Grade 12 Calculus and Vectors: Unit 3 - Curve Sketching

31 increasingandDecreasing
Functions
Ext
afindtheintervalsof increasinganddecreasing for function
fansans in son tooo o nooo positivethenthefunctionmusthavemo or asymptote
tetra
fansonsonso
Oonasonso




i
o anim o un o t t t
o on out no o
n to no sign t o o
no

Increasing

c o is coin ner
Decreasing


cn einer


Exa
Qproduce a graph with thefollowing properties
fans o c2.2 is o is
nano o 2 ins n o
fans cz.si
fan s a a s




Ex3
a produce the properties that definethe followinggraph
fin o


IX
s fi sis
a

fiasco c3as aas


É a
fearone
himfino
nigrato

r


3.2maximaandminima
localextrema akaextremevaluesakaturningpoints
Alocalmaximumoccursat a point onfansifthe ycoordinatesofallotherpointsinthevicinityarelessthantheycoordinateofthepointten
famchanges frompositive to zerotonegative ietinalso to piano to finaco
or
teaislocatedattheendpointofthegraph

, r e at
oalocalminimumoccursatapointon ifthe ycoordinatesofallotherpointsinthevicinityaregreaterthanthe y coordinate ofthepoint on
fin tens

finchangesfromnegativetozerotopositiveie tineasoto pianoto finaso
or
teaislocatedattheendpointofthegraph


thecollectionoflocalmaximaandlocalminimaarecalledlocalextrema

onfunctionhasanabsolutemaximumatapointma iftheycoordinateatthatpointisgreaterthanallotherycoordinates inthedomain
foundatlocalextrema or attheendpointsofthe
interval

onfunctionhasanabsoluteminimumat apointna ifthe ycoordinateatthatpointislessthananotherycoordinates inthedomain
Foundatlocalextremaorattheendpoints oftheinterval

oa criticalnumberof afunctionoccurswhen icano orteadoes
notexist na isacriticalnumber
Ifreanothencancanis a criticalpoint

s absolute




s




absolute
minimum
É local


s
f
ExI
aFindtheabsolutemaximumandminimumoffans n asu e on theinterval c s s
s
so wemustfindfanso fisimi

L
fanssus as
osusas
fiariza
fearizz

o antainas maximumoccursat
thereforetheabsolute
n n e andis isa
i thecriticalnumbersare theabsoluteminimumoccursat meandis is
naand n e

n sandus


Ex2
aFindandclassifythecriticalpointsoffinsn'tsusout
sowemustfindthecriticalnumberssincethereisnointervalforourdomainthenwedon'thave tobe concernedabouttheendpoints
fanssuitonas
onionas