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Formulas and Theorems for Reference
I. Tbigonometric Formulas
l. s i n 2 d + c , c i s 2 d : 1
sec2d
l * c o t 2 0 : < : s c : 2 0
I
+. s i n ( - d ) :
t , r s ( - / / )
= t r 1 s l /
:
s i n ( A*
B )
: s i t r A c o s B * s i l B c o s A
siri A cos B
siu B <:os,;l
9. cos(A + B) -
cos,4cosB
siu A siriB
10. cos(A
B) :
cosA cosB + silrA sirrB
2 sirrd t:osd
1 2. < ' o s 2 0- c o S 2( i - s i u 2 0 :
2 < ' o s 2o
I -
2 s i n 2 0
1 5.
13. tan d :
<:ol0 :
I
<. r f t 0
I
t a t t H
: s i t t d
(:ost/
sirr d
(:OS I/
ri" 6i
1 6. c s cd -
/ F
I
t l
r (. c o s [ ^
\ l
1 8.
: C O S A
2 1 5
Formulas and Theorems
II. Differentiation Formulas
! ( r " )
Q,:I'
] t r a - f g ' + g f '
g J ' - , f g '
,l'
, I
, i ;. [ t y t. r t )
l ' '( t t (. r) ) 9 ' (. , ' )
d , \
(sttt rrJ .* ('oqI'
t J , \
dr.
l('os
./ J
stll lr
{ 1a,,,t,
:r) -
o. t
1(<,ot
.r')-
(,.(,
.r'
Q : T
rl ,
(sc'c:.r'J
: sPl'.r tall 11
d ,
(<:s<t.r,; -
(ls(]
.]'(rot;.r
, t
fr("')
-.''
, 1
f r ( u " )
, l , ,
' t l l l r i
( l. t '. f
d , ^
I
t!. r '
J 1
r z
1(Arcsi' r) :
oT
I l 1 2
2t8 Formulas and Theorems
IV. Formulas and Theorems
1. Lirnits ancl Clontinuitv
A f u r r c t i o r ry :. f
( r )
i s c ' o n t i n u o u sa , t. r -
c i f :
i)
l'(a) is clefirrecl
(exists)
i i )
J i t l , / (. r ' )
e x i s t s .a n d
i i i ) h r u. l (. r)
. / ( r r )
Othelrvise..f is <lisr:ontinrrorrsat .r'-
rr.
Tire liniit lirrr l(r ) exisls if anclorrh'il iroth corresporrciirrgone-si<le<llinrits exist a,ncl
a,r'e
etlrtrl tlrtrt is.
lrgr,,l'(.r)
L .:..=
,lirn,
.l'(.r)
I' -
,lirl
./(.r)
- Intemrccliatc- \rahre Theroettt
A func'tion lt
.l
(r)
that is r'orrtinrrt.rrrsr-rrra t:krserlinten'a,l fo.
b] takes on every value
bct'uveerr
./(rr) arrd ./(6).
Notc: If
,f
is corrtiriuorlsorr
lrr.lr]
an<1.l'(a) ancl .l'(1r)difler in sigrr. then the ecluatiou
.l'(.,)-
0 has at leu,stotte soirttiotritr the opetr itrterval (4.b).
Lirrritsof Ilatiorial Frui<'tiorrsas .r + +:r;
if the <legreeof ./(.r') < thc clcglee of rt(r')
,. 2 ) ,.
l '. x ; r t r r 1 , l , ' :l i t , r
{ l
. r ' + r.. 1 " ' ] . )
/ / , \
l i r r r
i s i r r l i r r i t ei l t l r e
, l e g l e e
o l. / {. r ' )
t l r e
r l e g l e e
o f 1 7 (r )
. , - t r
9 . 1 /
l i r r r
/ ( ' ]
J
3. litl
it
. r ' + f - r / (. u J
Notc: The limit
. r r + 2 l l '
r. x i u l l l ) l ( ' : n l i l
L
) c
. r ' + + x. J ' ' -
a i
fiuite if the rlegteeof ./(.r:)
the degreeof .q(.r)
u,ill be the rtrtio of the leaclingc'ciefficientof .f(r;) to.q(r).
