Function basics
f (a) =b
a
composite function
A
-
-
domain
2
has
venge
5-
.
a
flglxl)
↳
Find the
inure of
by interchanging
for
solving
a
Zeros
f- Cx )
:
odd
functions
even
functions
of
-
aux
a
function
"
"
+
if , for
all
in
x
ft
D,
-
fl
are
are
abort the
Sym
about
x
-
)
-
-
-
x)
:
Fox)
f CX )
Sym
origin
the
occur where
y-axis
an
-
,
"
x
+
an
-
-
:
flex )
P¥tx
=
Tr I
Z
z
)
cos
-
'
-
X
is restricted
is
E
Ext
0 EX
to
restricted
to
-
Iz
ET
I LXEE
EXPONENTS
1)
GO
2)
z)
for
at l
,
ell
rational
I
=
-
-
a
am ÷
)
(am)
s
am an
-_
.
am
an
"
""
as
a
-
m
tf
+ u
4)
'
a
m
=
:
ah
=
-
aY=X
n
Ann
atm
Logs
In ( X
Inc
-
y)
't )
lnlxb )
Ini)
Incl )
Iim
was
-
-
-
-
-
is
-
-
Inlx )
lncx)
ttaxlnlx)
tiny)
-
inly )
flux
ylnlx)
undefined for
O
In ( x ) =D
XEO
):
fund
#
l NU
g
is restricted
ten 'x
a> I
value
#
flxto
basics
'
-
sink
functions
rational
elixir D ,
greatest integer
Zo
n
even
for
,
y
polynomial
form
is
if
absolute
and
andy
function
is odd
fix)
function
x
function
functions
more
or
a
-
-
tx
Xlnlx) Xtc
-
y log
'
,
a
X
log ax
inv
is the
.
of 2x
limits
for
In order
¥n=o
tin
the
fu )
Ec
tin
to exist ,
fins
:
-
is
-
A
or
if fox)
as
can
be
made
arbitrarily
large
c
near
fun
,
Foralimit-oex.hr/.ithrstobefini#
tinhorn
L
must
-
tin town
=
right
fiygfcxy-clorxin-o.fm#
fist's
If function f- ( x )
infinite
becomes
as
techie
'
eueuypolynom.ie/whosedegreeisIlbecoMescsas2#
infinite
becomes
x
abhin.LY?Yn9IEYn7uingxbiseaonsw
't
linn
.a=u¥÷*:.:÷÷
VeHiCptp
±
is
depending
A
polynomial
of
'
V. A-
for
values
=
x
which
in
undefined
is
If
1h
µ
=
H A
,
=
eEdin9↳eHic%
leading coefficient of Den
=
y
"
"
÷ . ..
.
'
.
÷ :c: : : :
"e
The
finna
line
x
f- Cx )
-
-
-
is
a
to
.
.
if
d
V. A
or
fifa
=
-
A
-
or
tin
X→
if
¥yHx5B
find "
=
at
-
D
ti -
or
+
a
fix )
=
-
as
" "
RULES
¥Pis¥%
ssxslicmlftx)
gun )
Igg
Hx)
!!÷!*k±↳=a.usiiH5¥):!÷÷I
3)
4)
Ijm
(
fix)
( f- ( x )
fish
+
-
gtx) )
gcx ) )
=
th )
fish
t
fix)
=¥z
y g
-
y
( x)
g.
=
B
(x)
=
+
B
the limit
7) If
D
-
and if
D
-
of
glx )
the "m
then
fine
at X - c
of fcx)
# glad
f- ( x )
E
g
WE had , and if
figg
fcx )
=
fine
hlx ) =L
,
their
=
'
Qd )
if
-
-
-
if
if
degree
degree
degree
of pox )
L
of Plex)
same
g
W
-
-
B
-
D
exists , such that
-
D
.
fishy g. ( x ) =D
fix ) E
figg
Such that
flying gcx D=
,
>
Qu)
.
acx )
.
then
then
fines
then
lings
Hfcs
=
In
,
an
¥4
PIM
Qcx ,
i.
O
-
-
A
bn ore
or
-
as
/ due
coefficients of highest power in
-
###
€
=L
"
,
-
E
Ts(Htx)×=¥up(×÷)×
Rationalfunvionthm
for
tiny
gcx )
exists at X
Squeezethhr
If
in
Phx )
and Atx)
degree
N€N④£Y
continuous
The
=
draw w/o
lifting
are
tx
ex )
X
-
't
are con
except
.
denominator
=
#
continuous
"
*
aEII"b×""+
afunaionnwi-i.ua
if
⇐
.
point
at each
f- ( x )
lim
C is
in the
domain
[a
interval
off )
fcc )
if
b]
,
it is
nonce
con't
not
point
-x→c
point of
discontinuity
=
con't
the
where
O
defined
is not
o
.
(
f- Cc ) exists
.
3
rath
in their domain
.
www.noise#z.,.feb
l
p
tummy
pencil
discount
1-
,
votive
continuous
everywhere
trig.in#rigexpomntiandogfunionsacwitateechpointin-heird0mci
TX ( his
with
functions
flx )
integer functions
greatest
pos
#
(
=
x
integers 2)
.
]
of
Kinds
at
Jumpdiscontinuity
right
integer
each
#
discontinuities
when left and
hand limits
exist
,
but
at
each
TX
where
x
is
defined
#
discontinuous
is
#
continuous
are
different
are
,
since
does
it
not have
limit
a
at
/o-#
because
lying
function
int
greatest
.
and
A
=
.
=
+
,
f-( x )
has
Theorems
1) extreme
Internet
*
(
e , is ]
Special
( there
is
a
vole
:
-1hm
Such that
:
c
If
f
in
3) Continuous functions
around
the
a
limit does
not exist
DNE
,
and f- ( x )
±
=
A
as
X→
at
or
X
→
a
( must
-
be
opposite
to be
directions
con't
.
)
continuous Functions
theorem
case
value
.
on
value
2) Intermediate
V A at X=a
a
,
#
r
discontinuity
-
when
lot
a
¥
Remouzbkdisco-tinitywhenfisyfcxg.fi
Infinite
the limit
B,
€M
function
a
given point
.
lying
integer
oscillating
If
oscillates
ex )
any
#
is
:
If f
tf
f
thin
4) Composition of continuous
-
-
:
on
int
the closed
.
[e
,
b]
interval
,
then
[ ails ]
f
attains
and M is
a
min
Value
2nd
2 max
flat EM
Such that
#
a
.
E
.
value
fcb )
,
somewhere
then there
interval
the
on
is at
least
one
#
L
in the
M
on
the closed interval
WI hee
Leib ]
the closed
on
is continuous
fu)
con't
con't
is
f (c)
'
-
.
and
fcz ) and
fly )
have
opposite signs
,
then
f
hes
a
zero
in
interval
that
O
If functions to
Fund
Cz , b )
2nd
g
are
Than
@ltxfgYY.7nYnYwitwIpIInIYfoitgf5ceYj.f gite
both continuous
at
x- c ,
then
so
are
the
.ec/b.sumsldiCfeeuus:fCxs+gcx
a. Constant
multiples K
-
f- Cx )
(K
functions
following
is
a
real #
:
)
c.
:
products
ft x) g (
-
x
)
f-( x )
)
d.
ex)
quotients g:
Provided
gusto
DIFFERENTIATION
""
fLx+oD
Iim
A
→
0
function is differentiable
The deriv of y
-
-
lion
or
sx → o
ox
fcx ) @
x
at
-
-
every
defined
a
for
x
f- ' (d)
limit
,
↳ slope
of
.
,
WHICH
THE
LIMIT EXISTS
)
)
of the diff
,
the curve
as
.
rates
oocnange
sx
tlwf
tin
the
I
as
n -70
aer
÷÷÷÷÷÷÷÷÷:÷÷
in
sx
YOURSELF
CHECK
is
quotient
'
,
represents the avg rate
the
the
INSTANTANEOUS
curve
'
,
IF
Dxy
,
for f at a
change of f from
difference
called the
is
slope of the
Ingenito
f- ( x )
y
demoted
,
of
RATE
CHANGE
OF
of
f
a
a
to
ath
point
a
that point
@
FORMULAS
:÷÷÷÷÷÷÷÷÷÷÷÷÷÷i÷
da
a
o
=
,
an
¥
uh
add
=
ax
,
nun
=
-
,
,
Inu
da
en
I dug
=
,
da
dat
ady
Cutt )
day
=
,
( Power R )
tag
u ±
da
,
af
daxv
sin
,
,
,
de
"
=
i
a
"
dax arcsinu
-
n
DIX
Ina
-
date
=
,
L
-
l
L U Ll
)
:÷÷i÷÷÷÷÷÷÷÷÷÷÷
day Col
dax
torn
da Cota
Seau
-
-
=
-
Csc
>
dear,
day
ctfu,
da
u
,
,
-
Sec
'
Csc
'
'
da
=
u
-
day
=
u
-
,
n
-
-
arccotu
arcsecu
dah
,
arccsc
=
-
¥uz¥×
daf
-
-
,
:
-
( ( ul
>
l
)
y If ( lui
,
>l
)
untasxddxcscu Eax
da
seen
seen
=
tenudfu
Fu log ax
,
,
=
-
cscucotu
=
,
da
Aha
logan
,
,
-
Tuan
-
NATURAL LOG
domain
is
function
is
the set of
cont
.
all
pos #
Increasing
1h10) does not exist
( HA IN
'
s
one
,
range
to
one
Es
+
as
)
conc down
!
