1. LINEARIZATION AND NEWTON’S METHOD Section 4.5

3. Linearization Algebraically, the principle of local linearity means that the equation of the tangent line defines a function that can be used to approximate a differentiable function near the point of tangency, The equation of the tangent line is given a new name: the linearization of f at a. Recall point-slope form of a line: y=m(x-x 1 )+y 1 The tangent line at (a, f(a)) can be written: y=f ’(a)(x-a)+f(a)

4. Linearization

5. So the equation of the tangent line at a = 1 is

6. (These are y-values…. Find the x that goes with it!

7. Tangent Line Equation:

9. Newton’s Method Finding a root for: We will use Newton’s Method to find the root between 2 and 3.

10. Guess: (not drawn to scale) (new guess)

11. Guess: (new guess)

12. Guess: (new guess)

13. Guess: Amazingly close to zero! This is Newton’s Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) It is sometimes called the Newton-Raphson method This is a recursive algorithm because a set of steps are repeated with the previous answer put in the next repetition. Each repetition is called an iteration.

14. This is Newton’s Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) It is sometimes called the Newton-Raphson method Guess: Amazingly close to zero! Newton’s Method: This is a recursive algorithm because a set of steps are repeated with the previous answer put in the next repetition. Each repetition is called an iteration.

15. Find where crosses.

16. There are some limitations to Newton’s method: Wrong root found Looking for this root. Bad guess. Failure to converge