' 2. r 2
r - x a l l r l ) l c : l l l l r
:
t ( ) , r '
5 r 2
Formulas and Theorems 2I
4. Horizontal ancl
\rt'rtir:al
As)'rnptotes
- A I i n e g - b i s n l u r r i z o n t a l a s v n i p t o t t ' < - r f t h e g r a p h o f q :. / (. r ' ) i f e i t h e r l i r r r l (. r ' ;
= l ;
,,r
.Itlt_
.f
(r) : b
- A l i r i e. t - e i s a v c r t i < ' a l a s ) ' r r r p t o t c o f t l i e g r a p h o f
t t
. f
(. r ) i f e i t i r e l
. ,. h r , . l (. , , )
, \ )
. / (. r ' )
A v c r a g c t r r r r lI r r s t a r r t i l l l ( - o l l sI l a t < ' o f ( ' l r a r r g t '
- A v t ' r a g t ' R a t c o f ( ' l r a t r g c : I f (. r ' 9. y r r ) a t t r i (. r ' l. q l
) i r l e l r o i t r t so r r t h e g l a i r l t < f t q -
. l ' ( t ).
t l t e r t t l t e a , v e l i r g ( )r i t t e o f c ' h a r r g eo f i l u - i t h r e r s p e c tt o. r ' o v c l t l r c i t r t c l r - a l
l r ' 1 1.. r
t ; i s
l!_r1'_l!,,) lr
ly
. l ' 1. l ' 9. r ' l , r ' ( ) l. r
- I t t s t a t r t n r i t ' o r r s R a t c o 1 ( 1 - l ' , l t r g , ' ,I 1 ( , r ' 1 y.. r / 9 )i s a l r o i r r t o r r t h e g r a l r l r o I r l , - ,. l ' (. r ) .t i u r r r
t h e i t r s t a u t A r r e o L l sr a t e o f c h i r r i g t , o f i 7 n ' i t h r t , s p t , r ' tt o. r ' a t , r ' 1 1i s. f ' ' (. r ' 1 ; ).
- Dcfirritiorr of t,lrc l)r.rir-ativt'
.f'
(.,)-lll
lEP,r'
t'(,,)
11,1,
!y)--ll:'J
T l r t ' l a , t t < ' rc l c f i r r i t i o t ro l t l r t ' < k ' t i r ' ; r t i v t .i s t l r t ' i r r s t a r r t i r l r ( ' ( ) u sr i r t t , o f c h a r r g t ' o f '. l
(. r ) u - i t l r
resltec:t to .t at .r -. (t.
G e o r r l e t r i t ' a l i r ' .t h t r < l e r i r ' : r t i v e o 1a f i t t l t ' t i 9 l t a t a l r , i l t t i s t l r t ' s l ' 1 r e , f t 1 e ' t a t r g < ' t t t l i t r t ' t ,
t h o g r a p h o f t h e f i r n c ' t i o n a t t l t a t
l i o i r r t.
' f h c
N r r r r r l r c r( ' : l s a l i r r r i t
2. l i n i
( 1 + r r) ;
n - \ )
8. Roller's
Theorerrr
If .l'is c't-rntituu.rttson
ln.0] arrrl
ciiff'elentiablt'on
(a.b) srrt'hthat.l'(rr).., l'(1,).
tht'n thcle'
is at leirst otte ttutttberc itr the opetr intelval (o.b) srrc'h
that.l/(r') -
9. Nlcan Valuc Thcorcrrr
I f / i s c o t r t i t n r o r t so t t
l n. l i l
a u c l c l i f f e l e n t i a b l e o n ( o. f ). t h e n t h e r e i s a t 1 t : a s to u t ' n u r r i l r e r
l / 1. \ I t ^ r
l i t i ( n. b ;. t t t l r t l t ; t t
' / " ' t
f ' 1 , I
t t
t I
- l i ' r
( r + 1 ) "
n + + a \ f l /
Formulas and Theorems
T l r t ' t ' x p o t r e t r t i : r l f u n c ' t i r ) u
c ' g t ' < l u ' s v e r v l a p i r l h. A S. r ' - + t c u , h. i l et h e f t t g a r i t h m i c , f u l r. t i o n
l/
.. lrr.r' glo\'s vt'r'r' skx.r,i-u'
a.s
.r' -) )c.