RULE
dlatyd-tu.at#orIf
PROPERTIES
Ii
:ma+m:i:i::::
'
,
CHA IN
RULE
ax
au
-
e
=
da ax =L nayax
#
*
'
(fcgcx)))
=
ftgcxl )
-
:÷÷÷
gud
"
DIFFERENTIATION
CONTINUITY
+
lftionfhbde.CI/Cthewfb6nttx=c
↳ If f- icy exists
°
Cannot
How
have hole
,
,
I;¥cfCx)
then
jump
,
vertical
fcc)
,
meaning
esymptope
function
a
-
-
at X=c
fails
at
con't
is
Line
a
X
-
CONTINUITY
DIFFERENTIABILITY
C
to
graph
cannot
at
der
.
be vertical
@
x
-
-
C
cannot
,
be
corner
or
cusp
c
discontinuity
discontinuity
jump
@
c
x
-
is
c
a
vertical
asymptote
C
on
the
graph off
graph of f
has
tangent
a
There
vertical
is
C
( diff
estimate
Symmetric
fleas
the der
of
Difference Quotient
=
b
h
)
a
function from
to estimate
'
=
z
cusp
cor ner
a
at
at
*
*
at
c
a
limy slopes from
left
can
F
# i
:#
*
you
@
tangent
to have
ti
hole lvemovzble
.
f
-
a
+
right )
graph using
the formula
fly =fu+h)-f
h
the derivative
[ fc2.in#-i-fCI-fy-h )
)
with t
#
as a
like E
'
set
IMPLICIT DIFFERENTIATION
°
°
when
sides
derive XZ
ex)
Z x
2
-
(
dy
MT
dy
MT
x
+
3y2
4¥
=
-
-
,
y)
defined
y but
x
then solve
,
explicitly
for
in
terms of
Xbrl
b
differentiable
1) If
f
is continuous
Adon
,
l
¥
find
2
+
,
6ydfh
=
,
-2×7=2
3g
f- ( x
'
to
)
y
'
( hey
'
form
when
w/ respect
Zxy
-
the
in
derivative
derive both
↳
defined
b
find the
to
bed
function
a
-
y
o
2x
X
-
DE RIV
let
OF
.
g CX)
INVERSE
be the inv
gt¥
the
der
.
of
ex)
f- (x) 4×3
a
.
what is
f- ( x)
-
f-
'
f
Cx )
-
'
+
x
(f
-
'
(
2) If f
xD
'
tu,
inverse
the
-
of flex )
FUNCTION
-
when
at
point
a
in the
reciprocal
of
the derivative
the
of
incl dec
tatted:
fun
.
then
,
the
at
,
Time
"
f-
'
is
?
x =3
b
.
f
'
o
f-
:
-
z
synthetic
'
-
'
derivative
(33=2
47=14×2+1
taxman
.
a
0=217+4×-16
x
the
what is
wise
fdivision
'
( x)
=L,
-
.
-
'
Cx ) when
is
f
"
mining
corresponding
off
so
incl dec
I
(x)
¥ ¥
is
the domain and
on
x
-
3
canoe
MOTION
position
o
°
function of
a
Position charging
↳
°
is
speed
velocity
of
time ,
our
Sct )
;
or
Xlt )
along
function
velocity
:
because it
function
acceleration
time
over
WHY
is
time
act)
is the instantaneous
the
deriv of ult )
axis
'
-
axis
y
is
moving
,
over
direction
bit w/o
.
t
.
This is the
Speed
is
derivative
always
positive
,
or
ult )
'
-
-
S Ct )
o
Hutton's
Mr t=
vt
charge
act )
-
ylt ) along
change of position function
object
the
.
s 't )
on
instantaneous
,
fast
how
gives
x
Vert
.
Line
moves
stopped
down
moves
up
#
same
for
acceleration
: : : : : : : : : : ¥.in?::.f:::n:::::::::T:::.
÷:f÷÷
DISPLACEMENT
TOTAL
DISTANCE
-
-
difference between
how
far
did
start
travel
you
+
positions
end
in total
act )
Pos
answer
negative
up
=
answer
or
right
down
-
-
ME AN VALUE
tf
f
in (
C
is continuous
e
,
b)
on
the
the
of
ROLL ES
let
is
at
f be
least
a
#
C
'
tangent
2nd differentiable
on
the open
line
at
point
a
( special
parallel
is
case
such that
f- Kc)
-
-
equal
to
the slope of
when
luiz
(2th
f-
'
Cc )
tin
=
f- ( Ctu )
Cc , b ] and
differentiable
on
O
-
x
"
x=2 ,
n
b
f
'
(2)
f- Cc,
w
@
-32g
'
number
µ
-
the open
interval
( c , b)
.
If f- (a)
=
Fcb)
,
then there
#
.
-
n -70
f- ( x)
-
the secant line
a
MVT)
of
the closed interval
on
or
RECOGNIZING LIM AS DERIVATIVE
ex )
,
tha
-
function
in ( z , b)
"
interval ( Eib ) then there exists
"
co
TH M
continuous
one
calls ]
interval
that
Such
Lo
TH M
closed
fthe slope
act )
=o
glo
wingly
V ( t) L O
left
down
left
or
act )
>o
so
f
'
Cx )
-
-
4×3 the
,
iche
of
the
tin
.
-
-
L' Hopital
forms
Indeterminate
°
If
tg×
lying
:
DNE
,
F.F.O.d.cs-cs.ci/ld,#
,
L' Hospital
doesn't
APPLICATIONS
work
of
lzpply
DIFFERENTIAL CALC
Extreme
Extremevzluetheoremo
f
slopes of
Orel
°
max
.
here
Iain
is
crit
local
where
points
ion
open
f'Cx )
-
-
O
or where
f
tf
b
NOT
extreme
only
occur
both
intervals
@ wit
#s
a.me?::si:::::i::s
If
fax)
you replace
becomes
the
a
( bite
FINDING EXTREMA
*
is
with vlx )
or
aux )
,
avg velocity tackle
it
.
continuous
on
e
closed internet
Cc ,b7
DIFFERENTIABLE
think
Max
Orel
curves
mzx
t
min
function
on
the
doesn't
,
then
interval
have to
have
mac
thin
f
hes
L
'
HE pi Tal 's
used to evaluate
°
↳
Rule
limits that
* do
yield
Oz
a
AJ
or
*
forms
Indeterminate
not
me
Butler
carry
on
rule
the limit
Restrictions :
flx ) and
fld)
are
differentiable on
open interval
an
gce )=o
=
If
glx )
this is true
↳ NOT THE
,
lximsabgftx
then
bg×
find
,
RULE ! !
Quoll ENT
(
if
take another derivative
"
king
:}
the line ×3
xd
YI
im
line
e.ix.
/
=3
x
By
line
-
=
1-
A
+
ex
-
+
as
nine;n¥ 57¥ Hot
.
.
sexism
a. "
.
cosh
-
-
o
x→Iz
X→
By
lime
Iz
=
will be
the
same
µ
O
=
as
v
cscx
dmx
-2cosrx
=
-
six
#
TE)
as
× → Ot
him
L' ttopital
*
#
limenx
Sioux -1=0
=
Lim
em,
L' tlopital
-
lion
the
,
to
×→
:
x>
lime
and
1=0
-
÷::÷i
xn
,
(F)
finna
a'
,
a
limit exists
provided the
,
Form
Formosa
E.
tf )
.
containing
X→
cscx
=
s÷m
↳
Ot
as
approaches
gets
"
By
=P
,
A
L'Hospital
0
,
toting fractions
TTeY÷z"£n
,
×⇐)
fin
.
try rewriting
him
*
Eu
-
zsinx
=
-2mm
tiny
exigua
-
sin x
ho+= Sinton
-
,
(8)
y;gy+
-
sink -0
xcorx
-
O
By C' HE Pitre
lion
not
=
O
Zsiwxcox
-
coax
-
xsinx
-
-
a
'
Form
o
-
-
ex
rewrite
Ot
'
-
)
introduce
finna HE (F)
lux
-
fine lny lying
each
-
.
¥
lxihfluy
as
-
Ot
By
L' Hospital
( Es )
exits
dim
X- O
-
X
seiaepmm.mg
run
Form
as
lying
six tx ( O
-
.
