FlashCardConstructionInstructions***THESECARDSAREFORCALCULUSHONORS,APCALCULUSABANDAPCALCULUSBC.APCALCULUSBCWILLHAVEADDITIONALCARDSFORTHECOURSE(INASEPARATEFILE).Theleftcolumnisthequestionandtherightcolumnistheanswers.Cutouttheflashcardsandpastethequestiontoonesideofanotecardandtheanswertotheotherside.Becarefultopastethecorrectanswertoitscorrespondingquestion!
COMMONFORMULAS/TRIGONOMETRY/GEOMETRY
Midpointformula
x1 + x22
, y1 + y22
⎛
⎝⎜
⎞
⎠⎟
Distanceformula(between2points)
d = (x2 − x1)2 + (y2 − y1)
2
QuadraticFormula
−b ± b2 − 4ac2a
PythagoreanTheorem
a2 + b2 = c2
sinθ =
opphyp
and yrand 1
cscθ
cosθ =
adjhyp
and xrand 1
secθ
tanθ =
oppadj
and yxand 1
cotθ
cotθ =
adjopp
andxyand 1
tanθ
cscθ =
hypopp
and ryand 1
sinθ
secθ =
hypadj
and rxand 1
cosθ
QuotientIdentity
tanu
sinucosu
QuotientIdentitycotu
cosusinu
PythagoreanIdentities
sin2 u + cos2 u = 1 1+ tan2 u = sec2 u 1+ cot2 u = csc2 u
AreaofaCircle/Circumferenceofacircle
A = πr2 C = 2πr
AreaofaParallelogram
A = bh
AreaofaTrapezoid
12h(b1 + b2 )
AreaofaTriangle
12bh
30-60-90triangle
1) Hypotenuseis2timeshortleg2) Longlegis 3 timesshortleg
45-45-90triangle
1) Hypotenuseis 2 timesleg2) Twolegsareequal
sin0
sin0 = 0
sin 30
sin 30 = 1
2
sin 45
sin 45 = 2
2
sin60
sin60 = 3
2
sin90
sin90 = 1
cos0
cos0 = 1
cos30
cos30 = 3
2
cos45
cos45 = 2
2
cos60
cos60 = 1
2
cos90
cos90 = 0
tan0
tan0 = 0
tan 30
tan 30 = 3
3
tan 45
tan 45 = 1
tan60
tan60 = 3
tan90
Undefined
sin(α + β) =
sin(α + β) = sinα cosβ + cosα sinβ
sin(α − β) =
sin(α − β) = sinα cosβ − cosα sinβ
cos(α + β) =
cos(α + β) = cosα cosβ − sinα sinβ
cos(α − β) =
cos(α − β ) = cosα cosβ + sinα sinβ
sin2θ =
2sinθ cosθ
cos2θ =
cos2θ − sin2θ 2cos2−1 1− 2sin2θ
Whatisa“solutionpoint”.
P.1
(x,y)pairthatmakesanequationswithanxandytrue
Howtofindxandyinterceptsofan
equation.P.1
x-interceptsety=0andsolveforxy-interceptsetx=0andsolvefory
Whatarethethreetypesofsymmetry?
P.1
y-axis(replacingxwith–xyieldingoriginalequation)x-axis(replacingywith–yyieldingoriginalequation)origin(replacingxwith–xandywith–yyieldingoriginalequations
Whatarethe3testsforsymmetry?
P.1
y-axisx-axisorigin
Howtofindthepointsofintersectionsof
twoequations?
P.1
Simultaneouslysolvingequations(elimination,substitutionorusingintersectfeatureofcalculator
Theformulaforfindingtheslope
betweentwopoints?P.2
y2 − y1x2 − x1
Whatarethe4typesofslope?
P.2
positive,negative,zero,undefined
Whatisthepointslopeformoftheequationofaline?
P.2
y − y1 = m(x − x1)
Whataretherelationshipsofslopes
betweenparallellinesandperpendicularlines.
P.2
parallellines(sameslope),perpendicularlines(negativereciprocal
slopes)
Howdoyoucalculateanaveragerateofchange?
