Newton Method
Derivation of the Method
If you want to find an value where a function, and the derivative can be calculated, Newton's Method is a good approach. This applet graphically steps through what is happening with the method.
Step 1 : The function where we want to find the root. In this case the root is
Step 2 : Solve for the derivative
Step 3 : Choose an value to start. Here we chose . This is an art with Newton's method.
Step 4 : Evaluate the function and the derivative at the value. Draw the point .
Step 5 : The length of the segment shown is . Note: it may be negative so it is not really a length.
Step 6 : Using the derivative, draw the tangent line. We will approximate the value of by finding where this line is .
Step 7 : The length of the red vector can be calculated from the lines slope.
The slope of the line, , is the rise over the run or . Solving for the red vector gives
Step 8 : An alternate method is to find the point on the line, using the slope of the line through two points. . Solving for gives the Newton method. This is the same as adding the red vector to the initial .
Step 9: Looking at the graph of the function it can be seen that the new values is close to the desired x value where .
Step 10: Repeat the operation using this new value to get a better guess of the desired value.
Illustration of the Newton Method
In this applet you can enter a function in the entry box and provide an initial guess of by moving the point on the x-axis.
Newton Method for Finding Roots of a Function
Activities
For the following functions experiment with different values of the initial guess.
= x^2 - 2
= 1/2 + sin(x)
= x - cos(x)
= 1/2 - nroot(x,3)
If[ x< 0 , sqrt( -x) , sqrt(x)]