4.5: Linear Approximations, Differentials and Newton’s Method

Greg Kelly, Hanford High School, Richland, Washington

For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.

y

x0 x a

f x f aWe call the equation of the tangent the linearization of the function.

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 a

y f a f a x a

y f a f a x a

L x f a f a x a linearization of f at a

f x L x is the standard linear approximation of f at a.

Important linearizations for x near zero:

1k

x 1 kx

sin x

cos x

tan x

x

1

x

1

21

1 1 12

x x x

13 4 4 3

4 4

1 5 1 5

1 51 5 1

3 3

x x

x x

f x L x

This formula also leads to non-linear approximations:

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.

Let be a differentiable function.

The differential is an independent variable.

The differential is:

y f xdxdy dy f x dx

Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change?

2A r

2 dA r dr

2 dA dr

rdx dx

very small change in A

very small change in r

2 10 0.1dA

2dA (approximate change in area)

2dA (approximate change in area)

Compare to actual change:

New area:

Old area:

210.1 102.01

210 100.00

2.01

.01

2.01

Error

Original Area

Error

Actual Answer.0049751 0.5%

0.01%.0001.01

100

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 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: 1

nn n

n

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 1

nn n

n

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

Newton’s 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 Newton’s Method is to help your understanding or to check your work.

It would not be allowed in a college course, on the AP exam or on one of my tests.

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 let’s do one on the TI-89:

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

Now let’s do one on the TI-89:

APPS

Select and press . Calculus Tools ENTER

Press (Deriv)F2

Press (Newton’s Method)3

Enter the equation.(You will have to unlock the 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.

Press to see

the summary screen.

ESC

Press to see each iteration.

ENTER

Press and then

to return your

calculator to normal.

ESC

HOME