b
e co
nsi
der
ed a
s an
appli
cati
on o
f m
athem
atic
s? (
This
is
a good
opport
unit
y t
o g
ener
ate
a deb
ate
on t
he
nat
ure
,
role
and
eth
ics
of
mat
hem
atic
s re
gar
din
g i
ts
appli
cati
ons.
)
5.7
C
once
pt
of
dis
cret
e ra
ndom
var
iable
s an
d t
hei
r
pro
bab
ilit
y d
istr
ibuti
ons.
Sim
ple
exam
ple
s only
, su
ch a
s:
1P
()
(4)
18
Xx
x f
or
1,2,3
x;
56
7P
()
,,
18
18
18
Xx
.
Ex
pec
ted v
alue
(mea
n),
E(
)X
for
dis
cret
e dat
a.
Appli
cati
ons.
E(
)0
X i
ndic
ates
a f
air
gam
e w
her
e X
repre
sents
the
gai
n o
f one
of
the
pla
yer
s.
Exam
ple
s in
clude
gam
es o
f ch
ance
.
Mathematics SL guide32
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
5.8
B
ino
mia
l dis
trib
uti
on
.
Mea
n a
nd
var
ian
ce o
f th
e bin
om
ial
dis
trib
uti
on
.
No
t req
uir
ed
: fo
rmal
pro
of
of
mea
n a
nd
var
ian
ce.
Lin
k t
o 1
.3,
bin
om
ial
theo
rem
.
Co
nd
itio
ns
un
der
whic
h r
and
om
var
iable
s hav
e
this
dis
trib
uti
on
.
Tec
hn
olo
gy i
s u
sual
ly t
he
bes
t w
ay o
f
calc
ula
tin
g b
ino
mia
l pro
bab
ilit
ies.
5.9
N
orm
al d
istr
ibu
tio
ns
and
cu
rves
.
Sta
nd
ardiz
atio
n o
f n
orm
al v
aria
ble
s (z-val
ues
,
z-sc
ore
s).
Pro
per
ties
of
the
no
rmal
dis
trib
uti
on
.
Pro
bab
ilit
ies
and
val
ues
of
the
var
iable
mu
st b
e
fou
nd
usi
ng t
ech
nolo
gy.
Lin
k t
o 2
.3, tr
ansf
orm
atio
ns.
Th
e st
and
ardiz
ed v
alu
e (z
) giv
es t
he
nu
mber
of
stan
dar
d d
evia
tio
ns
fro
m t
he
mea
n.
Ap
pl:
Bio
logy 1
.1.3
(li
nks
to n
orm
al
dis
trib
uti
on).
Ap
pl:
Psy
cholo
gy:
des
crip
tive
stat
isti
cs
(var
ious
pla
ces
in t
he
gu
ide)
.
Mathematics SL guide 33
Syllabus content
Top
ic 6
—C
alc
ulu
s 4
0 h
ou
rs
Th
e ai
m o
f th
is t
opic
is
to i
ntr
odu
ce s
tud
ents
to
the
bas
ic c
on
cepts
an
d t
ech
niq
ues
of
dif
fere
nti
al a
nd
inte
gra
l ca
lcu
lus
and
th
eir
appli
cati
on
s.
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
6.1
In
form
al i
dea
s of
lim
it a
nd
co
nver
gen
ce.
Example
: 0
.3, 0
.33
, 0
.33
3,
...
con
ver
ges
to 1 3
.
Tec
hn
olo
gy s
ho
uld
be
use
d t
o e
xplo
re i
dea
s o
f
lim
its,
nu
mer
ical
ly a
nd
gra
ph
ical
ly.
Ap
pl:
Eco
no
mic
s 1
.5 (
mar
gin
al c
ost
, m
argin
al
reven
ue,
mar
gin
al p
rofi
t).
Ap
pl:
Ch
emis
try 1
1.3
.4 (
inte
rpre
tin
g t
he
gra
die
nt
of
a cu
rve)
.
Aim
8:
Th
e deb
ate
over
whet
her
New
ton
or
Lei
bnit
z d
isco
ver
ed c
erta
in c
alcu
lus
conce
pts
.
TO
K:
What
val
ue
does
th
e kn
ow
led
ge
of
lim
its
hav
e? I
s in
finit
esim
al b
ehav
iour
appli
cable
to
rea
l li
fe?
