Using the bisection method, find the positive root of 2x(1 + x^2)^-1 = arctan x. Using
this root as x0; apply Newton's method to the function f(x) = arctan x: Interpret
the results you obtain.
This shows how to use the bisection method to find a root and apply Newton's method to the function.
Rootfinding and Optimization : Newton's, Secant, Bisection, False Position Methods
3. (Rootfinding and Optimization)
(a) Suppose that f is differentiable on [a, b]. Discuss how you might use a rootfinding method to identify a local extremum of f inside [a, b].
(b) Let f(x) = logx ? cosx. Prove that f has a unique maximum in the interval [3,4]. (NB: log means natural logarithm.)
(c) Approximate this local maximum using six iterations of the enclosure methods (Bisection and False Position) with starting interval [3, 4].
(d) Approximate this local maximum using six iterations of the two fixed-point methods (Secant and Newton). For Newton's Method, use = 3. For the Secant Method, use P0 = 4 and p' = 3.
(e) What is your best estimate for p. the location of the maximum?
(f) Provide the following two tables, comparing the four algorithms. The headin for the two tables should be the following.
(g) Plot the absolute error for all four methods on the same graph.
(h) What happens if you attempt to approximate the maximum by starting Newton's Method with p = 5?
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