Linearization and Newton

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Linearization and Newtons Method

• Section 4.5

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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 ym(x-x1)y1
• The tangent line at (a, f(a)) can be written
• yf (a)(x-a)f(a)

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Linearization
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So the equation of the tangent line at a 1 is
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Newtons Method
Finding a root for
We will use Newtons Method to find the root
between 2 and 3.
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Guess
(not drawn to scale)
(new guess)
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Guess
(new guess)
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Guess
(new guess)
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Guess
Amazingly close to zero!
This is Newtons 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.
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Guess
Amazingly close to zero!
This is Newtons 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.
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Find where crosses
.
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There are some limitations to Newtons method
Looking for this root.
Bad guess.
Wrong root found
Failure to converge
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