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. Our aim is to start with an initial guess and get closer to the point where the graph crosses the x-axis.
The results of applying Newton's Method appear underneath the graph. to the right of the graph. When we can't go any further (there is no change to the accuracy of the root), we stop.
You can choose 4 different functions, and choose your own starting value for the algorithm.
The heights of the vertical dashed lines are the function values at each point, and the solid segments are tangents to the curve at each point.
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. In fact, this one very slowly converges (which means it keeps trying to find the root but never quite gets there in a reasonable number of steps).
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The above animation shows what's really going on in Newton's Method.
Starting at the first "guess" on the `x`-axis, we go upwards until we hit the curve. The `y`-value here is the function value (and we find it's not zero, yet.)
We then travel down the tangent line back to the `x`-axis. This is our `x_1` value. We go up again to the curve to find the function value, and usually (we hope) 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: