6. Newton-Raphson method Very often, we will encounter the equation flit?) =0, which cannot be solved analytically. To solve it numerically, we may apply Newton Raphson method. The idea is simple and elegant. Consider the following diagram: (tinflall) Tangent of f (3:) at. :1: = 0.1 Initial guess /I., : r r I J," Better approximation root : (12 .r I ! Suppose we start with an initial guess it: = a1. Then, (12, the :r-intercept of the tangent of x) at 0,], can be computed by the following equation {provided that f'lall 79 0)? f'(a1) = slope of tangent of at} at a1 = w 0.1 0.2 _ Hall a? a1 101(11)- Obviously, we can apply this process as many times as necessary: _ an) an+1 an _ fwd\")- The sequence {on} given by this method will usually converge to the root 1", provided that the initial guess a1 is close enough to r. (a) By applying the Intermediate Value Theorem, show that equation 82\" = .7: + 6 has at least one root in the interval (0, 1). (b) State the Newton-Raphson method for solving 82\": = a: + 6. (c) By applying the Newton-Raphson method with initial guess a] = 1, approxi- mate the root with three iterations {correct to 4 decimal places)