TO
K:
Op
port
unit
ies
for
dis
cuss
ing h
ypo
thes
is
form
atio
n a
nd
tes
tin
g,
and
th
en t
he
form
al
pro
of
can
be
tack
led
by c
om
par
ing c
erta
in
case
s, t
hro
ugh
an
in
ves
tigat
ive
appro
ach
.
Lim
it n
ota
tio
n.
Example
: 2
3li
m1
x
x x
Lin
ks
to 1
.1, in
finit
e geo
met
ric
seri
es;
2.5
–2.7
,
rati
onal
and e
xponen
tial
funct
ions,
and
asym
pto
tes.
Def
init
ion
of
der
ivat
ive
from
fir
st p
rin
ciple
s as
0
()
()
()
lim
h
fxh
fx
fx
h.
Use
of
this
def
init
ion
for
der
ivat
ives
of
sim
ple
po
lyn
om
ial
funct
ion
s o
nly
.
Tec
hn
olo
gy c
ou
ld b
e use
d t
o i
llu
stra
te o
ther
der
ivat
ives
.
Lin
k t
o 1
.3,
bin
om
ial
theo
rem
.
Use
of
bo
th f
orm
s o
f no
tati
on
, d d
y x an
d fx
,
for
the
firs
t d
eriv
ativ
e.
Der
ivat
ive
inte
rpre
ted
as
gra
die
nt
fun
ctio
n a
nd
as r
ate
of
chan
ge.
Iden
tify
ing i
nte
rval
s o
n w
hic
h f
unct
ion
s ar
e
incr
easi
ng o
r dec
reas
ing.
Tan
gen
ts a
nd
norm
als,
an
d t
hei
r eq
uat
ions.
No
t req
uir
ed
:
anal
yti
c m
eth
od
s o
f ca
lcu
lati
ng l
imit
s.
Use
of
bo
th a
nal
yti
c ap
pro
ach
es a
nd
tech
no
logy.
Tec
hn
olo
gy c
an b
e u
sed
to
ex
plo
re g
raphs
and
thei
r der
ivat
ives
.
Mathematics SL guide34
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
6.2
D
eriv
ativ
e of
()
nxn
, si
nx
, co
sx
, ta
nx
,
ex a
nd
lnx
.
Dif
fere
nti
atio
n o
f a
sum
and a
rea
l m
ult
iple
of
thes
e fu
nct
ions.
The
chai
n r
ule
for
com
posi
te f
unct
ions.
The
pro
duct
and q
uoti
ent
rule
s.
Lin
k t
o 2
.1, co
mposi
tion o
f fu
nct
ions.
Tec
hnolo
gy m
ay b
e use
d to inves
tigat
e th
e ch
ain
rule
.
The
seco
nd d
eriv
ativ
e.
Use
of
both
form
s of
nota
tion,
2
2
d d
y
x a
nd
(
)fx
.
Exte
nsi
on t
o h
igher
der
ivat
ives
. d d
n
ny
x a
nd
(
)nf
x.
Mathematics SL guide 35
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
6.3
L
oca
l m
axim
um
an
d m
inim
um
po
ints
.
Tes
tin
g f
or
max
imu
m o
r m
inim
um
.
Usi
ng c
han
ge
of
sign
of
the
firs
t d
eriv
ativ
e an
d
usi
ng s
ign
of
the
seco
nd
der
ivat
ive.
Use
of
the
term
s “c
on
cave-
up”
for
()
0fx
,
and
“co
nca
ve-
do
wn
” fo
r (
)0
fx
.
Ap
pl:
pro
fit,
are
a, v
olu
me.
Po
ints
of
infl
exio
n w
ith
zer
o a
nd
no
n-z
ero
gra
die
nts
. A
t a
poin
t o
f in
flex
ion
,
()
0fx
and
ch
anges
sign
(co
nca
vit
y c
han
ge)
.
()
0fx
is
not
a su
ffic
ien
t co
ndit
ion
fo
r a
po
int
of
infl
exio
n:
for
exam
ple
, 4
yx
at
(0,0
).
Gra
ph
ical
beh
avio
ur
of
funct
ion
s,
incl
ud
ing t
he
rela
tio
nsh
ip b
etw
een t
he
gra
ph
s o
f f
, f
an
d f
.
