3. Rootnding [30 marks]. (a) Consider the equation :13eE : 1/5 . i. [10 marks] Graph 3; : ac and y : exp(a:) / 5 and estimate the value of the roots of f (:3) : 0 . (The original equation can be transformed into at) : U in many ways.) ii. [10 marks] Use Newton's method to compute the roots accurately. Test your nal answers and explain how you know they are right. (b) [10 marks] Consider the similar equation are\":+1 : 71 . This has a root at .7: : 1 . If we didn't notice that, and used an initial guess (to : 0.9, would Newton's method converge to the root at a: : *1? If so, would it converge quadratically fast or only linearly fast? Justify your answer. f (an) f'($n) Newton's Method: mn+1 : mn *