Using graphing software or g(x) = 0.1250.4x4 - 0.353 -1.852 +1.05 2.25 now consider the following possible behaviours of newton's method: converging quickly to a solution, getting faster as it goes. converging more slowly to a solution, with error that goes down by about the same factor each time. bouncing around a lot and perhaps reaching a solution eventually looking at the graph, try to predict what will happen if you use newton's method on g(z) above with each of the following start points: (a) ro 1 = (b) ro-0.5 (c) o -2.5 check your answers by applying newton's method and seeing what happens. (you don't need to hand in your entire working for the checking process -just run the method quickly and note whether the qualitative behaviour is what you expected, or not! )

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?