Where Is Newton Taking Us? And How Fast?

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1 Name: Where Is Newton Taking Us? And How Fast? In this activity, you ll use a computer applet to investigate patterns in the way the approximations of Newton s Methods settle down to a solution of the equation x 3 x = 0. To begin with, here is a picture of f(x) = x 3 x. 1. What are the solutions to x 3 x = 0? 2. What is the formula for x n+1 that Newton s Method would use in approximating solutions to this equation? Remember, Newton s Method takes an initial guess x 0 for the solution to f(x) = 0 and tries to continually improve it by using tangent lines to compute closer and closer approximations to the equation s true solution. But how does your first guess x 0 affect what happens in Newton s Method? Let s find out. Open the applet linked to on the Newton s Method activity webpage. You should see something like this:

2 The applet shows a graph of the function f(x) = x 3 x. Across the top there is a slider; this determines the starting value x 0 for Newton s Method. Dashed and dotted line segments illustrate the way in which Newton s Method finds the first 5 approximations x 1, x 2, x 3, x 4, x 5 after x 0 of a solution to f(x) = 0. These approximations are illustrated by dots on the x-axis and by the list of their values shown in the lower left corner of the applet. Use the applet to answer the following questions. 3. Where does the graph f(x) have a horizontal tangent? Type x 0 = 1/sqrt(3) into the Input box at the bottom of the applet. What happens, and why? (Explain your answer both in terms of the graph of f(x) and in terms of the Newton s Method formula for x n+1 we discussed in class.) 4. Say you chose x 0 be a number that is less than the negative root of f(x). Which root of f(x) would Newton s Method converge to? Can you explain why that is? 5. What if x 0 is a number greater than the positive root of f(x)? Which root does Newton s Method converge to, and why? 6. If you choose x 0 = 1/2, where s x 1? What does Newton s method converge to, in this case?

3 7. If you choose x 0 to be something between 1 and 0.6, what root of f(x) does Newton s method converge to? 8. Use the applet to decide which root Newton s method converges to for each of the following x 0 s: x 0 Root of f(x) x 0 Root of f(x) Let s do a math-related art project. On the last page of this activity you ll find a number line. In the applet, move the slider for x 0 around, and as you do that, color the number line with light and dark colors according to the following rules for what the applet shows the values of the x i s as being: If Newton s Method eventually converges to 1, but x 5 doesn t equal 1, then on the number line color the location of x 0 with dark red. If x 5 equals 1 but x 4 doesn t, then color the location of x 0 with red. If x 4 equals 1 but x 3 doesn t, then color the location of x 0 with light red. If x 3 equals 1, then leave the location of x 0 white. If x 5 equals 0 but x 4 doesn t, then color the location of x 0 with green. If x 4 equals 0 but x 3 doesn t, then color the location of x 0 with light green. If x 3 equals 0, then leave the location of x 0 white. If Newton s Method eventually converges to 1, but x 5 doesn t equal 1, then on the number line color the location of x 0 with dark blue. If x 5 equals 1 but x 4 doesn t, then color the location of x 0 with blue. If x 4 equals 1 but x 3 doesn t, then color the location of x 0 with light blue. If x 3 equals 1, then leave the location of x 0 white. So, for example, if you place the slider for x 0 anywhere to the right of 2.11, you can see that Newton s Method will eventually converge to the root 1, though the value of x 5 is bigger than 1. Therefore, the rules say that you should color all the points on the number line to the right of 2.11 with dark red. If x 0 is between 1.53 and 2.11, though, then the applet shows x 5 as equalling 1, so you should color the interval [1.53,2.11] with red. Using the applet and following the rules listed, try to color the entire number line. Once you ve colored the number line, notice that the darker the color is at a spot on the number Notice that the applet rounds the values of x 1 through x 5 to 2 decimal places. Your coloring doesn t have to be exact, but try to do a good job.

4 line, the longer Newton s method takes to converge to a root of x 3 x. Therefore, the right choice of x 0 can really speed Newton s Method up, while the wrong choice can slow it down. Also notice that your number line ends up with some jumbled colors in a couple of places (as you might have guessed, looking at your answer to Problem 7). Things get even messier (and more beautiful) if we venture into the realm of the imaginary numbers you learned about in algebra. It turns out that if you let x 0 be a complex number (one with both a real and an imaginary part) and use the Newton s Method formula you wrote down in Problem 2, then x 1, x 2, and so on will all be complex numbers, too, but they ll still (usually) approach one of the numbers 1, 0, and 1. (If you have a calculator or computer that can do arithmetic with complex numbers, go ahead and see for yourself; otherwise, take my word for it.) The complex plane is a coordinate system in which the x-coordinate of a complex number is its real part and the y-coordinate is the number s imaginary part; thus, the complex number 2 + 3i lies at the point (2,3) in the complex plane. If we color the complex plane with red, green, and blue, following rules similar to those you followed in Problem 9, then we get a picture that looks like this: The coloring you did in Problem 8 should roughly agree with the coloring the picture shows around the horizontal axis. In particular, the picture indicates that near x = 0.5 the colors get mixed together there is a red bulb attached to the blue region. If we zoom in on this a bit, we get the following picture:

5 Notice that there s a blue bulb at the end of this red bulb, and if you look closely at the tip of that blue bulb, there s a red one attached to that as well. These colored bulbs alternate forever, getting smaller and smaller and smaller... This repeating geometric shape formed by the colored regions is what s called a fractal. These shapes are often very beautiful, and they appear often in nature, video games, and math research today. Try doing a web search for fractals and see what you can find out. To find an online applet that generates the fractal we looked at above, visit djoyce/newton/newtongen.html. To generate the first fractal above, do the following: (1) enter 3 into the first box (which asks the number of roots); (2) fill out the second table like this: and (3) fill out the bottom table like this:

6 Now for some final challenges... Challenge #1. There s a certain value of x 0 near 0.4 at which Newton s Method just bounces us back and forth: x 2 is the same as x 0, x 3 is the same as x 1, and so on, as shown below: We can use the applet to find out about where this value of x 0 is, but where is it exactly? Challenge #2. Suppose your friend asks you how to use Newton s method to find solutions to sin x = 0, but you, with your perverse sense of humor, decide to give him an initial guess x 0 chosen carefully so that x 1 is exactly 2π units later (so x 2 will be 2π units after that, and the sequence of x n s will never settle down to a solution). What value(s) could x 0 have? (This problem has a messy numerical answer. For the ultimate in brownie points, use Newton s Method to approximate it to 5 decimal places of accuracy.)

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