.
combine the
Lim
x
'
x
-
( A. O)
enyfitxyoo )
fifth 't
Sioux
-
-
A)
Ot
By
=
×
L' ttopital
elwy
exponential
O
y
lime
xsinx
By
-
line ( tix) #
lime
Hot
tcorx
Hot
him
I
-
sinxtxcorx
(F)
coax =D
X→ Ot
By
lim
x→ot
e
o
L'ttopital
finna
-
-
-
x→ot
sinxtxcoox
L' ttopitzl
,
sine
Cosxtcosx
-
xsinx
=
E
=o
'
Om
O
o
X → Ot
-
-
l¥ho¥hy=l¥hf+
fractions
(8)
Sioux
Tsim,
* o
liar
-
-
In
x¥
lj.gg#lnCitx)
x→
-
ex 6)
=
lime
A
-
and then woe
"
ljlhrolnlltx)
o
-
"
variable
.
-
"
lion
X>
dummy
a
thx)
.
A
=
( T)
g- fine
fish
-
X→o
I
,
,
exalting Ctx
as
)
fraction
as a
lime
's
ooo
-
(O
xlnx
si ein
x→
00
form
A
=e
The :@ Haiti all!
T.EE#ESs
water pours into
related "
"
↳ connection
dependent
↳
Change
Volume
in
II
=
a
✓
date
I
Z
( 2rhd#
draw
.
.
a
relate
*
don't
S
.
=L
treaty)
we
-
s
is the
the water
when
use
an
ye
high ?
4ft
similar
cones
& &
4
TO
to time
don't
pit
#
s
Related Rates
coin
don't
r
-
-
-
=
date
*
need
can sub
r,
=
v
Kae
lo
-
2.
16
of the ladder
I are to slides
.
Choy
5ft from
neo
Hit
spy
at
g-
att
t
+
-
-
o
-
-
o
,x=s
x2tyZ=E
,
b/w
is
k¥236
,
-152311=7--46
Endzone
-
-
rocket
thru
the
velocity
vertically from
its
.
away
at 0.5rad Irwin
?
) ( O)
a
the telescope -1
velocity
jeni
,
-
At
⇐
what
so
is
at
-
-
telescope
a
rocket
angle
-43 I
is the
changing
rockets
otto, 0=43
Ee
'
't
-
on? :*
Find It
at
x
Idf
=ysee2OdItta¥e Itf
tawdry
-
,
×
=
-
-
certain moment the
a
ground
,ow
yen @
x
IF
.
-
-
landing pad
a
instruments show that the
I#ftIsec=dI
Xxdfxe-ifydu-q-fzzdtzdt.me
SG)
tracks
traveling
's
spy
=3
-
EELS-51¥
=
! III
located 10km
¥ae=o
-
-
-
is
Findlay
-
l
killer @ to
Tf
E's hide
at
=
velocity of the top
E- 16
}
IT
'
-
,
I
to determine
the
4
÷ IT
In
Izatt Im ;n= at
l
ex 2) A
3ft Is
at
Find the
is
increasing
Fg h
-
-
against
well The bottom
the well
of
leans
foot ladder
Tottymine
* its
} IT ( Fh) h
=
V
in
'
A
-
-
-
date ! !
Find
Eh
-
*
5ft
h=
until the end
( or
really
,
-
Jn
in
F
=
4h
solve
t
ft
-
level
'
equation
with respect
Plug in number
1)
is
-
water
we
ex
level
water
-
yet !
#s
how fast
.
date
-
*
to
info
in
Ee
went to know?
we
plug
variables
know?
differentiate
4.
changing
picture
do
what
what do
3.
*
f- Tired h
=
of
volume ! ! ! : ! !
-
rising
volume
=
height
tote rate
IMPLICIT DERIVATION
at
time
Thrive
10ft
4ft
radius
and
loft
of
cone
a
-
-
-
-
loser
C. s)
20km twin
n m
'
ex 3)
Train station
Izmit
!
-
Z
-
app
-
.
.
.
Y
-
om
e
TDs
#Booz
'
X
mph
120
2mi twin
1=0 , train
train
calc
10 miles E
is
,
is
,
20mi hes N
the distance b/w
rate at which
.
the trains
II.
=
is
changing
4=0
?
X =D
,
y
ctx -5120mi
z
hr
Iff
z
-
=
20
TF
22.361
or
I OTS
ok
ur
( lo) H2O )
-
-
xdaxz-iydyz-z.IE
are:
-
-
=
1800
t
two
(20×90)=1 org II
at
+
boo
-
Loos
-
-
coff da
are
IE
at
¥miEe%e÷÷
away
-
b) 10 min hour
-
jsf
,
'
mi
"
'
.
"
90mm
.
'
Ti
OTW mph
b/c he's
moving away
distance
is
positive
1011203+35 ( 90)
-
1200
t
3150
=
=
5553 DI
at
4350
=
II
at
119
one
5553 It
.
504
-
=
It
are
Intro to
Integrals
Area
:
-
yn
-
n÷iiEiin÷÷÷÷÷÷÷
upper bowel
¥"
"
'
index
ex )
lowerbound
6
E
i
-
-
i
I
-
-
-
-
,
X
-
O
t
eatin:c:p:
produces
Sum produces
fly
X
-
-
-
sub
int
,
is less
is
given
greater
#
of
Upper
.
2512ft !
.
(
1,2£
5 225
-
lower
81
""
.
.
E. C
T
"
lings Esl ""%
area
:
area under curve
under
one
needed to divide
shape
sum
÷÷¥÷÷"YY:::eyn
§
+
Iii
actual
sub intervals
+
""
at
ox
ra ,
lings
it , )
"÷#n
'
than actual
*
-
.
.÷
than the
a- o
"
÷
that
area that
not
2 in
area
1¥! !hfC
Has
-
width of
rectangle
a
n-sub.in/-ervahr5)
( iz
I
,÷i÷÷
.
.
÷÷:::÷.":÷÷÷÷÷
Lower SUM
ex
i
=
the
Twista : :: ::
) XZ
) E
j
=
UPPERandLOWER.SU#
Upper
ex
Formulas
Summation
It 2-13-14+5
= 21
25
Properties
summation
notation
Sigma
E
i
using
use
sub
n
these
to
-
int ,
correct set
get
-
ups
:
'
%I Euc
-
'm
"
g
iefi.in?.bCEci-iIE*h.aE
ziinn.in?.cii5czs*iheiiTanTYI
heed te
b cc
,
.is )
;
in
+
I
a-
n' Ein
function
'
Fn
.
§
3
-
.
-
-
"
z
'
'
tu
¥÷¥ -4¥
.
.
Reimann Sums
#
partitions
of unequal
widths
"
Find
area
under
box ) above the
graph of y
X
-
-
axis
int cz.io]
on
given data
using nippers
1) Sketch
h
¥¥
find
2)
left
right
Left
dx
:
sci )
right
different
are
7- (2) til
t
let f be continuous
the
'
A=?i)x
of
±
ex
A"
upper 5hm
( 2) + 7- Ll )
71
approximating the
represent
t ,
ith
the
11
t
a
area
pair of
icfcz
+
(
fcztzjtfc
Sums
Trapezoidal
=
@
" the Trapezoidal Rule
8)'t
'
z
I
a
curve 's
endpoints
rectangle
) area under flx)
As
under
consecutive
-
¥+1
( 2,6 ]
Ox
=
another rule
for
and x ;
where Xi
-
lolz)
12+301-22+7
=
calls ? The midpoint
on
t
:
612) -11013 )
midpoint Sums and
Midpoint rule
(3)
14+33-120=72
ft
=
the upper sum
'
'
z
)) ,
tfczt
-
=
n
I
IE ) -8Gt ZEIT
fats's )
Ii flu z ) if ( s E )
'
-
-
-
tcfzyztl
etc
Trapezoid
is
4
let
AItzfxzofx.cz
f
.
is most accurate
be
a
cont
.
function
on
lab]
AZBIzncfcxoj-2fcxdit-cfcxzs.n.tl/fxn-DtfCxn#
ex ) use
Trap
rule to
(0,1T )
'
ox
app the
.
using
4
area
under
fax)
-
-
time and abovethe x-axis
Trgsetoids
If (22+2)
Tie -1¥
-
-
=
,
Igffco) -12K¥ )t2fC
(o
'
t2fC¥ )
Fct Ztrz) to)
t
fit)
-
good enough for AP
on
the interval
.
lhtrotolntegrat.io#Basicsndef
.
*
oflx)
°
-
Zhtider
add -1C b/c
.
standard
if partitions
if meqvdwttths
what
notation
other situations
w/ constants
h
hot 's dx ?
f lxtdx-fghd xsg.FI?.*+cgcaxs.gcxDax- feocxsax-Jgx
:::i÷::..::::::÷
si
Basic
integration Rules
f.