P.2
f (b) − f (a)b − a
Whatistheslope-interceptequationofa
line?
P.2
y=mx+b
Whatistherelationshipbetweena
relationandafunction?
P.3
Functionhaseachxpointingtoonlyoneyvalue
Whatdoes“one-to-one”mean?
P.3
eachyvalueispointedtobyonlyonex-
value
Whatdoes“onto”mean?
P.3
rangeconsistsofallofY
Howdoyouproveagraphisafunction?
P.3
passestheVerticalLineTest
Whatarethe3categoriesofelementary
functions?P.3
a. algebraic(polynomial,radical,rational)b. trigonometricc. exponentialandlogarithmic
Whatistheleadingcoefficienttestforpolynomials?
P.3
a. evenexponentofleadingcoefficienti. leadingcoefficient>0up/upii. leadingcoefficient<0down/down
b. oddexponentofleadingcoefficientiii. leadingcoefficient>0downleft/uprightiv. leadingcoefficient<0upleft/downright
Whatisan“odd”function?
P.3
(symmetricaboutorigin)
Whatisanevenfunction?
P.3
(y-axissymmetry)
Whatistherelationshipofthedomainandrangeininversefunctions?
P.4
Thedomainsandrangesareswapped
Howcanyoudetermineifafunctionhasaninverse?
P.4
OriginalfunctionwillpasstheHorizontalLineTest
Howcanyouvisuallydetermineoftwofunctionsareinversesofeachother?
P.4
Thetwofunctionswillbereflectedabouttheliney=x
Whatarethedomainsandrangesofarcsin?
P.4
Domain:−1 ≤ 𝑥 ≤ 1
Range!!!≤ 𝑦 ≤ !
!
Whatarethedomainsandrangesofarccos?
P.4
Domain:−1 ≤ 𝑥 ≤ 1
Range0 ≤ 𝑦 ≤ 𝜋
Whatarethedomainsandrangesofarctan?
P.4
Domain:−∞ < 𝑥 < ∞
Range!!!< 𝑦 < !
!
𝑎!
P.5
1
𝑎!𝑎!
P.5
𝑎!!!
(𝑎!)!
P.5
𝑎!"
(𝑎𝑏)!
P.5
𝑎!𝑏!
𝑎!
𝑎!
P.5
𝑎!!!
(𝑎𝑏)
!P.5
𝑎!
𝑏!
𝑎!!
P.5
1𝑎!
𝑙𝑛𝑒!P.5
x
𝑒!"#P.5
x
Whatarethedomainsandrangesof𝑙𝑛𝑥?
P.5
Domain:(0,∞)
Range(−∞,∞)
Whatarethedomainsandrangesof𝑒!?
P.5
Domain:(−∞,∞)
Range0,∞)
Whatistheformulaforfindingasecant
line?1.1
Msec =f (x + Δx)− f (x)
Δx
Whatistheconceptofalimit?
1.1
Iff(x)becomesarbitrarilyclosetoasinglenumberLasxapproachescfromeithersidethelimitoff(x),asxapproachesc,isL
Whatisagenericdefinitionofatangentline?
1.1
Alinethattouchescurveatonepoint
Whatarethe3conditionsthatneedto
bemetforalimittoexist?
1.2
a. limx→a+
f (x)exists
b. limx→a−
f (x)exists
c. limx→a+
f (x) = limx→a−
f (x)
Whatarethe3conditionswherealimitfailstoexist?1.2
a. unboundedbehavior(verticalasymptote)b. limitfromtheleftnotequaltothelimitfromtherightc. oscillatingbehavior
Whatis“well-behaved”function?
1.3
limx→c
f (x) = f (c)
Whatarethe3basictypesofalgebraic
functions?
1.3
a. polynomialb. rationalc. radical
Whataretechniquesforfindinglimits?
1.3
a. directsubstitution(plugnchug)b. dividingout(factoring)c. rationalizingthenumeratord. makeatable/graph
Whataretheindeterminateformsofa
function?
1.3
00or∞
∞
limx→0
sin xx
1.3
1
limx→0
1− cos xx
1.3
0
lim!→!
