Harold’sCalculusNotes
“CheatSheet”
20March2013
APCalculusAB&BC
Limits
DefinitionofLimit
Letfbeafunctiondefinedonanopeninterval
containingcandletLbearealnumber.The
statement:
lim
→
meansthatforeach
0thereexistsa
suchthat
|
|
if0 |
,then|
0
Tip:
Directsubstitution:Pluginf(a)andseeifit
providesalegalanswer.IfsothenL=f(a).
TheExistenceofaLimit
The limit of f(x) as x approaches a is L if and
onlyif:
→
and
lim
→
1
→
TwoSpecialTrigLimits
1
0
→
Derivatives
DefinitionofaDerivativeofaFunction
Slope
NotationforDerivatives
lim
,
,
,
TheConstantRule
ThePowerRule
TheGeneralPowerRule
∆
∆
lim
∆ →
,
,
0
1(think
TheConstantMultipleRule
Copyright © 2013 by Harold Toomey, WyzAnt Tutor 1)
1 TheSumandDifferenceRule
1
2
PositionFunction
VelocityFunction
AccelerationFunction
TheProductRule
TheQuotientRule
,
0
TheChainRule
Sine
cos
Cosine
Tangent
Secent
Cosecent
Cotangent
ApplicationsofDifferentiation
Rolle’sTheorem
fiscontinuousontheclosedinterval[a,b],and
fisdifferentiableontheopeninterval(a,b).
Iff(a)=f(b),thenthereexistsatleastonenumberc
in(a,b)suchthatf’(c)=0.
MeanValueTheorem
IffmeetstheconditionsofRolle’sTheorem,
then
′
lim
lim
→
L’Hôpital’sRule
→
GraphingwithDerivatives
→
0 ∞
, , 0 ∗ ∞, 1 , 0 , ∞ , ∞
0 ∞
lim
TestforIncreasingandDecreasing
Functions
Find‘c’.
lim
∞ ,
0
lim
→
→
,
⋯
1. Iff’(x)>0,thenf isincreasing(slopeup)
2.Iff’(x)<0,thenfisdecreasing(slopedown)
3.Iff’(x)=0,thenfisconstant(zeroslope)
Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor 2 TheFirstDerivativeTest
TheSecondDeriviativeTest
Letf’(c)=0,andf”(x)exists,then
TestforConcavity
PointsofInflection
Changeinconcavity
1. Iff’(x) changesfrom– to+atc,thenf hasarelative
minimumat(c,f(c))
2.Iff’(x)changesfrom+to‐atc,thenfhasarelative
maximumat(c,f(c))
3.Iff’(x),is+c+or‐c‐,thenf(c)isneither
1. Iff”(x)>0,thenf hasarelativeminimumat(c,
f(c))
2.Iff”(x)<0,thenfhasarelativemaximumat(c,
f(c))
3.Iff’(x)=0,thenthetestfails(See1stderivative
test)
1. Iff”(x)>0 forallx,thenthegraphisconcave
upward
2.Iff”(x)<0forallx,thenthegraphisconcave
downward
If(c,f(c)) isapointofinflectionoff,theneither
1.f”(c)=0or
2. f” does not exist at x = c.
AnalyzingtheGraphofaFunction
x‐Intercepts(ZerosorRoots)
y‐Intercept
Domain
Range
Continuity
VerticalAsymptotes(VA)
HorizontalAsymptotes(HA)
InfiniteLimitsatInfinity
Differentiability
RelativeExtrema
Concavity
PointsofInflection
f(x)=0
f(0)=y
Validxvalues
Validyvalues
Nodivisionby0,nonegativesquareroots
x =divisionby0orundefined
lim →
→ andlim →
→ → ∞andlim →
→ ∞
lim →
Limitfrombothdirectionsarrivesatthe sameslope
Createatablewith domains,f(x),f’(x),andf”(x)
If ”
→ ,thencupup∪
→ ,thencupdown∩
If ”
f”(x)=0
ApproximatingwithDifferentials
Newton’sMethod
Findszerosoff,orfindsciff(c)=0.
TangentLineApproximations
FunctionApproximationswithDifferentials
′
∆
Integration
BasicIntegrationRules
Integrationisthe“inverse”ofdifferentiation.
Differentiationisthe“inverse”ofintegration.
