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ReviewforMidterm2,ConceptsLetfbecontinuouson[a,b].
DefinitionsLetfbeafunctionwithdomainD.fhasanabsolutemaximumonDat x = c iff (x) ≤ f (c) for all x in D .fhasanabsoluteminimumonDat x = c iff (x) ≥ f (c) for all x in D .fhasarelativemaximumat x = c ifthereexistaninterval( r , s ) containingcsuchthat f (x) ≤ f (c) for all x in both D and ( r , s ).fhasarelativeminimumat x = c ifthereexistaninterval( r , s ) containingcsuchthat f (x) ≥ f (c) for all x in both D and ( r , s ).ExtremeValueTheoremIffiscontinuousonaclosedinterval[a,b],thenfhasanabsolutemaximumandanabsoluteminimumon[a,b]
Iffhasalocalmaximumorminimumvalueat and f '(c) exists,then f '(c)= 0 .Supposefiscontinuouson[ a ,b ].Howdowefinditsmaximumandminimumvalues?
1. Checkwhere f '(x)= 0 orwhere f '(x) doesnotexist.2. Check f (a) and f (b) .
MeanValueTheoremSupposefiscontinuousontheinterval[ a ,b ]anddifferentiableontheinterval(a ,b ) .Thenthereexistsatleastonepointcin(a ,b ) suchthat f '(c) = f (b)− f (a)
b− a .
Theaveragerateofchangeinfover[ a ,b ]isequaltotheinstantaneousrateofchangeatsomepointin(a ,b ) .Theorem
x = c
If f '(x) > 0 foreveryxinaninterval( r , s ) thenfisstrictlyincreasingontheinterval( r , s ) .TheoremIf f '(x) < 0 foreveryxinaninterval( r , s ) thenfisstrictlydecreasingontheinterval( r , s ) .TheoremIf f '(x) = 0 foreveryxinaninterval( r , s ) thenfisaconstantfunctionontheinterval( r , s ) .TheoremIf f '(x) = g '(x) foreveryxinaninterval( r , s ) thenthereexistaconstantcsuchthat f (x) = g(x)+ c forallxin( r , s ) .
FirstDerivativeTest: Suppose x = c isacriticalpointforf.
If f ' changesfromnegativetopositiveatcthenfhasalocalminimumatc.If f ' changesfrompositivetonegativeatcthenfhasalocalmaximumatc.If f ' doesnotchangesignatcthenfhasnolocalextremumatc.
ConcavityandCurveSketchingExampleConsiderthefunctionfdefinedby f (x)= x3 .Notethat
f '(x)= 3x2 , f "(x)= 6xf "(x)> 0 for x > 0f "(x)< 0 for x < 0f ' is increasing on ( 0 ,∞ )f ' is decreasing on (−∞ , 0 )f "changes sign at x = 0
DefinitionThefunctionfisconcaveupontheinterval(a ,b ) if f ' isincreasingon(a ,b ) .Thefunctionfisconcavedownontheinterval(a ,b ) if f ' isdecreasingon(a ,b ) .Definition( c , f (c) ) isapointofinflectionifthegraphoffhasatangentlineat( c , f (c) ) andtheconcavityoffchangesat( c , f (c) ) .
SecondDerivativeTestSuppose f "iscontinuousnear x = c .1.If f '(c)= 0 and f "(c)< 0 thenfhasalocalmaximumat x = c .2.If f '(c)= 0 and f "(c)> 0 thenfhasalocalminimumat x = c .3.If f '(c)= 0 and f "(c)= 0 thenthetestfails.IndeterminateFormsandL’Hopital’sRuleRemember lim
x→0sinxx .
L’Hopital’sRule:Supposefandgaredifferentiable,f (a)= g(a)= 0 andg '(x)≠ 0 when x ≠ a Thenlimx→a
f (x)g(x) = limx→a
f '(x)g '(x) .
Fortheindeterminateforms 0
0,∞∞,thelimitofthequotientoftwo
functionsisequaltolimitofthequotientofthederivatives.Otherindeterminateformcanoftenbereformulatedtofit 0
0or ∞
∞.
RelatedRates,AppliedProblemSolvingAnofficebuildingislocatedrightonastraightriverbank.Apowerplantisontheoppositebank,1500feetdownstreamfromtheofficebuilding.Theriveris300feetwide.Ifwewanttoconnectthepowerplantandthebuildingbycable,whichcosts$1700perfoottolaydownunderwaterand$800perfoottolayunderground,whatistheleastexpensivepathforthecable?
ProblemSolvingSteps
1. Understandwhatisgiven.2. Chooseavariable(s).3. Obtainamathematicaldescriptionoftheproblem.4. Dothemath.5. Interprettheresults.
AntiderivativesTheansweris ⎡⎣ ⎤
⎦ ,whatisthequestion?
a) ddx ⎡⎣ ⎤⎦ = x7
b) ddx ⎡⎣ ⎤⎦ =
1x
c) ddx ⎡⎣ ⎤⎦ = 3cos2θ + 2sec2θ tanθ
d) ddx ⎡⎣ ⎤⎦ = xk
DefinitionIfF '(x) = f (x) forallx,thenFisanantiderivativeforf.SupposeF '(x) = f (x) forallx.ThenanyantiderivativeoffcanberepresentedbyF(x)+ c .Theseantiderivativesaredenotedbyf (x)∫ dx ,calledtheindefiniteintegraloff.
xk∫ dx = xk+1k+1 + c (PowerRule)
k f (x) dx∫ = k f (x) dx∫
[ f (x)± g(x) ]dx = f (x)dx ± g(x)dx∫∫∫
e−x2 dx∫ = ?
