Section 9.7 HW Problems.pdf - 9.7 Maclaurin and Taylor...

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9.7 Maclaurin and Taylor Polynomials657
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QUICK CHECK EXERCISES 9.7(See page 659 for answers.)1.Iffcan be differentiated three times at 0, then the thirdMaclaurin polynomial forfisp3(x)=.2.The third Maclaurin polynomial forf(x)=e2xisp3(x)=+x+x2+x33.Iff(2)=3,f (2)= −4, andf (2)=10, then the secondTaylor polynomial forfaboutx=2 isp2(x)=.4.The third Taylor polynomial forf(x)=x5aboutx= −1isp3(x)=+(x+1)+(x+1)2+(x+1)35.(a) If a functionfhasnth Taylor polynomialpn(x)aboutx=x0, then thenth remainderRn(x)is defined byRn(x)=.(b) Supposethatafunctionfcanbedifferentiatedfive times on an interval containingx0=2 and that|f(5)(x)| ≤20 for allxin the interval. Then the fourthremainder satisfies|R4(x)| ≤for allxin theinterval.EXERCISE SET 9.7Graphing Utility1–2In each part, find the local quadratic approximation offatx=x0, and use that approximation to find the local linearapproximation offatx0. Use a graphing utility to graphfandthe two approximations on the same screen.1.(a)f(x)=ex;x0=0(b)f(x)=cosx;x0=02.(a)f(x)=sinx;x0=π/2(b)f(x)=x;x0=13.(a) Find the local quadratic approximation ofxatx0=1.(b) Use the result obtained in part (a) to approximate1.1,

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