Exercise . G(x)-22/3. Find the Taylor polynomial of order 4, centered at x=8, fo

Question

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.

Explanation / Answer

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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