IntegralsArchimedesDiagram:
Archimedes used the method of exhaustion to find an approximation to the area of a circle. This is an early example of integration that led to approximate values of π. Example Let’s approximate the area below the graph of y= f (x) = 1− x2 between x = 0 and x =1 .
Upper Sums, Lower Sums, Midpoint Sums Riemann sum calculator: https://www.desmos.com/calculator/tgyr42ezjq
Sums1+2+ 3+ 4 + ...+ (n−1)+ n = i
i=1
n∑ = n (n+1)
2 .(Gauss)
12 + 22 + 32 + 42 + ... + k2 + ...+ n2 = k2
k=1
n∑ = n(n+1)(2n+1)
6
13 + 23 + 33 + 43 + ... + k3 + ...+ n3 = k3
k=1
n∑ = n2(n+1)2
4
RiemannSums,DefiniteIntegral
Howshouldweapproximatewithareasofrectangles?1. Weneedtopartitiontheinterval[ a ,b ]intosmallsubintervals.
2. Wemustthenusethefunctionftodeterminetheheightofeachrectangleanddecidewhethertocounttheareapositivelyornegatively.
DefinitionApartitionof[ a ,b ]isasetofpoints{ x0 , x1 , x2 , x3 ... xk−1 , xk , ..., xn−1 , xn } suchthata= x0 < x1 < x2 < x3 < ... < xk−1 < xk < ...< xn−1 < xn = b .
Ourgoalistoapproximateareaswitheverincreasingdegreeofaccuracy;sowewillwantourpartitionstodefinealargenumberofsubintervalswithsmallwidth.IfPisapartitionof[ a ,b ]thendeterminehowgoodthepartitionisbyconsideringthelengthofthelargestsubinterval.Somemorenotation:LetP = { x0 , x1 , x2 , x3 ... xk−1 , xk , ..., xn−1 , xn } beapartitionof[ a ,b ].Soa= x0 < x1 < x2 < x3 < ... < xk−1 < xk < ...< xn−1 < xn = b .Thesubintervalsare[ x0 , x1 ],[ x1 , x2 ],[ x2 , x3 ], ...,[ xk−1 , xk ], ...,[ xn−1 , xn ].LetΔ x1 = x1 − x0 , Δ x2 = x2 − x1 , ... , Δ xk = xk − xk−1 , ... , Δ xn = xn − xn−1
Thelengthofthelargestsubintervalisequaltomax{Δ x1 ,Δ x2 , ..., Δ xk , ... , Δ xn }
andisdenotedby P ,calledthenormofthepartitionP.Ifwewantourapproximationtobeaccurate,thenwewant P tobesmall(closetozero).Fortheheightsoftherectangleswewillchooseapointck from each[ xk−1 , xk ] and evaluate f ( ck ) .
WenowobtainaRiemannsum:
f ( ck )Δkk=1
n∑ .Whathappenswhen P → 0 ?
DefinitionSupposefisacontinuousfunctionon[ a ,b ].Thedefiniteintegraloffover[ a ,b ]is
limP →0
f ( ck )Δkk=1
n∑ andisdenotedby f (x) dx
a
b∫ .
Weread“theintegralofffromatobwithrespecttox”.
Formally,
limP →0
f ( ck )Δkk=1
n∑ = L means
for each ε > 0 , there exists δ > 0 such that
f ( ck )Δkk=1
n
∑ − L < ε whenever P < δ
Aslongasthenormofthepartitionissmallenough(normlessthandelta),itdoesn’tmatterwhatpointyouchoosefromeachsubinterval.TheRiemannsumwillbewithinepsilonofL.
Supposefisacontinuousfunctionon[ a ,b ].Theaveragevalueoffon[ a ,b ]canbecomputedintermsofadefiniteintegral.
Averagevalueoffon[ a ,b ]isequalto 1b − a
f (x)dxa
b∫
Howdowecompute f (x) dxa
b∫ ?
Wearegoingtoshowthatiffiscontinuouson[ a ,b ],then
f (x) dxa
b∫ = F(x)
b
a= F(b)− F(a) whereFisany
antiderivativeoff.
FundamentalTheoremofCalculus
I.Supposefisacontinuousfunctionon[ a ,b ].LetF (x)= f (t)dt
a
x∫ for a≤ x ≤b .ThenF '(x) = f (x) foreachx.Also
notethatF (b)= f (x)dxa
b∫ .
II.Supposefisacontinuousfunctionon[ a ,b ].LetG(x) beanyantiderivativeof f (x) .Then
f (x)dxa
b∫ = G(x)
b
a= G(b)−G(a) .
Definition
Supposefisadifferentiablefunction.Letdxbeavariablecalledthedifferentialdx.Thedifferentialdyisdy= f '(x)dx .
MethodofSubstitutiontofind f (x)dx∫
DefiniteIntegralsandAreasBetweenCurvesDefinition
Afunctionfisevenprovided f (−x) = f (x) foreachx.
Iffiseventhen f (x)dx−a
a∫ = 2 f (x)dx
0
a∫ .
Afunctionfisoddprovided f (−x) = − f (x) foreachx.
Iffisoddthen f (x)dx−a
a∫ = 0 .
AreasBetweenCurves
Findtheareaoftheregionenclosedbythecurvedefinedbyx− y3 = 0 andthelinewhoseequationis y= x .
(GraphandIntegrate)