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4.5: Linear Approximations, Differentials and Newton’s Method. We call the equation of the tangent the linearization of the function. 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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Presentation Transcript
slide3

We call the equation of the tangent the linearization of the function.

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

slide4

is the standard linear approximation of f at a.

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.

linearization example
Linearization Example

Find the linear approximation of f(x) = x2 at x = 1. Use the approximation to 1.12.

linearization example1
Linearization Example

Find the linear approximation of f(x) = x2 at x = 1. Use the approximation to 1.12.

linearization example2
Linearization Example

Use linearization to approximate

linearization example3
Linearization Example

Use linearization to approximate

approximating binomial powers
Approximating Binomial Powers

Approximating Binomial Powers: this is a special case of a general linearization formula that applies to the powers of 1+x for small values of x:

Approximating Binomial Powers

approximating binomial powers1
Approximating Binomial Powers

Use this formula to find polynomials that will approximate the following functions for values of x close to zero:

slide11

Important linearizations for x near zero:

This formula also leads to non-linear approximations:

slide12

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.

slide13

Let be a differentiable function.

The differential is an independent variable.

The differential is:

slide14

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)

slide15

(approximate change in area)

Compare to actual change:

New area:

Old area:

newton s method
Newton’s Method

Newton’s Method: numerical method of approximating the zeros of a function.

slide17

Newton’s Method

Finding a root for:

We will use Newton’s Method to find the root between 2 and 3.

slide18

Guess:

(not drawn to scale)

(new guess)

slide19

Guess:

(new guess)

slide20

Guess:

(new guess)

slide21

Guess:

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.

slide22

Newton’s Method:

Guess:

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.

slide24

There are some limitations to Newton’s method:

Looking for this root.

Bad guess.

Wrong root found

Failure to converge

slide25

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.

slide26

Approximate the positive root of:

APPS

ENTER

ENTER

APPS

Select and press .

Calculus Tools

If you see this screen, press

, change the mode settings as necessary, and press

again.

Now let’s do one on the TI-89:

slide27

F2

APPS

ENTER

ENTER

Press (Newton’s Method)

3

Press .

Now let’s do one on the TI-89:

Approximate the positive root of:

Select and press .

Calculus Tools

Press (Deriv)

Enter the equation.

(You will have to unlock the alpha mode.)

Set the initial guess to 1.

Set the iterations to 3.

slide28

ENTER

Press to see each iteration.

Press to see the summary screen.

ESC

slide29

Press and then

to return your calculator to normal.

ESC

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