Instructions {1] This assignment is optional and is due on March 16, 2013 in class. You mayr also turn it in during oioe hours or in class before the last day {March 15]. {2} You may complete the following problems to improve your score in the quiz category (up to II] additional points). {3] You must justify your answers and show your work to receive an}.r credit. Drawing a graph or saying \"By Calculator\" does not count as showing your work. {4} Please write your solutions neatly and in an organized way, so that a grader can read it and understand your solution {4} {5} A Taylor Polynomial Problem (5 points) The motion of a pendulum bob with mass m is governed by the differential equation mm\" + mgaine = o where L is the length of the pendulum arm, 9 is the acceleration due to gravity, and 9, in radians, is the angle between the pendulum arm and the vertical. {Forces other than gravity have been neglected.) At time t = ll, the pendulum bob is set in motion with 3(0) = 1 and 9(0) = 3. Assume that L = 16 ft and g = 32 ftla'sec2. Find the second degree Taylor poly nomial that approximates the angular position 33} of the bob near t = ll. The 'I'aylor polynomial provides an approximation to the solution of the nonlinear di'erential equation for the pendulum. In the previous problem, nd the third order of Taylor polynomial that approximates 3(t) for t near 0