[ fix )
+
gtx )]dx
fkblxktx-k-fflxldxfxndx-th.IT
,
+
-
-
C
integration of Trig functions
fcosxdx
other
-
-
Sime
trig
fsiwxdx
-
-
-
cosxtc
misc
.
ftanxdx.lu/seoxltCor-lnlwsxltCfexdx--ex+CSwexdx--l
nlsinxltC
faxdx
-
-
I
1ha
-1C
fsecxdx.lu/secxttarxltC
ftxdx.lu/xltCfcscxdx=lnlcscx-cotxl
fluxdx tC
-
-
xlux
Xtc
-
?
U SUB
sideris;;÷÷ ;
inoeneorabeeuaounaaounaionan:
3) solve
ex
,
the
)§xCx2tl
indef.am
problem from
here
'
b/c we're
working
differentials
XZTDS
(
-
+
'
C
-
du
-
l
'
Sruduorsutzdu
÷%tC
ex's
XZ
-
-
ex
U=x2tl
du=2xdx
z
.
⇐ +15
Ty
+
{
+
C
C
-
ex5)
.
4)
-
-
-
-
S ×f=sdx
-
fizxtldx
iz
did
Sdu
gateau
Z
z
.
-3
4×3
-
s
=x2dx
c
'
zu
't
u
j
-1C
12
7u
to
}
( 2x -11 )
3-
-
du=l2xZdx
duz=dx
u3z
-
u
du=2dX
't
I
l
4=2×+1
'
U
-
-
zfubdu
'
I
-
)SxCx2+D6d×
'
l
-11
-
'
!
-
ZC×3tzZ+c duz=×d×
-
okay
its
outside
Insite
-
3×204
I
5×1×2+139 dx
U=X3 -11
-
l
Lex
"
)f3xh¥dx
"
-
Hrx
l
exz
I
outside
inside
l
←
l
2X -15
f. Affair,P
l
'
l
l
'
5
I
-
1,2+5×+1
,
wit '
c
I
outside
Inside
l
Su "du
u
+
:n
expand
l
3×2
gzxts
T¥×+Td×
f
dU=2xchI
power rule
l
I
Htt
a-
-
.
l
) "dx
integration
"
-
orcsidefunot
Ptl
,
"
du
u
UK du
set
!!
"
+
6
+
c
C
-
z
s
I
+
C
-
exe
g. x' ( Rtl )adx
)
u
'
X
-
-11
fork
U
-
I
Caio U9du
If ,u÷
YI
UI
-
EEE
u÷ )
-
"
4G
,
-
( Sandu
ucolntc
-
46k¥
to
,
-
gsin23xws3xdx
exs)
E
:ii÷ :
I
.
-
a
interns
Bu
-
-
u
numan
.
XZ
,
dx
sea
solve
Saif do
7)
ex
W3
letuisx
-1C
a-
du=3dx
3
Singh
÷du=dx
to
"
'
a'
-
'
'
:c
'
f
-
LIMITS of INTEGRATION
DJ xCx2tD3d×
Ssiuzuiis
hee
w
-
dw= corudu
*
sometimes
"
2
(g) (
16
-
=z
-_
-
i
z
&
t
.
d
;
i
1
-1
tug!u
l
E, (
du
Zuk
3-
"
values
+
divide
2h41 ?
f
"
Te
-
SE
sets Xbdckicr
x'
-
I
→
x=s-
→
aa-
x
I
a
interns
I. ( Iste -3-2)
u=x
ITS ?CuI+u÷
"
2x
-
limits of
the
tzdu-dxlsohe.fr
,
,
change
du=Zd' X
-
-
-
:
and never
U'
'
o
#z
iimegrationtou
,
now evacuate
Z
II
'
-
)
way
inyegratei
-8
4141141
g-
in
?i÷÷¥
,
u
c
-
tzfwdwia.ie#.i isno:i g:gi .e. nm..r.ssiiEi
Edu=xdx
+
sina.sx.cc
Sian
:
dU=2xdx
C
+
'
z
-
6¥
=
'S
-
of
a
-6
Review
techniques
int
t
at this
power
-
ext
)
point
memorized
S.im#x::n
split
divide
-
it
usub
-
's:*
-
Fu
-
.du=
-
da
z
-
.
12=0
xztzxtl
(x
If
Zz
duty
ferox
i
-
.
I
J÷
u 't
zz
.
Zu 's
zutz I
)
,
du
i
314
Siu
'
I
z
Ez )
3-
in
-
-
sin
to
'
x
+25in
"
Yz
don't forget
-
+2
-
2
25in
-
'
J
fan
O
"
looks like
,
ht
C
l
T
,
tan
-3
I
w
3cg
+
Iz
look
@
resources
to see
~
restrictions
be
function
horizontal
¥)
I
) 't
dung
b/c
that
we can
does
line test
only
pass
-
'
( Xt Dtc
the
Square
due Ax
"
,
complete
next ,
I
'
-
-
,×÷r⇐
gas
(
l
,
=
the 2 ,
th 's irtheder.ro
l
good
4
311
x
-11541
Kil
lfifoutaheor
,
l 'm
'
XZ -12×-12
u=Z,
X=tz
-
see
J Ad
exr)
limits
4=1
,
branches
"x=¥xr
,
Zdu=3Xdx
Change
ten
,
Zxdx
break it
up
get
*aaiin÷÷
.
I Xu
u=
-
.
rule
↳
z
.
-
J !Y*dx
-
.
trig
"
ex 3)
Jsecxdx
Sseox
.
seoxttax
.
,
oh
secxttawxs.es#.: : amon
.
fcsaxdx-ihrlcscx-coexl.cc
s÷:%:¥i÷
Scscxdx
u
-
-
fsecxax
seoxttaux
dn=( seczxtseoxtarxdx
§#
lhkscxtcotxltc
-
lnlsecxttznxltl
=
)
ex
Staunch
,
du
when
In Ihl
=
-1C
f
wetaaeindeg
let
Int , wehaetocne
abs
dx
a-
du
.
SINA.TN/seu,ttamdtcBmemorne
u
-
doesn't
work
cosy
die
-
cosdx
-
this
Sioux
du
-
-
-
sinxdx
simxdx
faut
lnlul
in
::÷÷s:::
-
MORE
INTEGRATION
Transcendental functions
D
Sdf
example
exit
Inlultc
-
-
f. ( ¥
let
\
=L duh
"
Inch
be
cannot
e
=
"
a
AI
-
du
-
-
oh
:X
+
2lnlxl
chain rule
f3e'dx
+
3e×tC
+
-
"
d-
c
2
DX
Ina
divide by
=2×lu2
all
these
things
em
* can either
÷
ax
by derivative of exponent
divide
du
-
Tedx
-
Show
reverse
f.
u
-
-1C
ex
3.
HEY
du
bhlxl-i-tancaxy.cc#ETffzt3eYdx
,
for
"
du
-
Seudu
see# 1) dx
gtxdxtgseczcyxsdx
Inside function
¥
+
do
a-
-
rx
Gein
ax
,
I
-
usvb
Or
teth
dh=Ir×d×
shortcut ,
not both
2dU=
du
dx
-
-
-
2
of
-
Ix
"
dx
zerx
'
dx
-
Ix't
age
"
2e
du
=
"
exec
.
2e
)
J2
to
J
e3X
"
z
.
cosdx
Sioux
>
cosxdx
du
2h
+
zsinx
c
this
hsun
+C
-
'
they
divided
by the
deriv of
.
In
test
the
exponent
Fxerxtc
"""
a-
du
the
e.
2
a
problem
not reverse
it
zsinx.in#sx
ln2
is
produce
run
,
praoti@JdahzF.w
trig function formulas
other
ftp.fhffxz.tgaaatauua.com
-
-
sin
-
'
*
cease
issue
un
ginza
" ""
" "'
,
tudou.az
=
La
-
Sec
'
CE )
'
-
tzdu
C
fz
when
"
§dq
when X
1)
,
there trig functions
Sa?÷m
÷T¥r
let
u
-
-
Ix
,
=L
,
4=2
.
can .
SE.EE
-
-
Sec
Sec
SDI
+
'
cu )
¥sin"E
didn't
u=Zrz
E. Fire
-
Zdu=dx
Tsin
g
-
.
.
du=Izx
Zi 's
needa
dx
-
follow formula
IZ
'
u
Tz
'
z
IT
a
-
-
Seo 'cz
E
,
-
-
.
I
In
x=uz
Parts
Integration by
looking
hehe
int
at
look like
that
.
÷÷÷÷
'
fflxlgoxldx
÷:÷÷:* .÷
""
.
.
=
Udvtvdh
power
rule
'
hsub
rewrite with differentials
dcu.ir )
easy
memorized
th 's
in
respect
-6 the variable
.
