2413labI8 - Laboratory I.8 Newton's Method Goals The...

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1 Laboratory I.8 Newton’s Method Goals: • The student will learn how to solve equations using Newton’s Method. • The student will get an introduction to iteration and dynamical systems. Before the Lab: Newton’s Method is a technique which uses derivatives to solve equations that you can’t solve by hand (algebraically). An example is cos x = x . There are no algebraic tools which allow you to isolate x on only one side of that equation. We’ll have you solve that particular equation in the lab; we’re going to work as our example a simpler equation that we can solve by hand so that we know what we’re doing. We’ll do x 2 = 2, and in particular we’ll pretend we don’t know the positive solution, x = 2 . Solving x 2 = 2 is the same as solving x 2 – 2 = 0, so let’s graph y = x 2 – 2 and look where it crosses the x -axis: You can see the solution is just to the right of x = 1.4. Newton’s Method works by taking an initial guess and improving it. Let’s use as our initial guess x 0 = 2. We’re going to add a tangent line to y = x 2 – 2 to the picture: The tangent line hits the x -axis at x = 1.5. Notice that this is much closer to the exact solution than our original guess, x 0 = 2. This, in essence, is Newton’s Method: the tangent line to the curve at a point will hit the x -axis closer to the exact root than the starting point. Let’s check this with another picture. Our original guess of x 0 = 2 was “improved” to a guess of x 1 = 1.5. Let’s draw the tangent line to y = x 2 – 2 at x = 1.5 and see what we get:
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2 The new tangent line is so close to the function that we can hardly tell them apart. The intersection of the new tangent line with the x- axis is at x 2 = 1.4167; the exact value of the solution is
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2413labI8 - Laboratory I.8 Newton's Method Goals The...

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