Calculus 4.5

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4.5 Linear Approximations, Differentials and
Newtons Method
Palouse Falls, Washington State
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Palouse River, Washington State
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Isaac Newton shares credit with Gottfried Leibniz
as the developer of calculus.
Today we will look at Newtons method for
approximating the roots of a function.
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Chapter 4 Notes Section 4.5 Linear Approximations
and Newtons Method
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For any function f (x), the tangent is a close
approximation of the function for some small
distance from the tangent point.
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Start with the point/slope equation
linearization of f at a
The linearization is the equation of the tangent
line, and you can use the old formulas if you
like.
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Important linearizations for x near zero
This formula also leads to non-linear
approximations
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Differentials
When we first started to talk about derivatives,
we said that becomes when the
change in x and change in y become very small.
dy can be considered a very small change in y.
dx can be considered a very small change in x.
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Let be a differentiable
function. The differential is an
independent variable. The differential is
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Example Consider a circle of radius 10. If the
radius increases by 0.1, approximately how much
will the area change?
very small change in r
very small change in A
(approximate change in area)
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(approximate change in area)
Compare to actual change
New area
Old area
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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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We learn Newtons method of finding roots for
historical interest and to deepen our
appreciation of calculus.
Newtons method is sometimes tested on the AP
exam, and will therefore sometimes appear on a
test in this class.
There are much easier methods of finding roots
using our calculator.
Statue of Isaac Newton as a college student in
Trinity College Chapel at Cambridge University,
Cambridge, England
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Example
If you have the function graphed, you can find
the roots by using
Use the arrow keys to select the lower and upper
bounds, and press ENTER each time.
You will need to find each root separately.
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Example
An even quicker way to find roots is to use the
following when in the home screen
p