V
hard
nothing
else
add
fdlu-ut-fudutfvduznar.ie
(u
UV
Sud
fvdu
Jvdu Sadu
V)
-
-
-
-
-
+
ex
)
Jlnxdx
{ Xexdx
=
xlnlxl
-
X
t
C
U' X
rearrange
-
du
-
etch
-
-
du dx
'
-
§hdV=UV-fVd#
is
One
going
to
be a ,
be
du
FORMULA
forint
V=C×
by pots
JXe×dX=Xe× fe×d×
y
-
is
one
going
( this is
to
,
gxexdx
I
arbitrary)
Xex
=
-
exec
n m
How do
you
one ?
I
Ig
[L
A
l
10£
i'
v.
T
whichever
aigebratrige
C-
Polynomials
comes
fix
,
{ xcosxdx
)
ex
'
trig
-
,
-
u
two things
dv is
dv Cosxdx
=
-
algebra
b/c I b'infront
'
l
du >
box
Ctx
-
-
txdx
V' X
-
f m×=xyn×l ? )?dx-9
dx
dern
Sim
int
everything else
-
I
done
until
*
Sometimes
integral
""
news
"
-
-
31M€
U V
-
-
-
fxwsxctx-xsinxtcosx.ec
T
e
MUST REWRITE
All
the
( write
way
it out
THE
EQUATION
du
everytime )
31ns
-
3
-
2nd
nods the -
n'" t
you do
f ?1u×dx=XlnX/? Xl !
,
l
du
-fvdu←
fxcosxdx
§XcosXdx=Xsinx fsinxdx
-
,
'
both
-
✓
-
DX
-
-
integration
lnxdx
du
l
U :X
du
tog
I
'
multiplied
limits of
a- lux
,
)
set
f?
2)
(
's
algebra
-
#
I
)
E
#s
ex
I
know which
Cini N
-
TIC TAC TOE
If algebra
f.
A
H2
is
-
2x
) Cosxdx
X
assign
INT
X
}
I
-
and
problem
would
it to
assign
\
Cosx
columns
Should take
you
Stax
no other
,
2
you
T
-
DER
XZ
in the
-
\
CON
alg
down to
function really
zero
can
.
-
-
diagonals
°
x
-
take
sinx
integral
match The
62
-
°
-
Sioux
Colum
-
tc
always
-
varies
t
-
.
is not addition by
,
it
a
m - it
-
Obpper right
Take int
u=x2
-
due
(x
'
-
x
u
)
-
-
du
X
2x
parts
by
-
=
Dax
sioux
2x
-
-
42 Asim
f
+
to
1-
lower left
Cosxdx
f(
2x
du
l
v
( 2x
-
=
=
-
1) sinxdx
int
sinxdx
-
cost
1) cos x
-
b
-
Szcosxdx
zsinxtc
-
are
signlfaucor
left empty
integration by parts
check
V Sioux
du=2dx
-
to
) sinxt
x
Xx 1) cosy
2
the other side
of
aga
;
u
,
we
caeoma
tabular
method
TRIG SUB
:÷:÷÷:.÷÷÷÷÷÷:÷:÷
÷:*
÷:
this is only
exception for
case
,
Eez
:
try
a
non
put
u
bottom
to
the
non
the side,
not the bottom
Fez
you
sina.ua
want
case
sinks
,
u=
A
.
"
jhekufdffzmtinutan.mx
tanked
"
'
.ae
alcosot-ra.ua
b/c of
.
Square
¥12
wit
Sa26401
exlfiffxdx
¥
=
-
-
2
=
=
21
do
secret
&
fsecodo
=
Ex
x
pro
ai?:%:
are
labs
tangent lsec
a
at
grin
Etu
⑥
.
and
DO NOT WANT
:
wi
or
relations "ns
tanlsec
z
lhlsecottandltchaetotaw.it
In
back -6 "
Z Itc
+
-
since
CaseU?-aT
dx=2cosOdO
a
26,01=54*2
✓
Secta hip
,
anything w/
u
cannot
fdosdi.29.Y.de?.......fEeadxx#Eawaotaae
be
45650
do
-
4-
2
*
bksguariag
this
is amenorrhea
⇐ 5¥01
+
-
'
ix
rothe Number! !
21=3 Seco
thing
bottom
dx-3secotanoddx2-9-3-andl
a
bsol
t
eual
u
esf
o
xz.
F
I
f }a%j3
Ta)
-1C
roots
r
seatmate
z
"
ar-seaoaoawuyaanu.u.g.aj.%fecoea.%ea.es
a-
25in't
on
ttzxrfxtc
-
du
-
v
-
-
-
Seco
secotand
tano
f
9)
oixsecotauo
>
Asec Odd
use
Seco
-
-
secure
qgsecso
identity
-
acsecoeao
-
ssedo-9kcotao-91nlsecof.LY#o
parts
=
-
sseeo.dotsseoodo.IE
'z[ I
-
F
#
+
Int 's -1
'FFDtC
I .
IMPROPER
INTEGRALS
[ablxldx Jaguar
or
f:b
or
,
Area under
Y Lxz
-
-
by
bounded
axis
x
and
X' '
A
Iz
=
÷
¥↳ma
y
at tuts
numbers
.
=f ? Lxzdx
A-
=
=
-
A =L
t
→
A-
conc
e. o n
Kingsway ;
,
Lings
Diverges
=
.
int -4
-
Ini
-
°
as
valve
It
te
-
-
L
-
I
)
i
the
area is
always
less than I
t
line
let -1=0
III.
'
,
A-
txt ?
-
14117
doesn't
one
'
-
a
? txdx
In
Iti 's
gives it
-
-
k
'
faster , closing the
gap faster
One of these
exist and
dec
=
S
no
these
A-
-
# =L
n
i
n
A-
wax
I
-
t
I
-
-
as
=/ ?
txzdx
I
-
converges
If the limexistst
If the lim
there
is a
value,
function
the
converged
doesn't exist
=L
ex
1)
del
food,xE×dx
if
Converges
aigle
U
-
-
AV
X
Sofie
=
-
ye
so
find
-
e-
x
I
ie
.
't
f
%
e-
-
dx
Thx
xe-x.ie
-
ye
-
line
+
→
-
Isa
'
-
e
process
X
-
finna
eyes
o
"
xe
-
e
-
'
'
Ito
A
-
-
O
"
et
"
O
O
c' Hospital
P
Fool
xiao
-
break up line
can
×
-
converges
\
value
I.e
④
-
its
x
.
×
if
tic-tac-toe
-
go.xed.FI
-
,
,
-
-
te
-
-
te
t
-
-
t
-
-
e
t
e
te
)
,
-
(
o
-
1)
] tins
fins
-t=
-
=
-
as
,
lim
too
Dyitiopital
,
et =P
o=Ie
=L
E
f? txpdx
-HEP-E5#
=¥
=p
.
.
.pe
,
.
Discontinuities
So
CD.IE?gF7g
'
#
.
=L
I
-
,
P
5. ¥
ax
botnddais.ie
pal
fo
recall
^
=P
,
Pti
HH
ONCE
ONE
piece
STOP !
)
ex
f
O
f-
IT
is
x
#
"
ex
i
very
.
asymptote
.
FI
,
ashy Mpt
.
in
middle
of
Interval
DIVERGES !
'
de
-
as ,
creates
,
6
orgs,¥
.
Job AI
,
¥2
e-n-SIE.s-itia.is!¥
Iim
l
tins , so'a¥×
.
,
infinity ? ?
left
side
tim
+ ,
Sin
-
→
(t)
'
-
Sin
-
-
'
(O )
exbf.gs
IT
Iim
+→ I
z
-
=
-
o
IT
In
-2
converges
*
must
×o¥
novel
,
asympt
breakup
S
+
-
.
b/c
as
on
top
-
A on
btw
"
.
the
denote
lim
conipaei
"
ex#f¥ ?
.
t
Kind of like limits
.
a
known
to
S
If flx )
S
f
(x )
is 2
+
is
gcx) Io
so
does
.
°
S ? 4¥
Eg Cx ) and
diverges
converses
integral gcx)
waig
If f. Ck)
as
;
o
con
,÷¥:
.
p
-
Z )
I
the D
gox)
is
nomore
means
this converges
L
t.eu
.
ft
-
Test
by
Compare
5.
Finns
Z
gcx )
comparing to
dx
,
Show which
adding l means
dividing more
-
it
Cord
O
Testveview
X :#
sein
have to
u=Inx
we
du=¥dx
§ dud
=
lnlultc
lnllhlxlltc
-
"
9×+1 ) (
ax
x
/
!
no !
ti
)
Slope
steps
°
Fields
for
look
where the
slopes
along
x
=
lines tothe
graph
&
where
they
are
slope
is O
i
da!
:O
O
y
axes
If slopes depend
'
tangent
:
numerator
•
-
neg
.
or
on
X
or
y
positive
DOL7.tn?tmhIt tikT;sthetimetodoubleor
Double
Find
dy
at
dated
day
=
L
t
)
half
L
-
)
dates
xy
=
X
ALL
aFxy)
SUBSTITUTE !
