In this assignment we compute derivatives using a variety of differencing formulas. We have see the...

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In this assignment we compute derivatives using a variety of differencing formulas. We have see the three-point endpoint formula for approximating f'(x): 1 '(x) [3f(x) + 4f (x + h) f(x + 2h)] 2h and the three-point midpoint formula for approximating f'(x): f'(x0) [f(xo+h) f(x h)]. 2h We have also seen the three-point midpoint formula for approximating f"(xo): "(x) 12 [(x h) 2f(x) + f (x + h)]. h2 There is also a three-point endpoint formula for approximating f'(x): f'(x0) ~ h2 - [f(x) 2f (x + h) +(x+2h)]. Continued on page 2 Part A: Suppose f(x) = e* 1/2. Using the most accurate formula above and points x = 0.25, 0.5, 0.75, 1.0, compute approximations of f'(0.5), f'(1.0), "(0.5) and f'"(1.0). What are the absolute errors in your approximations? State which formula you prefer in each case. Part B: Suppose f(x) = ex - 1/2 for x = [0,1]. In Matlab, approximate '(0.5), '(1.0), "(0.5) and "(1.0) using equal node spacings h=2m, m = 2,3,4,.... and the best formula presented, using only points on the interval [0,1]. Tabulate or form a plot of the absolute errors as a function of h. Using big-oh notation describe the experimentally observed error in your approximations as a function of h. Your answer here should be of the form O(hP): what is p? Explain how you find p. In this assignment we compute derivatives using a variety of differencing formulas. We have see the three-point endpoint formula for approximating f'(x): 1 '(x) [3f(x) + 4f (x + h) f(x + 2h)] 2h and the three-point midpoint formula for approximating f'(x): f'(x0) [f(xo+h) f(x h)]. 2h We have also seen the three-point midpoint formula for approximating f"(xo): "(x) 12 [(x h) 2f(x) + f (x + h)]. h2 There is also a three-point endpoint formula for approximating f'(x): f'(x0) ~ h2 - [f(x) 2f (x + h) +(x+2h)]. Continued on page 2 Part A: Suppose f(x) = e* 1/2. Using the most accurate formula above and points x = 0.25, 0.5, 0.75, 1.0, compute approximations of f'(0.5), f'(1.0), "(0.5) and f'"(1.0). What are the absolute errors in your approximations? State which formula you prefer in each case. Part B: Suppose f(x) = ex - 1/2 for x = [0,1]. In Matlab, approximate '(0.5), '(1.0), "(0.5) and "(1.0) using equal node spacings h=2m, m = 2,3,4,.... and the best formula presented, using only points on the interval [0,1]. Tabulate or form a plot of the absolute errors as a function of h. Using big-oh notation describe the experimentally observed error in your approximations as a function of h. Your answer here should be of the form O(hP): what is p? Explain how you find p.