LINEARIZATION AND NEWTON’S METHODSection 4.5

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)

Linearization

So the equation of the tangent line at a = 1 is

(These are y-values…. Find the x that goes with it!

Tangent Line Equation:

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.

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)

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)

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)

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.

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.

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

There are some limitations to Newton’s method:

Wrong root found

Looking for this root.

Bad guess.

Failure to converge