To find the roots of equations like
Let's do an example.
To use Newton's method, you write the equation in the form
Because the method requires that we do the same computation many times, until it converges, I recommend that you use an Excel spreadsheet; the rest of my answer will contain instructions on how to do this.
Enter a good guess for x into cell A1. For this equation, I will enter 2.
Enter the following into cell A2:
=A1-(EXP(A1) - A1^3)/(EXP(A1) - 3*A1^2)
Please notice that the above is Excel spreadsheet language for
Copy the contents of cell A2 into A3 through A10. After only 3 or 4 recursions, you can see that the method has converged on
We can use the Intermediate Value Theorem to see that each pair has at least one point of intersection.