1. Consider the function f(x) = 4x, for 0

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Extracted text: 1. Consider the function f(x) = 4x, for 0<><1 2,="" and="" f(x)="2(1-x)," for=""><><1. express="" f(x)="" in="" terms="" of="" the="" fourier="" sine="" series,="" i.e.,="" f'="" (s)="" -="" σ.="" (sin="" πx).="" n="1" for="" n="1,2" and="" 3,="" find="" f"="" (x).="" show="" f(x).="" f'="" (x),="" f²="" (x)="" and="" f="" (x)="" on="" the="" same="" plot.="" for="" n="1," 2="" and="" 3,="" find="" (="" s,="" (f(x)="" –="" f"="" (x))2="" dx)/="" ,f="" (x))2="" dx).="" what,="" if="" anything,="" can="" we="" conclude="" from="" the="" approximations="" f="" (x).="" f*="" (x)="" and="" f="" (x)="" of="" f(x)?="" 2.="" is="" it="" possible="" to="" write="" sin="" (37x)="" as="" a="" linear="" combination="" of="" sin(2nx)="" and="" sin(tx)="" for="" all="" values="" of="" xe0,1?="" prove="" your="" answer.="" 3.="" for="" xe0,1.="" express="" sin="" 7x="" a+="" br+="" cx="" +="" ex="" find="" values="" of="" a,="" b,="" c="" and="" e="" by="" ensuring="" that="" the="" error="" (or="" the="" residual,="" r)="" defined="" by="" r(x)="-sin" 7x+="" a+bx+cx="" +ex'="" equals="" zero="" at="" x="0,">