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LINEARIZATION AND NEWTON’S METHODSection 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: y=m(x-x1)+y1
• The tangent line at (a, f(a)) can be written:
y=f ’(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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(These are y-values…. Find the x that goes with it!
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Tangent Line Equation:
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Newton’s Method
213
2f x x Finding a root for:
We will use Newton’s Method to find the root between 2 and 3.
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Guess: 3
213 3 3 1.5
2f
1.5
tangent 3 3m f
213
2f x x
f x x
z
1.5
1.53
z
1.5
3z 1.5
3 2.53
(not drawn to scale)
(new guess)
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Guess: 2.5
212.5 2.5 3 .125
2f
1.5
tangent 2.5 2.5m f
213
2f x x
f x x
z
.125
2.5z .125
2.5 2.452.5
(new guess)
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Guess: 2.45
2.45 .00125f
1.5
tangent 2.45 2.45m f
213
2f x x
f x x
z
.00125
2.45z
.001252.45 2.44948979592
2.45 (new guess)
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Guess: 2.44948979592
2.44948979592 .00000013016f
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.
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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: 2.44948979592
2.44948979592 .00000013016f
Amazingly close to zero!
Newton’s Method: 1n
n nn
f xx x
f x
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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nx nf xn nf x 1n
n nn
f xx x
f x
Find where crosses .3y x x 1y 31 x x 30 1x x 3 1f x x x 23 1f x x
0 1 1 21
1 1.52
1 1.5 .875 5.75.875
1.5 1.34782615.75
2 1.3478261 .1006822 4.4499055 1.3252004
31.3252004 1.3252004 1.0020584 1
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There are some limitations to Newton’s method:
Wrong root found
Looking for this root.
Bad guess.
Failure to converge
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