Newton's Method Interactive Graph
by M. Bourne
The graph below allows you to explore the concept of Newton's Method for finding the roots of equations.
The results of applying Newton's Method appear to the right of the graph.
You can choose 4 different functions, and choose your own starting value for the algorithm.
Things to do
First function: `f(x)=2x^2-x-2`
Try different positive and negative starting values.
Use the "minus" and "plus" buttons next to the slider to nudge close to the local minimum at `x=0.25`.
Second function: `f(x)=1-x^2+2^x`
Once again, try different values for the starting point.
We don't always get the closest root. Try starting near `x=1`.
Third function: `f(x)=3x^3-9x^2+5x+2`
A cubic will generally have one local maximum and one local minimum. Investigate what happens if we choose values near those points.
Fourth function: `f(x)=x^2`
In this case, we have a root which is also the minimum. Newton's Method works best when the slope is a reasonably high value near the root. But in this case, we can see that even after 12 steps, we are not very close to the root.
Change start value:
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The animation shows what's really going on in Newton's Method. Starting at the first "guess", we find the value of the function (which is not zero.) We then travel down the tangent line back to the `x`-axis. This is our `x_1` value. We find the function value, usually it is closer to 0. We repeat the process until we get acceptably close to the root (usually by looking at whether the decimal places in the root are still changing.)
The examples used in this graph applet were first introduced on the preceding page: