ERROR ESTIMATES IN TAYLOR APPROXIMATIONS

is the interval jx 9j 0:5.) We use Taylor’s inequality with n = 1. We have f0(x) = 1 2 x 1=2 and f00(x) = 1 4 x 3=2 = 1 4x3=2. We first need an upper bound M for jf00(x)j= 1 4x3=2 on [8.5,9.5]. The largest value occurs when the denominator is the smallest, so the actual maximum is 1 4(8:5)3=2. This is our best choice of M but is rather ugly. ................
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