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Evenfunctions

Letf(x)bearealvaluedfunctionofarealvariable.Thenfisevenifthefollowingequationholdsforallxinthe

domainof

f:

Geometrically,thegraphofanevenfunctionissymmetricwithrespecttotheyaxis,meaningthatitsgraph

remainsunchangedafterreflectionabouttheyaxis.Examplesofevenfunctionsare|x|,x2,x

4,cos(x),andcosh(x).

Oddfunctions

(x)=x3isanexampleofanoddfunction.Again,letf(x)bearealvaluedfunctionofarealvariable.Thenfisoddif

thefollowingequationholdsforallxinthedomainoff:

Geometrically,thegraphofanoddfunctionhasrotationalsymmetrywithrespecttotheorigin,meaningthatits

graphremainsunchangedafterrotationof180degreesabouttheorigin.Examplesofoddfunctionsarex,x3,

sin(x),sinh(x),anderf(x).

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Basicproperties

Thesumoftwoevenfunctionsiseven,andanyconstantmultipleofanevenfunctioniseven. Thesumoftwooddfunctionsisodd,andanyconstantmultipleofanoddfunctionisodd. Theproductoftwoevenfunctionsisanevenfunction. Theproductoftwooddfunctionsisanevenfunction. Theproductofanevenfunctionandanoddfunctionisanoddfunction. Thequotientoftwoevenfunctionsisanevenfunction. Thequotientoftwooddfunctionsisanevenfunction. Thequotientofanevenfunctionandanoddfunctionisanoddfunction. Thederivativeofanevenfunctionisodd. Thederivativeofanoddfunctioniseven. Thecompositionoftwoevenfunctionsiseven,andthecompositionoftwooddfunctionsisodd. Thecompositionofanevenfunctionandanoddfunctioniseven. Thecompositionofanyfunctionwithanevenfunctioniseven(butnotviceversa). TheintegralofanoddfunctionfromAto+Aiszero(whereAisfinite,andthefunctionhasnovertical

asymptotesbetweenAandA).

TheintegralofanevenfunctionfromAto+Aistwicetheintegralfrom0to+A(whereAisfinite,andthefunctionhasnoverticalasymptotesbetweenAandA).

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PerformingIntegralandPlottingusingsymbolictoolboxinMATLAB

syms xy

y=cos(x)+5*cos(2*x)

% how t o make i nt egr al i n mat l ab% not e t hat i nt egr al wi l l be f or y wi t h r espect t o x

int(y,-pi,pi)

%i f you want t o pl ot a symbol i c equat i on

ezplot(x/pi,y,[-3*pi 3*pi]);hold on;

ezplot(x/pi,cos(x),[-3*pi 3*pi]);

hold on;

ezplot(x/pi,5*cos(2*x),[-3*pi 3*pi]);

grid