how would you overcome the main problem in newtons methodSol

Solution

Sometimes we are presented with a problem which cannot be solved by simple algebraic means. For instance, if we needed to find the roots of the polynomial , we would find that the tried and true techniques just wouldn\'t work. However, we will see that calculus gives us a way of finding approximate solutions. A Simple Example Let\'s start by computing . Of course, this is easy if you have a calculator, but it is a simple example which will illustrate a more general method. First, we\'ll think about the problem in a slightly different way. We are looking for which is a solution of the equation . The problem is that it is difficult to generate a numerical solution to this equation. But remember in the section on approximations , we saw how to approximate a function near a given point by its tangent line. The idea here will be to actually solve the approximate equation which is easy since it is a linear one. If we think for a minute, we know that is between 2 and 3 so let\'s just choose to use the linear approximation at . We know that so that . The linear approximation is then Notice that the linear equation is easy to solve. We will then approximate the solution to by the solution to which is . If you have a look on a calculator, you will see that . So you can see that we have found a fairly good approximation. We can understand what we have done graphically. We are looking for a solution to which is where the graph of crosses the x-axis. We approximate that point by the point where the tangent line crosses the x-axis.