NEWTON'S METHOD This slide show consists of graphs of various demonstrations of Newton's Method. When viewing the slides, the following keys are operational: HOME takes you to the first slide in the sequence you selected END takes you to the last slide in the sequence you selected UP ARROW takes you to the previous slide in the sequence you selected F9 immediately quit the program These keys do NOT operate like that while you are reading this document. A. Method - Example 1 2 This shows how the method would work on f(x) = x , starting from x = -2, and looking for x = 0. First f(x) is drawn, then the vertical line x = -2 is added, then the tangent line to f(x) at x = -2 is drawn, and so on. B. Method - Example 2 This applies the method to f(x) = sin(x), looking for x = 0, and starting with x = 11ã/32. C. Problem - Example 1 This is the same as B, except the initial value is taken as ã/2 - .02. A solution will be found, (at x = -ã), but not the one that was sought (x = 0). D. Problem - Example 2 This is the same as B, except the initial value is taken as ã/2. A solution is not found because the tangent line is horizontal, and never crosses the x-axis. Consequently there is no second estimate for x. E. Problem - Example 3 3 This applies the method to f(x) = x - x , looking for x = 0, and starting with x = 1/û5. No solution is found because the estimates oscillate. However, if a better starting value is used, the desired root can be found. F. Problem - Example 4 This applies the method to f(x) = û³x³, looking for x = 0, and starting with x = 1. No solution is found because the estimates oscillate. However, even if a different starting value is used, the desired root can never be found. G. Problem - Example 5 This applies the method to f(x) = x exp(-x), looking for x = 0, and starting with x = 2. No solution is found because the estimates diverge from x = 0. However, if a different starting value is used, the desired root can be found. H. Problem - Example 6 This deals with the quartic function 4 3 2 f(x) = x - 20x - 25x + 500x + 1 . It is easily shown that f(-4) = -863 and f(4) = 577, so there is a root in the interval (-4, 4). The purpose of this example is to first show the numerical values associated with the starting choices -4, 4, -3, 3, -2, and 2, and then give a graphical interpretation of these choices. This should indicate why it is always sensible to sketch the function before using Newton's method. When you have finished reading this document, press Q to quit.