5.TheChebyshevpolynomialsTj(x)=cos(jarccosx),j=0,1,...,areorthogonal

withrespecttotheweightfunction(1x)1/2on[1,1];thatis,Tj(x)Tk(x)(1

x)1/2dx=0,ifjk.UsetheGramSchmidtprocessappliedtothelinearlyinde-

pendentset{1,x,x}toconstructthefirstthreeorthonormalpolynomialsforthis

weightfunctionandshowthattheyareindeedscalarmultiplesofT0,T1,andT2.

5- The Chebyshev pelynemials "IS-[1} = ces arcces x},_j = 6,1,- _ _ , are erthegenal with respect tn the weight functien [I sir\": en [1, I]; that is, I]. lellklxl [I stillI'll2 d: = D, ifj a: l:_ Use the CramSchmidt precess applied tn the linearlyI inde- pendent set {1, 1', 1:1]- te censtruct the rst three pdhenprmsl pelynemials Fer this weight functien and shew that they are indeed scalar multiples of T6' T1' and T2