(1+ 𝑥)!!
1.3
e
Whatarethe3conditionsthatneedtobemetforcontinuity?
1.4
a. f(a)definedb. lim
x→af (x)exists
c. f(a)= limx→a
f (x)
Whatistheconceptofa“continuous”function?
1.4
whenagraphcanbedrawnwithoutliftingthepencil
Whatistheconceptof“everywherecontinuous”?
1.4
continuousovertheentirenumberline
Whatare3typesofdiscontinuity?
1.4
a. holeb. infinite(verticalasymptote)c. jump
Whatistheconceptofa“one-sided”
limit?
1.4
whenonlythelimitfromtheleftorthelimitfromtherightofx=cisdefined.
Whatare5typesoffunctionsthatarecontinuousateverypointintheir
domain?
1.4
a. polynomialfunctionsb. rationalfunctionsc. radicalfunctionsd. trigonometricfunctionse. exponentialandlogarithmic
WhatdoestheIntermediateValue
Theoremstate?
1.4
Iffiscontinuousontheclosedinterval[a,b]andkisanynumberbetweenf(a)andf(b),thenthereexistsatleastonenumbercin[a,b]suchthatf(c)=k
Whatisaverticalasymptote?1.5
Verticallinethatisapproachedbutnevertouched(endbehavior)andisaresultofthedenominatorofarational
expressionbeingundefined
Howcanyoudeterminethedifferencebetweenwhenaholeexistsanda
verticalasymptoteexists?1.5
Ifyoucancancelafactoroutofdenominatoritisahole
Whatisahorizontalasymptote?
1.6
Horizontallinethatisapproachedbutnevertouched(endbehavior)andisaresultofthedenominatorgrowingfaster
thanthenumerator
lim!→!
𝑐𝑥!
1.6
0
lim!→!!
𝑐𝑥!
1.6
0
lim!→!!
𝑒!
1.6
0
lim!→!
𝑒!!
1.6
0
!!
! 𝑥 > 0
1.6
1
(sneakytechnique)
!!
!! 𝑥 < 0
1.6
1
(sneakytechnique)Whatarethe3testsfordetermining
horizontalasymptotes?
1.6
numexponent>denexponent,no
asymptotenumexponent<denexponent,y=0numexponent=denexponent,
y= leadingcoefficientleadingcoefficient
Whatisthedefinitionofthederivativeofafunctionusinglimits?
2.1
f '(x) = limΔx→0
f (x + Δx) − f (x)Δx
Whatisanalternateformofthederivativefunctionusinglimits?
2.1
𝑓! 𝑐 = lim!→!
𝑓 𝑥 − 𝑓(𝑐)𝑥 − 𝑐
Whatisthedifferencequotient?
2.1
f (x + Δx) − f (x)
Δx
Whatarethe3caseswhereaderivative
failstoexist?
2.1
a. anypointofdiscontinuityb. cuspc. verticaltangentline
DifferentiationRules:ConstantRule
2.2
ddx[c] = 0
DifferentiationRules:SimplePowerRule
2.2
ddx[xn ] = nxn−1
DifferentiationRules:ConstantMultipleRule
2.2
ddx[cf (x)] = cf '(x)
DifferentiationRules:
SumandDifferenceRules2.2
ddx[ f (x) ± g(x)] = f '(x) ± g '(x)
ddx[sin x]
2.2
cos x
ddx[cos x] =
2.2
−sin x
𝑑𝑑𝑥 [𝑒
!]
2.2
𝑒!
Whatisthestandardpositionfunction?
2.2
𝑠 𝑡 = −16𝑡! + 𝑉!𝑡 + 𝑆!
