The evaluation of 1+cos (z-r)2 0.5 is approximately equal to (radians). For is inaccurate in floating-point arithmetic when example, if 3.16, then flg(316)) = 0.5908 using 4 decimal digit, idealized, chopping floating-point arithmetic. Note that the correct value of g(3.16) is 0.499985, so this computed approximation has a relative error of ap- proximately 18% The fourth order (n = 4) Taylor polynomial approximation for f(z) = cosx expanded about a = is 24 (a) Substitute the above Taylor polynomial approximation for cos r into the formula for g(x), and simplify in order to obtain a polynomial approximation for g(x) when x (This polynomial approximates g(z) very well when r is close to since the Taylor polynomial approximation is very accurat e when is close to .) (b) Show that the above floating-point computation of g(3.16) is unstable. Use the notation and the definition of stability given in Handout 7 to show this. Hint: consider a perturbation of the data x 3.1G+ , where 13 1 is small. Use the polynomial approximation to g(z) in (a) to determine a very accurate approximation to the exact value of r = 1 + cos and show that for all small values of , the exact (x-)2 , value of r is not close to the computed floating-point approximation of 0.5908 (c) If x = 1.41 (radians), then A(g( 1.41)) = 0.3871 using 4 decimal digit, idealized, c floating-point computation is stable hopping floating-point arithmetic. Show that this