E r p o t r e r t t i a l f r r u c ' t i o r r sl i k e u -. 2 ' r t r! /
r , , ' l l r. ( ) \ \ - n t o l. e r : r p i c l l y a s. r + : r t h a r r a n ) , p o s i t i v e
l)()\'('1<if .r.
'1.'ht'fitttt'tiott
i/
hr.r' gr'o\'s sl<lu'eras .t
x tltiil a1\r lotx,orrstarrt
lt1;lvrr<1niai.
\ \ i ' s a r '. t h a t a s. r '
) c :
l. I t. tt g l ' ) \ \ ' : l ; r - 1
l l r i r r r, / i , rI i l l i r r r
l ( r \
\ , r ' i l l i r r r
l t | ' )
{ t.
r
. r z / { , r ' ) . r . . l (. r ' )
f i. l
( r ' ) g l t x l s f h s t e r t h a t r a (. r ' ) a s. r ' - +
) c. t h e r r q ( , r ' ) g r ' o w ss l o l r , t r t l u. r n. l ' (. r. )
A S. r. + r c.
. / (. r ) a r r < lr 7 (. r ' )
g r o u , a t t h e s a r n t ' r a t t , a s
. r ' + r i f l i r , r
L
l 0
( t r i s f i r r i t e a n c l
,. , \
q (. r , )
rrouzt'r'o
).
F o l t ' x a n r l l l t '.
- r ' g t r x l s l ; r s t c r t l r a r r. r. : ii l s. r , +
r c s i r r r. r ,l i r r r
: r ,
. t
'
. r ' l g r ' , , 1 ' sl i r s t c l t l r a r r h r. r ' : r s. r. : r c s i r r <. e1 i , , , -
x
3.. r ' : + 2. r 'g l ( ) \ \ ' si r t t l l , s i r l r r t 'r ' r r t r ' , r s. , , 1a s. r. )
' r 2 l 2
c sirr<.r'
,] ,i{
I
T i r f i r l < ls o t t t t ' o f t h e ' s t ' l i t r r i t s i r s , r '
. \ ' ( ) l l n r i r v l r s ( ' t h e g r a p h i n g t a l r. r r l a t. , r '. \ I a k e s u. c , t l a t
a l r a l ) l ) l ( ) l ) l i a t c r. i t ' u - i r r gr. l - i t r r l o r i -
i s r r s c r l.
I r r r - t ' r ' s cFr r r u ' ti o r r s
1Li. CourlraringRatcs
of C'hatrgc
i. I f. / l r r r l 1 7i r l t ' t u , o f r r r r <. t i o u s
s r r <. ht h a t
. l ' ( q (. r. ) )
. r f o r e - , \ ( ) 1. . . 1 ,i n t i u , r l o r r r a i r ro l q.
a r r t L .q (. 1 ' (. r ' ) )
. r '. l i r r i r r t h c ' r l o l r a i r r o f . f. t h e r r.. f ' a r r d 1 7a r e i r r v e l s t ' f i u r r. t i o n s
t i l e i r c h o t l r c r.
A f t l r r r ' 1 i o r r
. f h t l s r t t t i t r v t ' r s r ' l i r t t t t i o u
i f a r r r l o n h. i f r i o l r o r i z o r r t a l l i u e i r r t c r s e r r , t s i t s
g r a l r l r u r o l t ' t l r i r r ro r r < ' ( r.