Opti
miz
atio
n.
Bo
th “
glo
bal
” (f
or
larg
e x
) an
d “
loca
l”
beh
avio
ur.
Tec
hn
olo
gy c
an d
ispla
y t
he
gra
ph
of
a
der
ivat
ive
wit
ho
ut
expli
citl
y f
ind
ing a
n
expre
ssio
n f
or
the
der
ivat
ive.
Use
of
the
firs
t o
r se
con
d d
eriv
ativ
e te
st t
o
just
ify m
axim
um
an
d/o
r m
inim
um
val
ues
.
Ap
pli
cati
ons.
No
t req
uir
ed
:
po
ints
of
infl
exio
n w
her
e (
)fx
is n
ot
def
ined
:
for
exam
ple
, 1
3yx
at
(0,0
).
Ex
ample
s in
clu
de
pro
fit,
are
a, v
olu
me.
Lin
k t
o 2
.2,
gra
ph
ing f
unct
ion
s.
Mathematics SL guide36
Syllabus content
C
on
ten
t F
urt
he
r g
uid
an
ce
Lin
ks
6.4
In
def
init
e in
tegra
tion a
s an
ti-d
iffe
renti
atio
n.
Indef
init
e in
tegra
l of
()
nxn
, si
nx
, co
sx
,
1 x a
nd ex.
1d
lnx
xC
x,
0x
.
The
com
posi
tes
of
any o
f th
ese
wit
h t
he
linea
r
funct
ion axb
.
Example
:
1(
)co
s(2
3)
()
sin
(23)
2fx
xfx
xC
.
Inte
gra
tion b
y i
nsp
ecti
on, or
subst
ituti
on o
f th
e
form
(
())
'()d
fgxgxx
.
Examples:
42
22
1d
,si
nd
,d
sin
cos
xx
xx
xx
xx x
.
6.5
A
nti
-dif
fere
nti
atio
n w
ith a
boundar
y c
ondit
ion
to d
eter
min
e th
e co
nst
ant
term
.
Example
:
if
2d
3d
yx
xx
and
10
y w
hen
0
x, th
en
32
11
02
yx
x.
Int:
Succ
essf
ul
calc
ula
tion o
f th
e volu
me
of
the
pyra
mid
al f
rust
um
by a
nci
ent
Egypti
ans
(Egypti
an M
osc
ow
pap
yru
s).
Use
of
infi
nit
esim
als
by G
reek
geo
met
ers.
Def
init
e in
tegra
ls,
both
anal
yti
call
y a
nd u
sing
tech
nolo
gy.
()d
()
()
b agxxgb
ga
.
The
val
ue
of
som
e def
init
e in
tegra
ls c
an o
nly
be
found u
sing t
echnolo
gy.
Acc
ura
te c
alcu
lati
on o
f th
e volu
me
of
a
cyli
nder
by C
hin
ese
mat
hem
atic
ian L
iu H
ui
Are
as u
nder
curv
es (
bet
wee
n t
he
curv
e an
d t
he
x-ax
is).
Are
as b
etw
een c
urv
es.
Volu
mes
of
revolu
tion a
bout
the x-
axis
.
Stu
den
ts a
re e
xpec
ted t
o f
irst
wri
te a
corr
ect
expre
ssio
n b
efore
cal
cula
ting t
he
area
.
Tec
hnolo
gy m
ay b
e use
d t
o e
nhan
ce
under
stan
din
g o
f ar
ea a
nd v
olu
me.
Int:
Ibn A
l H
ayth
am:
firs
t m
athem
atic
ian t
o
calc
ula
te t
he
inte
gra
l of
a fu
nct
ion, in
ord
er t
o
find t
he
volu
me
of
a par
abolo
id.
6.6
K
inem
atic
pro
ble
ms
involv
ing d
ispla
cem
ent s,
vel
oci
ty v
and a
ccel
erat
ion a
. d d
sv
t;
2
2
dd
dd
vs
at
t.
Ap
pl:
Physi
cs 2
.1 (
kin
emat
ics)
.
Tota
l dis
tance
tra
vel
led.
Tota
l dis
tance
tra
vel
led
2
1
dt tvt.
