AP CALCULUS NOTES
SECTION 5.7 NEWTON’S METHOD
From algebra, we have the ability to solve a linear as well as a quadratic function. implies
There are also formulas, albeit complicated ones, for finding roots of 3rd degree and 4th degree equations, but no such formulas exist for finding roots of polynomials of degree 5 or higher or transcendental equations. For such equations, solutions can be approximated using Newton’s Method.
A.) Newton’s Method: is an application of the idea of linear approximation that is used by most calculators to find roots. 1.) Obtain an initial rough estimate, , using your calculator. 2.) If , then is a root. If , then we find the tangent line at and solve for its root (call it ).
3.) If , then is a root. If , then we continue the process of finding a tangent line at and finding its root.
With each succession of values our improved approximation is given by
Ex.1.) Starting with , find the 3rd approximation to the root of the equation .
With Newton’s Method, the approximations converge toward the desired root pretty quickly. In other words, the numbers become closer and closer to r (root) as n becomes large.
So how do we know when to stop? The rule of thumb that is generally used is that we can stop when successive approximations agree to 8 decimal places.
For the remaining problems, you can use the program Newton which does the process for you. 1. Type into .
2. Type into .
3. Run Newton.
4. Input your initial approximation.
5. Use the answer as your next approximation until you reach agreement to 8 decimal places. Ex.2.) Use Newton’s Method to find .
Ex.3.) Use Newton’s Method to approximate the solutions to .
Situations in which Newton’s Method Fails:
1.) when for some n.
2.) when an approximation falls outside of the domain of f.
3.) when successive approximations...
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