0
0
Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor 3 TheConstantMultipleRule
TheSumandDifferenceRule
ThePowerRule
If
,
1
1
1then
ln| |
If
,
then
TheGeneralPowerRule
,
1
1
1
2
SummationFormulas
1 2
6
1
1
4
∆ ,
ReimannSum
‖∆‖
DefinitionofaDefiniteIntegral
Areaundercurve
lim
‖∆‖→
∆
∆
SwapBounds
AdditiveIntervalProperty
TheFundamentalTheoremofCalculus
(SeeHarold’sFundamentalTheoremofCalculus
“CheatSheet”)
TheSecondFundamentalTheoremof
Calculus
′
(SeeHarold’sFundamentalTheoremofCalculus
“CheatSheet”)
Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor 4 MeanValueTheoremforIntegrals
Find‘c’.
1
TheAverageValueforaFunction
IntegrationMethods
1.Memorized
See1‐pager ofcommonintegrals
Set
, then
2.U‐Substitution
_____
_____ ____v _____
_____
_____
Pick‘ ’usingtheLIATERule:
L‐Logarithmicfunctions:ln , log ,
.
I‐Inversetrig.functions:tan
, sec
,
A‐Algebraicfunctions: , 3 ,
.
T‐Trigonometricfunctions:sin , tan ,
.
E‐Exponentialfunctions: , 19 ,
.
3.IntegrationbyParts
where
arepolynomials
Substutution:
Identity:1
Substutution:
1
Identity:
7.TrigSubstitutionfor
8.TableofIntegrals
sin 6.TrigSubstitutionfor√
.
Case1:Ifdegreeof
thendolongdivisionfirst
Case2:Ifdegreeof
thendopartialfractionexpansion
4.PartialFractions
5.TrigSubstitutionfor√
sec Substutution:
tan Identity:1
CRCStandardMathematicalTables book
Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor 5 TI–NspireCASiPadapp
TI‐89Titaniumcalculator
RiemannSum
MidpointRule
TrapezoidalRule
Simpson’sRule
Googleofmathematics.Showssteps. Free.
www.wolframalpha.com
WolframAlphaiPhone/iPadapp
9.ComputerAlgebraSystems(CAS)
10.NumericalMethods
11.WolframAlpha
PartialFractions
Condition
CaseI:Simplelinear(
where
arepolynomials
anddegreeof
degree)
CaseII:Multipledegreelinear(
CaseIII:Simplequadratic(
degree)
degree)
CaseIV:Multipledegreequadratic(
degree)
TypicalSolutionforCasesI&II
|
TypicalSolutionforCasesIII&IV
NumericalMethods
|
∗
lim
‖ ‖→
RiemannSum
⋯
and∆
and‖ ‖
∆ Types:LeftSum,MiddleSum,RightSum
where
̅ ∆
MidpointRule
∆ ∆
̅
̅
where∆
̅
⋯
̅
and ̅
ErrorBounds:|
Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor |
,
6 TrapezoidalRule
∆
2
2
2
⋯
2
where∆
and
ErrorBounds:|
∆ |
∆
3
4
2
4
4
2
Whereniseven
and∆
∆ and
Simpson’sRule
ErrorBounds:|
InfiniteSequencesandSeries
|
lim →
Example:(
2.GeometricSeriesTest
3.p‐SeriesTest
4.AlternatingSeriesTest
5.IntegralTest
6.RatioTest
7.RootTest
8.DirectComparisonTest
9.LimitComparisonTest
⋯
⋯
1
1
1
→
1
1
onlyif| | 1
where istheradiusorintervalofconvergence
lim
⋯
Similartoanarithmeticseries
ConvergenceTests
TermTest
⋯
GeometricSeries(finite)
1.
(Limit)
,
, …)
,
ArithmeticSeries(infinite)
PartialSum
Sequence
GeometricSeries(infinite)
⋯
(SeeHarold’sSeriesConvergenceTests “Cheat
Sheet”)
”
” ” ” ” ” ” ” Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor 7 TaylorSeries
PowerSeries
⋯
PowerSeriesAboutZero
⋯
0
MaclaurinSeries
Taylorseriesaboutzero
!
TaylorSeries
!
!
TaylorSerieswithRemainder
∗
1 !
where
andlim →
BinomialSeries
1
1
1
CommonSeries
!
1
ln 1
1
sin
cos
sinh
1
1 !
2
!
1
2
1
1
1 !
2
2
1
2
2
∗
0
2 …
!
2!
3!
4!
1
!
| |
1
1
Euler’sEquation Copyright © 2013 by Harold A. Toomey, WyzAnt Tutor ⋯
⋯
1
| |
2
1
cosh
arctan
1
1
| |
1
2
3
4
5
3!
5!
7!
9!
2!
4!
6!
8!
3!
5!
7!
9!
2!
4!
6!
8!
3
5
7
9
1
cos
0
sin
⋯
⋯
⋯
⋯
⋯
⋯
8