n In 3. Let's analyze the following test problem: use Newton's method to approximate a root of f(x) = sin(x), starting with an initial guess of x1 = 2. (a) (2 points) Based on what we've seen in class: (i) Do you expect Newton's method to converge to a root of sin(x)? (ii) If so, at what rate do you expect Newton's method to converge? (b) (2 points) Fill in the error column of the table. Use a driver script to generate the data! Jerror] 2.000000000000000 4.185039863261519 3 2.467893674514666 4 3.266186277569106 5 3.140943912317635 6 3.141592653680804 (c) (3 points) Determine the order of convergence a associated to this test problem, using the last three rows of table below. Show your work to get the final computation. Continue to use the driver script! 1 2 (d) (1 point) The convergence rate you found doesn't match any of the predicted convergence rates for Newton's method we studied in class. What geometric feature of the graph of f at the root x* = suggests that the convergence rate might be different from the ones we studied in class? n In 3. Let's analyze the following test problem: use Newton's method to approximate a root of f(x) = sin(x), starting with an initial guess of x1 = 2. (a) (2 points) Based on what we've seen in class: (i) Do you expect Newton's method to converge to a root of sin(x)? (ii) If so, at what rate do you expect Newton's method to converge? (b) (2 points) Fill in the error column of the table. Use a driver script to generate the data! Jerror] 2.000000000000000 4.185039863261519 3 2.467893674514666 4 3.266186277569106 5 3.140943912317635 6 3.141592653680804 (c) (3 points) Determine the order of convergence a associated to this test problem, using the last three rows of table below. Show your work to get the final computation. Continue to use the driver script! 1 2 (d) (1 point) The convergence rate you found doesn't match any of the predicted convergence rates for Newton's method we studied in class. What geometric feature of the graph of f at the root x* = suggests that the convergence rate might be different from the ones we studied in class