-4.9canbesubstitutedifcalculatinginmetersinsteadoffeet
ddx[tan x] =
2.3
sec2 x
ddx[csc x] =
2.3
−csc xcot x
ddx[sec x] =
2.3
sec x tan x
ddx[cot x] =
2.3
−csc2 x
DifferentiationRules:
ProductRule
2.3
f (x)g '(x)+ g(x) f '(x)
firstdsecond+seconddfirst
DifferentiationRules:QuotientRule
2.3
g(x) f '(x)− f (x)g '(x)
g(x)2 bottomdtop–topdbottomover
bottomsquared
DifferentiationRules:ChainRule
2.4
f '(g(x))g '(x)
douterdinner(don’ttouchthestuff)
DifferentiationRules:GeneralPowerRule
2.4
nun−1u '
ddx[sinu] =
2.4
(cosu)u '
ddx[cosu] =
2.4
(−sinu)u '
ddx[tanu] =
2.4
(sec2 u)u '
ddx[cotu] =
2.4
−(csc2 u)u '
ddx[secu] =
2.4
(secu tanu)u '
ddx[cscu] =
2.4
−(cscucotu)u '
𝑑𝑑𝑥 [ln 𝑥]
2.4
1𝑥 , 𝑥 > 0
𝑑𝑑𝑥 [ln|𝑢|]
2.4
𝑢!
𝑢
log! 𝑥
2.4
1𝑙𝑛𝑎 𝑙𝑛𝑥 𝑜𝑟
𝑙𝑛𝑥𝑙𝑛𝑎
𝑑𝑑𝑥 [𝑎
!]
2.4
𝑙𝑛𝑎 𝑎!
𝑑𝑑𝑥 [𝑎
!]
2.4
𝑙𝑛𝑎 𝑎!𝑑𝑢𝑑𝑥
𝑑𝑑𝑥 [log! 𝑥]
2.4
1
𝑙𝑛𝑎 𝑥
𝑑𝑑𝑥 [log! 𝑢]
2.4
1𝑙𝑛𝑎 𝑢
𝑑𝑢𝑑𝑥 𝑜𝑟
𝑢!
𝑙𝑛𝑎 𝑢
𝑑𝑑𝑥 [𝑒
!]
2.4
𝑒!𝑢!
Whatistheexplicitformofanequation?
2.5
whenanequationissolvedforonevariable
Inversefunctionshavewhattypesofslopesatinversepairsofpoints?
2.6
reciprocalslopes
ddx[arcsinu] =
2.6
u '1−u2
ddx[arccosu] =
2.6
−u '1−u2
ddx[arctanu] =
2.6
u '1+u2
ddx[arccotu] =
2.6
−u '1+u2
ddx[arcsecu] =
2.6
u 'u u2 −1
ddx[arccscu] =
2.6
−u '
u u2 −1
Whatisarelatedratederivativeusuallytakenwithrespectto?
2.7
time
Whatistheformulaforthevolume
ofacone?
2.7
𝑉 =𝜋3 𝑟
!ℎ
Whatistheformulaforthevolume
ofasphere?
2.7
𝑉 =43𝜋𝑟
!
Whatisaanothernameforatangentlineofapproximation
called?
2.8
linearapproximation
Whatmethodusesatangent
linetoapproximatethey-valuesofafunction?
2.8
Newton’smethod
Whatisa“maximum”?
3.1
f(c)>allf(x)onaninterval
Whatisa“minimum”?
3.1
f(c)<allf(x)onaninterval
Whatisthedifferencebetweencritical
numbersandcriticalpoints?
3.1
criticalnumbersarex-valuesandcriticalpointsare(x,y).Criticalnumbersarefoundwhen f '(c) = 0 orwhere f '(c)doesnotexist.
Whattheoremstateiffiscontinuousonaclosedinterval[a,b],then
fhasbothaminimumandamaximumontheinterval
3.1
ExtremeValueTheorem
Wheredoesthederivativefailtoidentifypossibleextrema?
3.1
endpoints
WhatdoesRolle’sTheoremstate?
3.2
iff(a)=f(b)thenthereexistsatleastonenumbercin(a,b)suchthat f '(c) = 0
WhatdoestheMeanValueTheorem
state?
3.2
f '(c) = f (b) − f (a)b − a
WhataretwomajorsimilaritiesbetweenRolle’sTheoremandtheMeanValue
Theorem?
3.2
Functionmustbe1)continuousand
2)differentiable
Whatismeantby“increasing”interms
ofaderivative?