- If
.l
is t'itlrt't ittt t'eilsilg or' <it'treasirrg in arr intt:r'val. tfien
f'
|as a1 i1.,r'r.sefilrc:ti,'
o r. t ' t t h r r t i r r t t ' r ' t ' a l.
l. h
I
i s t l i l f i ' r t ' r r t i a ] r l t ' a t t ' v t ' t ' r -
l r o i r r t o r i
a r r i r r t e r v a l I. a r r c l , f ' (. , t )
I 0
o r r I. t 1 e 1
! l
l
r (. ,
I i s t l i f T t ' r < ' u t i t r l r l t ' a t e v e r r ' l r o i n t o f t h e i n t e r i o r o f t h e i n t e r v a l
l ' ( I )
a r r r l
, t ' l l l. r I )
l
r. t t
Formulas and Theorems
I x P r r r rr l t t i l s r , 1 r
' '
' _ - l
1' I'htr t'xllorlt'utial futtctit.rti !/
t'' is the irlverse function
of t7:
- I ' l r t ' c l o r n a i t t i s t h c s e t r l f a l l r t ' a l r l t l r l l l ) e l ' s.
<. l r <
D C.
' l ' l u ' r i t n g t ' i s
t l r t ' s e t o f a l l l l o s i t i v e
n t t n t l l e l s.!
, l
( L t '
l l
. , r ' i s < ' o n t i r t l r o r r s .i n c ' r ' e r r s i r r g.
a t t d ( o n ( ' i r v e r t l t f b l a l l. r :.
t t.
i i t ] ' _ , '
. , i x a t r t l
l 1 t l t _r '
T. , l t t
r
. r. .f i r r '. r .- > 0 l l r r ( r ' ) -. r '
f i r r a l l. r '.
P r o l r t ' r ' ti t ' s o [ ] t t. r '
' l ' l r c
r k r r r r i r i uo 1 r 7 l r r , r ' i s t h t : s e t to f a l l l t o s i t i v c
t r u t t t l i e r s ,. r ' > 0.
' [ ' l r t ' r i r r r g t ' o f
i 7
. h r. r ' i s t l i e s c t o f a l l r t ' a l l r t r t t t ] r e r s. x <
l / <
: r '
:1. r. lrr.r' is <.orrlirrrrorts.itr<'r'e'asirrg.
urrrl corrcavtr clou,tt cverYrvltertl r-rttits tlclrltrin.
- l r r ( r r | )- l t r r r I l t r
1 i.
- l t r l , t
f l , I
l t r , r l r r/ ,
( ;. 1 1 11 1 l ,.. 1 ' 1 1 v1 1
i7 hr.r '- 0 iI 0 .: .r'.- I arrrllrr.r'> 0
if .r > 1.
E.
, l l l t.
l t r. r ' - * : r
t r r t r l
,. l t ] l i
l t t. r ' -
1).l.g,,.r'
il;
2 0. 1 - tl p c z o i t l i r l I l r r l t '
If ir f\urt.tiorr.fis c'outiuuorrs
orr tlrt't'krsecl
inte't'val
[4.b]
where fo.b]
has ]reenpartitioned
i r r t r r l s t t l r i r t t t , r ' r ' t r l s
I. r ' 1.. r ' r j .l , r. i 2 ]..... [. r : , , r.. t : , , ] .e n t : l t
o f l e n g t h ( b - a ) l n.
t h e n
r l t r
I
f t , ) r /. r '
= - ; ; [. / ( , 0 )
(. r r )+ 2 / (. r z ) +... + 2 J (. r ' , , r ) +. / (. r " ) ]
.
. t , ,
Tlrt. T'ralrezoiclal Rrrlt' is tlre avelage of the left-hancl and riglrt-hancl R,iemann sulns.
Formulas and Theorems
Y"t".lty,
Sp..a,
"t
The vclocity of an object tells how fast it is going and in which direction. Velocity is
an instantaneous rate of change.
The spceclof an obiect is the absolute value of the velocity, lr(t)I.
It tells how fast
it is going disregarding
its direction.
The speeclof a particle irrcrcascs(speedsup) when the velocity and acceleration have
thersarrresigns. The speed
clecreascs
(slows down) when the velocity and acceleration
have opposite signs.
