Newton's Method

II.  Homework

III.  Newtons Method

Consider finding the root of

y = e^x - 4x

Try as you may, there is no algebraic technique that finds this root.  We will approximate the solution as follows:

A graph shows that a solution lies between 0 and 2.

Our initial guess is x = 1.  Now draw a tangent line through (1,f(1)).  Next see where the tangent line crosses the x-axis.  The tangent line is a close approximation to the curve for nearby values, hence the x intercept of the tangent line is close to the x-intercept of the curve.  The tangent line has equation

y - f(1) = f'(1)(x - 1)

The x intercept occurs when y = 0, hence

-f(1) = f'(1)(x - 1)

solving for x,

x = 1 - f(1)/f'(1)

This x will not be the true root, but will be a better guess than x = 1.  We will use this (call it x₂)  as our second guess. Next play the same game:

x₃ =  x₂ - f(x₂)/f'(x₂)

Continue this process to get

x_n+1 =  x_n - f(x_n)/f'(x_n)

For our example this expression is

x_n - (e^x_n - 4x_n)/(e^x_n - 4)

Use the solver to do this:

x₁ = 1, x₂ = 0, x₃ =.3333, x₄ = .3572, x₅ =.3574, x₆ = .3574

We see that .3574 is the root accurate to 4 decimal places.  

Exercise:  Estimate sqrt(5) using Newton's method.  Hint:  Find the root of x² - 5.

IV.  When Newton's Method fails

1.  If our first guess  (or any guesses thereafter) is a point at which there is a horizontal tangent line, then this line will never hit the x-axis, and Newton's Method will fail to locate a root.

2.  If our guesses oscillate back and forth then Newton's method will not work.

3.  If there are two roots, we must have a firs guess near the root that we are interested in.

4.  If there are no roots, then Newton's Method will fail to find it.  (This can be frustrating when you are using your calculator to find a root.