,
cannot
draw
graphs
arbitrarily
5. I separable Differential
Defofdifeq
an
:
order
x
factor
YOU
der to
differential
of
LI
-
-
equation
written
be
can
in
form
=fCx ) gly) :
A
Product
of
SEPE RATION
of Exponential
The rate
in
.
y factor
t
MUST SHOW
Law
higher
on
FirHep
.
that has more than one derivative in it
equation
depends
Eq
change of
Ky
,
quantity
a
where he is
either
'D
Change
directly
growth ( decay
( Application of diffeq )
proportional
to that
quality itself
#
If
Tips :
date
=
Ky
.
make sure
do wit
then
y
correct
-
Ce
''
t
where
time frame
round cells
unless
(
=
growth
the
initial amount
given
needed I told
:÷÷g..
:
is
present
(
y
intercept )
'
TIPS
to
check
COMMON sense
K
c
ii÷iiem
constant
Original
#
o
o
general
specific
find
°
a
soliton still
solution
specific
exponential e
-
In
has
uses
solution
Iii:
name . .
( unidentified
point to
a
for that point
both sides
when stuck
with
Euler's method
use
minus
-
Pref't
?§
×
Ox
(that Ipt )
µ
,
#
Steps ( depends
Asking :[
a*=y
y
x.
lo ) =3
problem
n.E.u.se :c:
¥
.
initial
3rd
Stp
<
.
6
-8
2
:
DON'T
0
NEED
To Do
3
work
.ie?!ii:?.e
:
,
4.312 -1.2 (4.312
-
42)
-
5.1424-1.215.1424
-
-
"
5.1424
6.09888
67
practice
n
f' (
)
x
daily =y2
( 2x
-12)
.
Let
y
,
A)
)
value
step
FRO
-
on
Find the
-
flex) be the particular
b) Euler
's method
ftx.tn
Lingo
,
app
initial
solution ,
.
condition
Start @
x=o
,
2
of
flo )
steps
of
=
-
I
equal
FCI)
.
.eu#iIxd:ixihEii.a:.cxiuo:f.i. . .*i. . IfF
in
L' Hospital
life
time
x so
situation
I
fcx ) -11=0
lifo
-
-
¥2
o
c)
By
htt, k-zjczct.HN
fLEIz
-
Sim
-
L' Hospital ,
I'Fo f'w
'
fytzdy =f2xtz
oh,
seperatebes !
y=x42xtC good place
tea
force
to
Plug
,
solve
,
that pt
Iim
x→
ft )
'
( 210) -12)
-
cos O
(
=z
=L
-
Ly
-
y=
=x2 -12×1-1
-1×2-12×+1
-
Particular
soldier
size
The
Logistic
date
issue
"
good
only
:
C-
often
"
"
Rios :
for
jYTIYI.fest.me
,
No
population
Inc
b/c
A>o
can
Infinitely
finite resources
.
-
-
equilibrium
growth
.
.
" ""
carrying
much
can
hold )
doff
)
env't
I have
If
Kyl l Ia) toytogetaiiys
'
dye
on
dy
=
side
tay
-
k
-
ta
+
ta In
(
ft 4) don't
;!4I%;
-
Ia (
a
variable
iffy
'
part:L
off
In
rule
1)
I
yI=
'
Inly
'
e
Eat
.
Int
ta
'
-
1h14
A)
separation
'
ly )
division
-
(y
a- y
-
-
#
af
one
Ky ( AAI)
'
de
than
A
.
.
.
if
you
yisdec
.
I
also
the lion
=
,
-
as
gotta get rid
it
Jaffa
t
'
-
-
at
more
Attepemozo
variables
=
dy
A
now .
of
separate
.
.
eheuguillibrium
capacity
the
to approach
#
constant
( how
yisdec
↳
A
stable
II:;I9s
K>O
Predators emit factors
y
then
30
Yo
overpopulation
=
-
-
'
At
=
ttc
=
Kt
-
f Ia at
-
-
+
C
e
that
the
must
.
by
A
exponential
""
Ce
Ce
-
"
cannotattune
.mn?.7IeIoyTy.as=TydtfIa
.
=
hey population
can't
1=134
y
I
=
-
-
A)
A
CA
1
F-
+
=L
Cy
y
y=o
I
=
-
Ia
y
BA
y
B
-
-
y
Y Ce
( l
Ce
"
-
-
-
Ce
-
t
Kt
"
-
)
A
=
A
K'
you
:n:
is
of
can
growth
og
-
constant
#
save time
"he
.¥%:÷
,
manipulate
The value
carrying
Cap
""""
own
°
c-
memorize
so
Iat¥
-
-
.
A
range
¥ypfh
drlh.nu/gent01
tho
t
y=¥
::IiiaIIIe%%
The
A
-
P
t
find
when
A
population
growing the fastest
is
Iz
example
dat
sole
.3yC4
=
ylo )
Y Ice
-
Y
V
)
day
=L
-
I
zy (
IT )
-
I
-
'
I
(
I
c
-
(
ex
-
)
I
general
soul .com
cos
z
-
4
=
=-3
I
'
y
ne
Ice
=
y
-
-
I
2)
+3 e-
If
'
-
Zt
the deer
population
k=
-
a) find deer pop Pct ) if Po
DI
at
dy
.
Ice
=
b) how
too
I -19 e
-
too
t
-
-
FIE
100cL
-
-
C)
to recon
pop of
-
-
y=Y÷ae
4cg
-
at
e m
c
1000
=
c
-
10-00
=
I
Foo )
take
-
Lo
=
-
a
sooner
4T
-44=2
-
44=1
rut
e
-
-
y
l
-
Ge
Cooder
(=
=
9 e
t
1000 deer
'
-
Kt
%
-
(
-
=
Ia )
-
does it
long
Soo
t
Lily ) (
=
Ft
y
Kyl
=
A-
4
.4t
t
=
a
-
-
Inka )
1=5.493
yr
c -
ex
3)
Suppose
die
-
date
Zp
-
.
oozp
'
E
-
ace
.
to
logdilfeq
.
a) if
weeks
Pco )
-
5
Pct) ?
lion
1-
-120
→
A
↳
100
DP
Tde
-
pop develops
a
=
=
Fop
Fop
-
(
Tooo DZ
I
-
top)
A=y
make sure
in
this zone
,
b) If PCO )
PCO )
-
'
-
?
60
120
?
100
too
Is
carrying
Cap
.
J
Sequence S
Simply
D : nature
#s
terms
R:
in
nth term
things generated
list of
a
by
{ an}
a rule
usually Hats
integers @
Ipos
I
when
o,
bin
doesn't
o
( line
sequence
i
Harb
work
.
in )
÷
If
If
things
things
be
a
seq of
red numbers
{ an }
ants then
,
.
.
) # terms
in
explicit fund (rule
3)
set
off
-
-
-
a
diverges
to
{ an } dirges
then
.
tf
lining
=
C
,
finite
a
reel #
then
,
{ htt } ?
2,3
to
-
*
as
!
between
oscillates
{
an
fixed
2
4
,
.
.
.
①
htt
. .
don't
} concierges
trust initial
settle
what
its
matter
'
#s
to C
ou that
,
numbers
then
,
{ an }
lol
oscillation
digs by
sequences
Limitofaseqlet
alien
limos
Properties of
Limits
of
bn
anic ,
-
-
tf
k
is
linhas can
things
't bn)
anbu
'
=
limp
LIK
can
lines In
LK
=
=
Ya
CL
.
,
c
is
R
hints
If
an
=L
L
is
a
k€0 , but 0
then
-
-
bn
Pos integer
Iim
there exists
An E
Iim
Cn E bn
Cn
{ an }
if
sequence such
flu) an
Lines
integer
an
for
all
n
>N
,
→ as
u,
an =L
N such that
HefS ?
If stat
{ 2539=1
then
=L
l
nhs
*
heed 2
eet
related to
a
given
sequences
sequence
that
can be
{
How
# and
function such that
n
and
a reel
)=C
lying flex
SHE
=
.
=L
<
linings an
htt
=
A
They
If
an
braces
in
the list
O
an
hustlers
sequence
2)
n÷i::i::
i. a . .
let {an }
.
4) Inside
"""
an
:
expressed
2h -12
} 7=0
do
( get
if Hart
0
odd # s ?
-
{
2h
{
2h
-
13%1
-113%
,
If Hart
I
If Stato
is
that
for
all
then
a
f
Sequences
Possibilities
or
,
bound
lower
the
is
{
of
Hinman
If Iim
an
}
an
Let
thereon
Ao
a
real
be
a
sequence
#s
na;jIg% !! !
-
due
'
-
nos
then {and
,
diverges
/
Dec
Inc
How to
Monetarism
a
Laz Las
,
a.
