MATH 1300 Lecture Notes

Variables,functionsandgraphsopenandclosedintervalnotation•closedintervala←✗Eb,[a,b],bothendpointsincluded•openintervala<✗<b,la,b),bothendpointsexcludedset-BuilderNotation{✗ES:...}{×:a_<✗<b}=[aib]target→whatyouintendtheoutputtobeDefinitionbycases→functionsthataredefinedbyagroupofequationsBoundvariablesoccurin:•thedefinitionofafunction•thedefinitionofasubset•limits•thedefinitionofaderivative•summations•definitionofanintegral•notationsofordersofmagnitude&asymptoticequivalence•Taylor'stheoremGreatestIntegerfunction→greatestinteger"y"suchthatyEx•LXJ→floorfunction•if✗ispositivetonotaninteger,thenLxtisthewholenumberthatyougetwhenyouwrite"x"asadecimalandignorewhat'safterthedecimalpointLeastIntegerfunction→leastinteger"y"suchthaty>_✗•5×7→ceilingfunction••if✗ispositivetonotaninteger,then5×7isthewholenumberthatyougetwhenyouroundupsignfunction•putsoutthesignofthefunction•sgna)={!¥!0✗=0 Variables,functions,4graphscont.detafunction,×>0•81×1=1when✗=040when✗=/⑦=81×1=§✗=0•testforwhateverisaddedorsubtractedfrom✗•parentfunctionalwaysgoesupby1Shifting/Stretching•y=fad-1C-shiftsthegraphupifcispositive-shiftsthegraphdownifcisnegative•y=flx+c)-shiftsthegraphtotherightifcisnegative-shiftsthegraphtotheleftifcispositive•y--cflx)-stretchesverticallywhen/¢>I-shrinksverticallywhen144-flipsgraphoverthex-axiswhencisnegative•y=flex)-stretcheshorizontallywhen14<1-shrinkshorizontallywhen14>1-flipsgraphoverthey-axiswhencisnegativeOddv5.Evenfunctionftx)=-fad:txP=-1×3):c-xp=x2•passesfromnegativetopositivewhenitisodd•signjustdisappearswhenitiseven:=✗2•signcomesoutsidewhenitisodd:1-XP=-(✗3)-whenyouaddaconstant,itbecomesneither