3. The acr:cier:rtionis thc irrstantarreousrate of change of velocity
it is the derivative
c-rfthe veloc:ity that is. o(l) :
r"(t). Negative acceleration(deceleration)means
that
t[e vgloc:ity is dec:r'easirrg.
Tlie acceleration gives the rate at which the velocity is
crharrging.
Therefore,
if .r is the displacernentof a rnoving objec:tand I is time, then:
i) veloc:itY
u(r) :
tr (t\ :
ii) ac'creleration
: o(t): ."'(t): r'/(/)
#.
:
i i i ) i ' ( / )
[ n ( t 1 , t t
iv) .r(t)-
[
,,31a,
Notc: T[e av('ragc velclcity of a partir:le over the tirne interval frorn ts to another time f. is
Averagevel;c'itv:
T#*frH#:
"(r] -;'itol.
wheres(t) is the p.sitionof
the partic:leat tinre t.
- The avetage value of /(r)
on
[a.
ir] is
f (r) d:r.
Arca BctwtxrriCtrrvt,s
If ./ ancl
g are continuousfuncrtionssuch that /(:r)
2 s@) on
[a,b],
then the area between
,. b
I
I l r e c r r r v e s
i s
/ l /
( " ,I
q ( r l ) d r.
J a
26
Formulas and Theorems 225
28
Volume of Soiids of R.evolution
Let /
be nonnegative and continuous on
[a,.
b]. and let R be the region bounded above
b y g :
/ ( r " ). b e l o w
b y t h e r - a x i s , a n d o n t h e s i d e sb y t h e l i n e s
r : : n a n d r : b.
When this region .R is revolved about tire .r'-axis.it gerreratesa solid (having circular
f o
crrosssec'tions)u'hosevolume V -
{j'(.,'l)
/ t t
Volunrcsof Soli<lswith Knowrr Cross Scctions
l. of area A(:r:). taken pt'r'lierrcli<'ular tcl the r-zrxis.
d r.
A(37)
taken perpt'rrriicrrlarto the 37-axis,
29. SolvirrgDifferential Equations: Graphically ancl Nurnerrir.all.l'
Skrpc Fieicls
Af ever'1'poirrt (.r.r7) a differetrtial ecluatiorrof the folrrr
f t, .i/) gives the slope of tht'
nernber of the farnily of solutit.rnsthat
c:onta,insthat poirrt. A slope fielcl is a, gra,lrhictrl
represent:rtiotrof this family of curves. At eac:hpt-rirrtirr the plarre.
a short s()gnlentis rlrau'n
"vhose
slope is eclualto the value of the clerivativer
at that poirrt. I'hese scgnrerrtsare taugcnt
to the sohrtion's graph at the poirrt.
The slope fielcl allows you to sketc:hthe graph of ther
solution cul've even though you rlo rrot
have its ec|ration. This is clc-rneby starting at arry point (usuallv the point given bv the initial
c'ondititin).and moving fron one poirrt to the next in the direc'tionirrdicntedby the segrncnts
of the slope fielcl.
Somc t'trlc'ulatorshavtt built in operations
fbr drawing slope fields; fcir calculatorsrvithorrt
tiris
feature tlrere are l)rograms available fbr drawing thern.
30. Soiving Diffelential Equations b)' Separatirrgthe Variables
There are lnAny technicluesfor solving differential equations. Any
differential equatir_rnvou
may be asked to solve ott the AB Calculus Exam can be solved by separating
the variables.
R,ewrite the equatioll as an erluivalent equation with all the r
and dr terrns on otle side arxl
all the q
and d37terrns ou the c-rther.Antidifferentiate both
sides to obtain an e(luation
without dr or du, but with orte c'onstantof inteqration. Use the initial condition to evahrate
this constant.
of area
r h.
z.
Fol cross sections
',llttttt'
.[rr"
^rr,
Fbr <'rossse<rtions
v.ltrttc' -
.[,"
^r,,
