newton-raphson as calculus

8/3/2019 Newton-raphson as Calculus

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4.5: Linear Approximations, Differentials

and Newtons Method

Gre Kell Hanford Hi h School Richland Washin ton


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For any functionf(x), the tangent is a close approximationof the function for some small distance from the tangent

point.y

x0 x a!

f x f a!We call the equation of the

tangent the linearization of

the function.

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The linearization is the equation of the tangent line, and

you can use the old formulas if you like.

Start with the point/slope equation:

1 1y y m x x ! 1x a! 1y f a! m f ad!

y f a f a x ad !

y f a f a x ad!

L x f a f a x ad! linearization offat a

f x L x} is the standard linear approximation offat a.

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Important linearizations forx near zero:

1k

x 1 kx

sin x

cosx

tan x

x

1

x

1

21

1 1 1

2

x x x ! }

1

3 4 4 3

4 4

1 5 1 5

1 51 5 1

3 3

x x

x x

!

} !

f x L x

p

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.

y

x

(

(

dy

dx

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:

y f x!

dxdy dy f x dxd!

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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?

2A rT!

2dA r dr T!

2dA dr

rdx dx

T!

very small change in A

very small change in r

2 10 0.1dA T!

2dA T! (approximate change in area)

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2dA T! (approximate change in area)

Compare to actual change:

New area:

Old area:

2

10.1 102.01T T!

2

10 100.00T T!

2.01T

.01

2.01

T

T

!

Error

Original Area

Error

Actual Answer

.0049751} 0.5%}

0.01%}.0001}.01

100

T

T!

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Newtons Method

21

32f x x! Finding a root for:c mputer may nothaveenough memory toopen theimage, or theimagemay havebeen corrupted.Restartyour computer, and then open thefileagain.I fthe red x stillappears, you may havetodelete theimage and then insertit again.

We will use Newtons

Method to find the

root between 2 and 3.

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in. Ifthe red x stillappears, you may haveto deletetheimage and then insertit again.

Guess: 3

21

3 3 3 1.52

f ! !

1.5

tangent 3 3m fd! !

21

32

f x x!

f x xd !

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

21

2.5 2.5 3 .1252

f ! !

1.5

tangent 2.5 2.5m fd! !

21

32

f x x!

f x xd !

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 fd! !

21

32

f x x!

f x xd !

z

.00125

2.45z !

.00125

2.45 2.449489795922.45

! (new guess) p


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Guess: 2.44948979592

2.44948979592 .00000013016f !

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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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

Guess: 2.44948979592

2.44948979592 .00000013016f !

Amazingly close to zero!

Newtons Method:

1

n

n n

n

f xx x

f x !

d

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 xd

1

n

n n

n

f xx x

f x !

d

Find where crosses .3

y x x! 1y !

31 x x!

30 1x x! 3 1f x x x! 23 1f x xd !

0 1 1 21

1 1.5

2

!

1 1.5 .875 5.75.875

1.5 1.34782615.75

!

2 1.3478261 .1006822 4.4499055 1.3252004

3

1.3252004 1.3252004 1.0020584 ! 1}p


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There are some limitations to Newtons method:

Wrong root found

Looking for this root.

Bad guess.

Failure to converge

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Newtons method is built in to the Calculus Tools

application on the TI-89.

Of course if you have a TI-89, you could just use

the root finder to answer the problem.

The only reason to use the calculator for Newtons Methodis to help your understanding or to check your work.

It would not be allowed in a college course,

on theAP exam or on one of my tests.

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APPS

Select and press .Calculus Tools ENTER

If you see this

screen, press

, change the

mode settings as

necessary, and

press

again.

ENTER

APPS

Now lets do one on the TI-89:

p

3 1f x x x! Approximate the positive root of:


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Now lets do one on the TI-89:

APPS

Select and press .Calculus Tools ENTER

Press (Deriv)F2

Press (Newtons Method)3

Enter the equation.

(You will have to unlockthe alpha mode.)Set the initial guess to 1.

Press .ENTER

3 1f x x x! Approximate the positive root of:

Set the iterations to 3.

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Press to see

the summary screen.

ESC

Press to see

each iteration.

ENTER

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T

Press and thento return your

calculator to normal.

ESC

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