1. §3.6 Newton’s Method. The student will learn about
Newton’s method of approximating roots and
tangent line approximations.

2. Introduction to Newton’s Method
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 through Newton’s Method gives us a way of finding approximate solutions.

3. An Easier Example to Start
Let’s start by computing the √5. This is of course easy with your calculator but stay with me for this.
First we rewrite the problem as an equation f (x) = x 2 – 5 = 0
Newton’s method is an iterative method. That means that you must first pick an initial value for the solution and then the method will yield a better value. The method may be repeated as often as necessary to get the accuracy needed.
What would be a good initial value for √5?
OK we will use 2.

4. An Easier Example to Start
Before we continue let’s look at the method.
Consider the drawing. If x Is the root and x n is an approximation then x n + 1 is a better approximation.
Using the tangent line slope
And solving for x n + 1 yields 4

5. Back to our Example
We were trying to find √5 using f (x) = x 2 – 5 = 0 and x n = x 0 = 2 and f ′ (x) = 2x.
Probably not too impressed! Let’s find x 2.
With just two iterations we have accuracy to
FACT: Newton described this method in a book he wrote in 1669! 5

6. Example 2
Approximate the solution to cos x = x in the interval [0, 2].
First we rewrite the problem as f (x) = cos x – x = 0
We will let x 0 = 1 (midpoint of the interval) and we know f ′(x) = - sin x - 1
If we were to repeat the process we would get x 2 = x 3 = accuracy to 9 places!
A bit tedious BUT if you know a little programming your calculator or computer can do this easily. 6

7. Example 3 Approximate using x 0 = 1
OK it’s a silly example (Do you know the solution?) but stay with me while I make a point.
The computation is easy with x 0 = 1, x 1 = - 2, x 2 = 4, x 3 = - 8, x 4 = 16, etc.
So the method fails. But, it fails spectacularly! 7

8. Failure Newton's method makes no guarantee on convergence.
Indeed, convergence depends on the starting point and on the shape of the function. 8

9. Your Calculator
Calculators basically only know how to add and multiply.
So, how does it find ? Let’s use Newton.
Notice that the operations involved in the iteration are addition and multiplication and you computer can do that!
Use x 0 = 2 and approximate √5 with two iterations. 9

10. Your Calculator
Use x 0 = 2 and approximate √5 with two iterations. 10

11. Tangent Line Approximations
From our definition of derivative we know that
When h is small.
If we multiply both sides of the above by h, we get
Δy is the exact change in y
dy = h · f ′ (x) is an approximation of Δy and is called the differential. 11

12. Tangent Line Approximations
Summary
Approximate change exact change
Another useful form: 12

13. Tangent Line Approximations
Let’s use this form for a practical problem.
Approximate √5 using the differential above.
With x = 4, h = 1,
Does this look familiar? 13

14. Summary. We learned how to use Newton’s Method to solve equations.
We developed the approximation formula using the differential dy,
f (x + h) – f (x) ≈ h · f ′ (x) = dy

15. ASSIGNMENT
§3.6; Page 66; 1 - 9, odd.