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)