3.3
f '(x) > 0 forallxin(a,b)
Whatismeantby“decreasing”intermsofaderivative?
3.3
f '(x) < 0 forallxin(a,b)
Whatismeantby“constant”intermsof
aderivative?
3.3
f '(x) = 0 forallxin(a,b)
Whatdoes“strictlymonotonic”mean”?
3.3
Whenafunctioniseitherincreasingordecreasingonentireinterval
Whatdoesthefirstderivativeteststate?
3.3
a. if f '(x) changesfromincreasingtodecreasingatx=cthen f '(c) isarelativemaximumb. if f '(x) changesfromdecreasingtoincreasingatx=cthen f '(c) isarelativeminimumc. if f '(x) doesnotchangesignsatx=cthenf '(c) isaneitherarelativemaximumorrelativeminimum
Howdoyouusethesecondderivativetodetermineconcavity?
3.4
a. if f ''(x) > 0 ,forallxinanintervalfisconcaveupwardb. if f ''(x) < 0 ,forallxinanintervalfisconcavedownward
Whatare“pointsofinflection”?
3.4
where f ''(c) = 0 or f ''(c) isundefined(whereagraphgoesfromconcaveupwardtoconcavedownwardorviceversa
Howdoyouusethesecondderivativetodeterminerelativeextremausingcritical
numbers?
3.4
a. if f ''(c)> 0 ,thenf(c)isarelativeminimumb. if f ''(c) < 0 ,thenf(c)isarelativemaximumc. if f ''(c) = 0 thenusemustusetheFirstDerivativeTest
Inoptimizationproblemswhat
istheequationthatistobeoptimizedcalled?
3.6
primaryequation
Whatisadifferentialequation?
3.7
anequationthatcontainsaderivative
Whatistheequationfor
atangentlineofapproximation(linearapproximation)?
3.7
𝑦 = 𝑓 𝑐 + 𝑓′(𝑐)(𝑥 − 𝑐)
0𝑑𝑥
4.1
C
du =∫
4.1
u+C
kf (x)dx∫
4.1
k f (x)dx∫
[ f (x)± g(x)]dx∫
4.1
f (x)dx ± g(x)dx∫∫
xn dx∫ =
4.1
xn+1
n +1+C
cos x dx =∫
4.1
sin x +C
sin x dx∫ =
4.1
−cos x +C
(sec2 x)dx∫
4.1
tan x +C
sec x tan x dx∫
4.1
sec x +C
(csc2 x)dx∫ =
4.1
−cot x +C
csc xcot x dx∫
4.1
−csc x +C
𝑒!𝑑𝑥
4.1
𝑒! + 𝐶
𝑎!𝑑𝑥
4.1
(1𝑙𝑛𝑎)𝑎
! + 𝐶
1𝑥 𝑑𝑥
4.1
ln |𝑥|+ 𝐶
Tochangeageneralsolutionintoaparticularsolutionwhatis
needed?4.1
aninitialcondition
𝑎!!!!! iswhattypeofnotation?
4.2
sigmanotation
𝑐!
!→!
4.2
𝑐𝑛
𝑖!
!→!
4.2
𝑛(𝑛 + 1)2
𝑖!!
!→!
4.2
𝑛(𝑛 + 1)(2𝑛 + 1)6
𝑖!!
!→!
4.2
𝑛!(𝑛 + 1)4
!
LeftRectangleRule
4.2
𝑏 − 𝑎𝑛 (𝑓(𝑥!) +⋯ 𝑓 𝑥!!! )
RightRectangleRule
4.2
𝑏 − 𝑎𝑛 (𝑓(𝑥!) +⋯ 𝑓 𝑥! )
Thedefiniteintegralastheareaofaregion
4.3
𝑓 𝑥 𝑑𝑥!
!
𝑓 𝑥 𝑑𝑥!
!
4.3
0
𝑓 𝑥 𝑑𝑥!
!
4.3
− 𝑓 𝑥 𝑑𝑥!
!
𝑓 𝑥 𝑑𝑥!