E
.
Eas
az
.
Strictly
.
.
.
.
Inc
Inc
1) Suit
.
-
,
find
Su
next
.
If
a
'
a.
>
Az > as
Zaz
saz
.
-
.
.
.
.
strictly dec
dec
2)
.
decline
term
Lo
before
one
-
is dec
it
Sh
Sn
.
if
Cline
Lt
'
dec
it
ratio test )
i'
no
7) take derivative
examples
-
method
An
=
hn÷;
l
-
is
htt
-
method
I
htt
@ ti)
(
h2 -12in -12
n
-
-12
(
e
)
3h -12
-
70
n+ ,
Inc
method )
!
h!
n
e-
.
!
( htt )
=
e-
E
deriv
en
÷
htt
=
n -11
-
2
E
n
)!
m2 -12N)
2t
-
ah
Cut#tY+4
h2
Lt ) inc
l ldec
an
-_
method
Inch )
damn ))
!
-
-
the
-
eh
POS
b/c
L ,
So
F. E
gets
diverging
-
h
> O
In
in
,
.
for 211
-
to Ll
>o
SO Inc
veles
Series
Finite
U
t a
i
-
-
S
=
,
su
z
series
a
(l+th)
sequence
last term
)
t, tf
I -11 gtfo
'
,
'm
.
partial
use
can
we
,
t
t
=
in
%
It I,
'
of terms
+
E
-
52
first
or
z ta z
For infinite
Si
sum
defined
-
,
slowing
denoted
sums
Sh
by
Infinities
down
•
I,
k
=
Ak
a
,
ta
z
t
.
.
-
index of
.
A
t
k
W
term
K
Thru
=
W
I
general
term
summation
:
is
and
of
Seq
→
very
-
-
#
,
sum
so
few
talk
=
ebrl
Cour
-
-
-
The
limit
L
then the
series
converges
L
#
t
-
div
sums
that
can define
we
.
integrals
¥2
sum
of
&
an
polynomial
an
two
sum of
The
then
a
notes
,
If
has
sum
sum
series central
* review improper
extra
partial
a
have that
said to
→
→
Sn }
{
If
=
9¥}
if
gcn ) by
the
more
a
converging
convergent
series
both
can
series
cow
and
or
is
a
a
div
converging
divergent
series
series
(the first finite # s
tests
and
degree
then
and
gcn )
I
of
bln )
hln )
are
poly nom
exceeds
the series
cow ;
.
is
.
that of
otherwise div
a
divergent series
do
not det
the
cow
of
a
)
Srs
Netty
Test for
1)
then the
the
and has
{
if
↳
In
then
Sn }
an
If
converges
things an F
O
II.
then
,
,
is the
sum
converges
an
things
diverges
an
S
properties
as
the series
of
I can
-
n
Lingnan
then
,
that doesn't
=o,
'
-
-
-
-
*
Jan
CA
o
h
the
mean
=
has
a
-
series
ATB
I
( an
E
sum
-
as
n
}
be
convergent
-
Ibn
and
to
have
i
E ( ah ibn)
converges
partial hums Thru
Sn
If
of partial
{
hthtgne
the series
to
cow
for Divergence
as
if
series
of
1)
corn
Euan
↳ * bit
series
partial sums
series also
sum
↳ the limits
2) If
"
Test
sequence of
,
infinite
Convergence
If the
Converges
''
lov
-
then the
line L
-
bn )
A
-
-
-
B
i
series
have
said to
is
and
that
Shml
( if seq
#
then sum
,
series
very few
*
-
-
#
find
me
con
=
a
for ( telescoping
sum
of portici
ex
if the lim
n
Su
-
-
the
fines
Sn
-
-
due
div
telescopes
→
closes
¥7
,
get
can
we
"
,
s
=
-
-
-
for
fat en
I
,
-
in
-
o
.
Su
sum
→
-
=
-
-
-
l
I
ti
ti
-
-
I
=
-
I
I -11--0
no
,
-
ninth F Eti
Sn
'
-
Sh
AL htt ) -113 ( n)
htt )
-
th
-
things
"
Su
-
this
Intl
I In
"
o
due
top
because
of
*
'
:{
S"
-1
bottom
funk
piecewise
owes not
=
,
limit
due
div
like perfidious
looks
-
=
sums
→
-
I
partial
itself
into
k¥+1)
a
diverges w/
div
as
Sz
→
C
geometric)
Sums
t)
-
→ as
S
18
II.
Show
ex )
Thru
+
CK
'll
=/
-
+
.
htt I
lim
→
A
Su
=L
.
H
.
-
tu )
,
you
tell
can
it is
if
telescoping
do
'
n
Ctf fast
-
"
the
,E?
,
+
D=
'
e,
b/c
#
we
get
e
you
P
.
F
and
somethin )
Subtracts
from the
other
Geometric
series
ex )
a
e.
Lark
K
-
-
%o
O
Irl
diverges
II
quotient
of
ato
Converges
ex ,
0.090909
express
"
8h)
Ir
sum
Ll
WII
Foo (
"
-
-
:
+
-
-
r
=
+
Too
too
II.
teal
ex
,
E.
'
=
I
:
'
of ( f)
Sun
FE
Ig
topic
-
but
th!
b/c
ht!
L
I
>I
diverges
14
-
* off
k
.
'
I
=
'
.
'
-
EE's
'
'
Stat tern
2
praetor
.
tha
Cow V
Conn
.
-
by
by p
DCT
-
-
test ,
.
.
-
.
.
*
l
Effy
,
it to power
N
k
+
Foo
.
an
ooo
.
Tooo
.
.
-
)
"
÷÷
sum :
"
Foo
l
I
fool 'Too)
term :S
Starling
%ooo
Integers
2
in common
,
sum
+
W
as
=
storing
term,
Integral
the nth
when
from ( I
,
A)
if the
,
the series
term
II
integral
diverges
exit Determine
related
is
converges
-
o
-
-
-
integral
blips ? Fat
b' Fa 457
'
¥z+
-
,
4 tan
binges
-
telescoping
can
] thesesums
thx 5- ¥41
Fae
=
II
'
'
,
4
-
-
4 tan
Ill )
can
we
E,
decreasing
if the integral diverges
i
,
find
-
4.
Iu
F-i
IT
if psi
converges
if
diverges
find
if
exists
converges
a
P>
LIE
¥41
converges
the
II.
also
integral
find
2)
ex
,
#
OLPE
I
diverges
→
range in which the
l ( converge)
L
l
-1¥
sum
,
the bounds of the sum
¥3
,
P =3
sum
LIE
,
the p
-
( ouvrages
series
ELIE
Taz
LZ
-
La
f? #
,
Series
'
between
,
-159 fix) ax
ex
¥z+
:
test
lies
,
ex s)
)
by
Seres
f? fix)dx LEAK
p
test
Boun#¥rgeet
"
Fe
,
also
-
the
,
and
IT
by
II.
can
harmonic
'
Iz
have
t
-
4 tan lb
)
=
,
pseries
dx
,
( b)
-
converges
we
'
'
positive , continuous
series
4 tan Xlb,
fins
-
then the
that is
test
ok
,
,
geometric
not
not
,
use
function
a
cord
lim
+,
to
ti
Chi
=
T
4)
find if
which
¥122 -11T
bines
du
"
as
→
th ( D )
I ↳ improper
b' Fos [tlnlbl
-
the interval
in
EEE ?Italy
↳ reduces to
Sba
and
it resides
,
's
dorc
{ In ,z , ]
diverges
integral
,
makes sense that it
diverges
Limitfomparisontest
Directcomparisoutest
Ibn for
let Olan
II.
1. if
all
bn converges then
,
larger
2.
Ewan
diverges
I.ibn
then
,
smaller
than DCT)
exact
Suppose
converges
an > 0
nlingsfabn)
smaller
an
If
( more
n
diverges
Then
larger
the two
converge
or
,
SO and
bn
series Ian
both
L
where
=L
is
positive
and
finite
Ibn either both
and
diverge
NME
-
summation
all
bounds cannot start
p test
it
bounds are
to
goes
otherwise
other
all
,
infinity
positive
from
both
tests
o
ble
ways
Start at
can
¢
o
Root
Ratio
Given
If
If
L
l
L
=L
L
If
the
L >
I
,
the series
,
the test
n5
5
,
powerful
less
-
limos
IanI
powerful
mLku '
stay
limits
the
! ,n
By
fin
If
0
=
counts
1
=
L' to
If
L
If
L > I
Alternating
let an > O
%
.
