Exercise . G(x)-22/3. Find the Taylor polynomial of order 4, centered at x=8, for This is a lengthy process. We break it into steps (A) We chose to center the polynomial at 8 because every coefficient is rational. This property follows from a Precalculus interpretation of fractional powers. Recall that . For n a counting number and b0, b/n is the positive n-th root of b, and . For p, q non-zero integers and b > 0, bp/q = (bi/ For review, express each of the following as a rational number: 81/3, 84/8 and 84/3 (B) Find the first four derivatives of G(x). (C) Evaluate G(x) and each of the derivatives at 8. Each value is rational. (D) Enter the Taylor polynomial.
g(x)=x^2/3 g(8)=4
g\'(x)=2/3.x^(-1/3) g\'(8)=1/3
g\'\'(x)=-2/9.x^(-4/3) g\'\'(8)=-1/72
g\'\'\'(x)=8/27.x^(-7/3) g\'\'\'(8)=1/432
g\'\'\'\'(x)=-56/81.x^(-10/3) g\'\'\'\'(8)=-7/10368
the taylor polynomial of degree 4 centered at x=8 is
4+1/3(x-8)-1/144(x-8)^2+1/2592(x-8)^3--7/248832(x-8)^4
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