!
withpointcbetweenaandb
4.3
𝑓 𝑥 𝑑𝑥 + 𝑓 𝑥 𝑑𝑥!
!
!
!
𝑘𝑓(𝑥)𝑑𝑥)!
!
4.3
𝑘 𝑓 𝑥 𝑑𝑥!
!
𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥!
!
4.3
𝑓 𝑥 𝑑𝑥 ± 𝑔 𝑥 𝑑𝑥!
!
!
!
TrapezoidalRule
4.3
𝑏 − 𝑎2𝑛 [𝑓 𝑥! + 2𝑓 𝑥! +⋯ 2𝑓(𝑥!!!)
+ 𝑓(𝑥!)]
FundamentalTheoremofCalculus
4.4
𝑓 𝑥 𝑑𝑥!
!= 𝐹 𝑏 − 𝐹(𝑎)
MeanValueTheoremForIntegrals
4.4
𝑓 𝑥 𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎)!
!
Averagevalueofafunction
4.4
1𝑏 − 𝑎 𝑓 𝑥 𝑑𝑥
!
!
SecondFundamentalTheoremofCalculus
4.4
𝑑𝑑𝑥 [ 𝑓 𝑡 𝑑𝑡] = 𝑓 𝑥
!
!
NetChangeTheorem
4.4
𝐹! 𝑥 = 𝐹 𝑏 − 𝐹(𝑎)!
!
un du∫ =
4.5
un+1
n+1+C
𝑘𝑓 𝑥 𝑑𝑥
4.5
𝑘 𝑓 𝑥 𝑑𝑥
𝑓 𝑥 𝑑𝑥!!! (evenfunction)
4.5
2 𝑓 𝑥 𝑑𝑥!
!
𝑓 𝑥 𝑑𝑥!!! (oddfunction)
4.5
0
duu
=∫
4.6
ln u +C
au du∫ =
4.6
1lna⎛
⎝⎜
⎞
⎠⎟au +C
sinudu =∫ 4.6
−cosu+C
cosudu∫ =
4.6
sinu+C
tanudu∫ =
4.6
− ln cosu +C
cotudu∫ =
4.6
ln sinu +C
secudu∫ =
4.6
ln secu+ tanu +C
cscu =∫
4.6
− ln cscu+ cotu +C
sec2 udu =∫
4.6
tanu+C
csc2 udu =∫
4.6
−cotu+C
secu tanudu =∫
4.6
secu+C
cscu cotudu∫ =
4.6
−cscu+C
du
a2 + u2=∫
4.7
1aarctan u
a+C
dua2 − u2
=∫
4.7
arcsin ua+C
duu u2 − a2
=∫
4.7
1aarcsec
ua+C
Whatisadifferentialequation?5.1
anequationthatincludesaderivative
WhatisEuler’sMethod?
5.1
anumericalapproachtoapproximatingtheparticularsolutiontoadifferential
equation
Whatisthesolutiontoaexponential
growthordecayproblem?
5.2
𝑦 = 𝐶𝑒!"
Whatiskinahalf-lifeproblem?
5.2
ln (12)𝑡 = 𝑘
Whatistheprocessofcollectingall
termswithx’sandy’sonoppositesidesoftheequalsigncalled?
5.2
separationofvariables
Howdoyoufindtheareabetweentwo
curves?
6.1
[ f (x) − g(x)]dxa
b
∫
DiskMethod
HorizontalAxisofRevolution
6.2
π [R(x)]2 dxa
b
∫
DiskMethod
VerticalAxisofRevolution
6.2
π [R(y)]2 dyc
d
∫
WasherMethod
HorizontalAxisofRevolution
6.2
π ([R(x)]2 − [r(x)]2 )dxa
b
∫
WasherMethod
VerticalAxisofRevolution
6.2
π ([R(y)]2 − [r(y)]2 )dyc
d
∫
Volumeofsolidwithknowncrosssectionperpendiculartox-axis
6.2
A(x)dxa
b
∫
Volumeofsolidwithknowncrosssectionperpendiculartoy-axis
6.2
A(y)dyc
d
∫
