Using the information in the picture,  a) find all 'a' values in which Newtons Method convereges lineraly or quadratically b) find all 'a' values in which Newtons Method diverges. This is an unsolved example from lecture

Certainly, let's work through both parts of your question step by step.

a) To determine the values of 'a' for which Newton's Method converges either linearly or quadratically, we need to consider the behavior of the method. Newton's Method involves iterative steps using the formula:

where .

1. Linear Convergence:
   For linear convergence, we need at the root where the method is converging. In other words, the derivative must be less than 1 in absolute value. So, let's find :

   For linear convergence, . Since is a positive number (starting with ), this condition simplifies to:

   So, for linear convergence, must satisfy .

2. Quadratic Convergence:
   For quadratic convergence, we need at the root where the method is converging. In other words, the derivative must be equal to 0. So, we have:

   For quadratic convergence, can be any real number.

So, for part (a):
- Newton's Method converges linearly for .
- Newton's Method converges quadratically for all real values of .