Show that if xn+1 is the x-intercept of the tangent to f at (xn, f(xn)), then Xn+1 Xn- f(xn) f'(xn) ✓ provided that f'(xn) 0. (1) 2. Let k be the final digit in your student number and let y = 3k + 2. The purpose of this question is to use Newton's method to obtain an approximation to √√7, so let f(x) = x² - y. (a) Rewrite the equation f(x) = 0 to express x in terms of y. (b) Let the initial guess x0 = k + 1. Use this value of xo and equation (1) to calculate x1 and x2. (c) Compare your value of x2 to the value you get for √ by using a calculator. Without changing the initial guess, how could you use Newton's Method to obtain a better approximation? [3 marks]

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