NEWTON

1
NEWTONS METHOD

• Section 3.8

2
When you are done with your homework, you should
be able to

• Approximate a zero of a function using Newtons
  Method
•  

3
Appollonius lived in 250BC. He studied conic
sections. Name the three conic sections and how
Appollonius characterized the ellipse.

1. The three conic sections are the circle, ellipse
   and hyperbola. Appollonius characterized the
   ellipse as the perpendicular of a flat plane with
   a cone.
2. The three conic sections are the circle, ellipse
   and hyperbola. Appollonius characterized the
   ellipse as the perpendicular of a flat plane that
   is parallel to the side of a cone.
3. The three conic sections are the parabola,
   ellipse and hyperbola. Appollonius characterized
   the ellipse as the perpendicular of a flat plane
   that is parallel to the side of a cone.
4. The three conic sections are the parabola,
   ellipse and hyperbola. Appollonius characterized
   the ellipse as the perpendicular of a flat plane
   with a cone.

4
NEWTONS METHOD

• Newtons Method uses tangent lines to approximate
  zeros
• The assumption is that the function and the
  tangent line cross the x-axis at about the same
  point

5
NEWTONS METHOD FOR APPROXIMATING THE ZEROS OF A
FUNCTION

• Let where f is differentiable on
  an open interval containing c. Then, to
  approximate c, use the following steps.
• Make an initial estimate that is close to c.
  (Make a graph!)
• Determine a new approximation
  .
• If is within the desired accuracy,
  let serve as the final approximation.
  Otherwise, return to step 2 and calculate a new
  approximation.
• Each successive application of this procedure
  is called an iteration.

6
Use Newtons Method to approximate to five
decimal places.

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