(Solved): EXAMPLE 1 (a) Approximate the function f(x)=x by a Taylor polynomial of degree 2 at a = 64. (b) ...

EXAMPLE 1 (a) Approximate the function f(x)=?x by a Taylor polynomial of degree 2 at a = 64. (b) How accurate is this approximation when 63 ? x ? 65? SOLUTION (a) f(x) = ³??x = x¹/3 f'(x) = F"(x) f""'(x) = 1,- ( ²3 ) 3x -x-5/3 10 - (-/-) 27? and so = 4 The desired approximation is Thus the second-degree Taylor polynomial is f'(64) T?(x) = f(64) + (x64) + 1! + 48 f(64) = 4 |R?(x)| ? 5|x - 6413 8178/3 f'(64) = f"(64) = 5 ??x ? T?(x) = 4+ (x ? 64) - - - 48 f"(64) 1 4800 -(x - 64)² (x64) 1/9216 ? (x-64)². (b) The Taylor series is not alternating when x < 64, so we can't use the Alternating Series Estimation Theorem in this example. But using Taylor's Formula we can write f"""(z) R?(x) = (x - 64)³ = 3! 9216 ( x ? -64)² where z lies between 64 and x. In order to estimate the error we note that if 63 ? x ? 65 then -1 ? x 64 ? 1 so |x - 64| ? 1 and therefore Ix - 641³ ? 1. Also, since z ? 63, we have 28/3638/3> 62840 Z-8/3 (x-64)3 = 3! 5(x - 64)³ 81z8/3 62840 < 0.0000010 Thus, if 63 ? x ? 65, the approximation in part (a) is accurate to within (rounded to seven decimal places).