Refer to the three approximations derived for each of the four functions in Exercise 26. For each function use \(\mathrm{R}\) to construct a line plot of the error terms \(E_{2}(x, \delta)\) and \(E_{3}(x, \delta)\) relative to the error term \(E_{1}(x, \delta)\). That is, for each function, plot \(E_{2}(x, \delta) / E_{1}(x, \delta)\) and \(E_{3}(x, \delta) / E_{1}(x, \delta)\) versus \(\delta\). The lines corresponding to each relative error function should be different. What do these plots suggest about the relative error rates?

Exercise 26

Use Theorem 1.13 with \(p=1,2\) and 3 to find approximations for each of the functions listed below for small values of \(\delta\).

a. \(f(\delta)=1 /(1+\delta)\)

b. \(f(\delta)=\sin ^{2}(\pi / 4+\delta)\)

c. \(f(\delta)=\log (1+\delta)\)

d. \(f(\delta)=(1+\delta) /(1-\delta)\)

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P Theorem 1.13 (Taylor). Let f be a function that has p+1 bounded and continuous derivatives in the interval (x, x+6). Then f(x+8)= k=0 Sk f(k) (x) + Ep(x, 8), k! where 1 Ep(x, 8) = (x+8) f(p+1)(t)dt: = p! SP+1 f(p+1)() (p+1)! for some = (x, x + d).