The
"
L 1) an
-
,
-
I
1) lion
has
alt
"
series
fan
,
then the
and
=
the first
neglected term
Snl 4 Rnltlantil
remainder
Lintjmphlanflninsaslanltn
=L
series
the
is
inconclusive
divergent
is
series
absolutely convergent
is
Test
§
and
L 1)
-
series
"
"
an
,
if the
an
=
O
following
2)
is
the
If
conditions
ant , Ean
,
are
for
met
ad
next
term
n
( strictly decreasing)
-
involved in Zppx the
of the remainder
to
Suns
then or
less
is
Sn
,
by
-
Ling
alternating
satifies the
Rn
IS
,
htt
,
series
Remainder
condition an
-
-
the test
,
-
a
,
the
,
-
If
pitch
L l
L
L
let
,
powerful
"
converge
.
Ian
series
DIVERGENT
powerful
A. S
Given the
same
in other tests
CONVERGENT
-
less
,
n
L
divergent
more
"
let
inconclusive
is
→
,
*
absolutely convergent
is
is
the series
,
less powerful
more
Ian
series
-
* Must
be
convergent
Absolute
conditional
convergence
-
If the series
the
series
the
I tant converges then
Ian also converges
def
,
def : Ian
is
absolutely
'
-
,
converges
cow
Et 's
by AST
It
'
-
a
.
.
6
.
but It
,
sums
not
I
w/o
can be
always
¥
infinite
true
series i
changing
depends
the value
on
of
say
that
the sum
whether the
=
.
same
*
sum
series
is
absolutely
pay attention
to
"
(t )
C 1)
" '
-
If
rear r
every
"
th
converges
,
II.
only
or
:
ex )
can
so we
conditionally converges
conditionellyconvergentflsabsc.am
convergent
8 lanldiveges
Elam
diverges
"
Rearrangemeatofseriesorearranged
this is
bit
Elan)
if
.
,
o
conditionally convergent
'
'
finite
is
Ian
tangs
convex
t
,
:
not true
is
converse
convergence
-
we
th
=
T
I
-
(t
arrange
'
-
-
'
=
z
=L (
ts
E)
-
i
tu
In 2
.
.
I,
tytfz te) ztfs
-
-
t
the
Et 's
-
'
-
z
-
in
+
the
negative
pattern
=ln2
t
.
positive
terms
'
-
'
=
-
to
the
t,
t
t
-
to) iz
-
'
To
's
.
.
.
-
'
.
'
s
z (U2
-
SiniIzul=l-l)
.
( OS ( ITU)
=
L
-
l
)
if
NEED
THEOREMS
TO KNOW
* Must mention
differentiability
+
Extreme
If
once
function
a
f
continuous
is
closed
the
on
interval Calls ] , then
f
Mist chain
a
maximum and minimum Leech at least
)
→
diff → coin
Intermediate
If
,
,
.
value Theorem
continuous
is
then it also takes
only
cont
Only requires
so
function f
a
interval
→
continuity
value Theorem
needs
any
on
ft b)
between fla ) and
value
continuity
the closed interval Cz ,
to
use
prove
something
b)
takes
and
and
flat
values
f Cb )
end of the
at each
at some point within the interval
MUST
something
=
↳
ex)
Roller Thm Special
ft 3) Lo Lfc
-
-
2)
:
-
,
ft c )
-
-
for some
O
C,
2L CC 3
Mean value Theorem
If
a
function f
continuous
is
on
a
-
( a , b) then there exists
,
point
a
Entea
in the open
c
internal cab )
such
°
Moles
→
→
"
between
use
me
'
Cc )
.
.
e
b
-
f
to prove
f
=
''
=
Something
Something
z
x
-
Rex )
=
en
proving f- txt
If
f-
ex )
is cont
-
-
look
O
,
the same
values with
3C XL 7
Lagrange
o
)
→
'
interval
←
a
"
to prove
f
f
levels
'
the
on
-
that
f
f-
differentiable
Calls] end
y
3ExE7
on
,
-
vote ! !
diff
and fl 3) =f( 7)
,
Error
series
f "Z)
( n ti )
!
(
x
-
c)
ht '
Ex
¥:÷; :*: ::: ::
-
-
I txt XZ
"
-1
.
.
.
TX
"
imeueaom tina "
-
cos x
-
l
-
Ii
t
¥,
c-
t
.
.
.
for
DIII
,!
on
Area
volume
t
*
Area between
a=fdCrigmfunr-leb-funnion)dyforeyt
A
-
( top funk
-
simplify
the
for
bottom fun ) ax
Find intersections b/w the two
°
•
fab
-
integral
before
a
EX E b
functions if limits of
-
putting
in
}X
is
pos
VAM
INT
hit
between
area
a. curves
Curves
2
.
area under
x
axis
Iff! ,Dn ! ! I !Y¥p9
is
neg
"
Irion
of axis
given
calculator
a
#
it
"÷sa"¥"
volume
T
?÷: ::c:: :::c
.
.
y
Tip
:
Square
washy
-
1) draw
radius
Vitra
w/ holes
picture
,
ID the
before
's
subtracting
:÷ ÷ ÷:÷ ÷
formulas
geometric
-
d
circle
diagonal
-
-
rhombus
square
region
÷÷÷÷÷:÷÷÷÷÷÷÷÷÷:÷÷:::
5) Square radii
x.mn
sax.
:
,
S2
:
:
equils
Ed xdz
Tlr
or
E
'
Ff
base
¥si÷:÷÷:::::i:::÷÷
" "
'
=
d'
Z
circular
g
1-1
+
-
a
1/-2
Shell preferred
vertical
-
x
G
t
-
writ
-
-
y
when
#
cross
diameter
xy plane
w/
DIW
SHELL
E
,
S2
sections
ow
Arc
length
Smooth
If
curve
f- Cx ) is
a
-
-
DISTANCE
function
smooth
curve
on
S=fabf¥oh
volume
-1
Length -1 Area
coil
is
are
diff
calls ]
,
,
VELOCITY
.
ACCELERATION
.
arc
Ieng
th
is
velocity
day
( position )
acceleration
Fx ( velocity )
positive !
m;÷÷÷ii÷÷÷::.
speed
=
IVI
ty)
sus:
r
:÷:÷÷÷.÷::*
is
function
to
ax
's
goer
S.A=2xfabfbx)f¥Wd÷
r
t to axis
oeu
.
average
velocity
finalposition-inu
total time
COMMON ERRORS
Quotient
DO
SIMPLIFY
NOT
rule
&Y%+g"7awYn
-
high
d
low
°
d low
high
-
Don 't
-
below
square
parentheses
use
°
Chain
ANSWERS
NUMERICAL
esp
-
in
polynomials
I
gwlient
the
connect
o
rule
do not close
-
Hgh, ))
Cx ) fight ) ) g.
f- ( x)
o
'
f-
-
'
Cx )
-
-
-
Fox)
o
-
°
Siu Lxz )
-
-
'
f- Cx )
cos cxz
-
e
" "
'
F W" e
) 2x
-
-
-
Fox )
:
brevet points
" ''
4
-
that
wi
are
-
defined lilies
Totaldistanutrarelledtdisplaumeutp
article
mores
Ja
horizontal line
a
on
with
time t
at
velocity
ult )
given by
"
displacement
:
-
UCH dit )
crit #
-
Olt f 'll)
toankl
fab
-
-
if f- Cc) DNE
'
O
or
,
it is
only
local extrema
lull )l At
o
Iff
''
us
-
Candi ate for
a
-
' '
o
Parlicleiuotiou
The
°
direction
charges
pantile
when the
only
velocity changes from
positive
to
negative
+
vice
versa
Just
→
because ult)
of the
" need
=o
particle
DOES
is the
Not MEAN
hBsE
MUST CHECK ! !
DIRECTION CHANGE ! !
of the
recoils
,
way ,
*b¥÷Yge%÷F
-
when alt )
Speed
Inc
→
don't
solve , write
→
use cell
→
to
only
.
daxlvct) )
value
>O
,
of speed
Lt) is
UH
.
)
>o
if it is not
inc end
c- )
is
acts )
or
co
and
ul Cz ) co
-
asked for ble if
you
calc
wrong you
're done
for
de
Jwstlfyextremao
AI
MI
value
°
of
Make sure
↳
↳
C is
you
continuous
diff
.
on
net
always the midpoint
state
on
conditions
the closed
open interval
internet
for
o
↳
o
or
S
c
'
absolute
Try endpoints
for
one
x c
XC
or
>
c
table of values
-
-
relative extrema
extrema
✓
dvhebyw2sh#
htt
(
Definitions
'
Kha
-
ffCb
)
run
correct ( RHD
:
'
-
crust
b)
fans
values
of
0
Cosa
since
trig
tard
l
O
O
I
F
Ez
z
Stone
'
z
Fs
→
E
' '
t
o
as
"
we
.
rate
a
don't
of
usually
change
asiana
trig integrals
Jaffna
du
=
"""
÷:
FE
is
it
,
"
Es
avg
la#urdu
-
'
.
is
statistical
inv
-
O
.
is
-
at
5in
-
tan
'
""
ka
'
-
'
+
f-
"
C
+
c
"
use
this