Linearization and Newton’s Method. I. Linearization A.) Def. – If f is differentiable at x = a, then the approximating function is the LINEARIZATION of.

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Presentation transcript:

Linearization and Newton’s Method

I. Linearization A.) Def. – If f is differentiable at x = a, then the approximating function is the LINEARIZATION of f at x = a. B.)

C.) Note: This is just like just different notation!!! D.) Graphically: We call the equation of the tangent the linearization of the function.

E.) Ex.- Find the linearization at x = 0 of

F.) How accurate is it?

II. Differentials A.) Def.: Let y = f (x) be a differentiable function. The DIFFERENTIAL dx is an independent variable. The DIFFERENTIAL dy = f’(x)dx, where dy is dependent upon the values of f’(x)dx. B.) Ex. – Find the differential dy in each of the following:

C.) Ex. – Find dy and evaluate dy for the given values of x and dx in the following:

III. Estimating Change A.) Graphically:

B.) Let y = f (x) be a differentiable function at x = a. The APPROXIMATE change in the value of f when x changes from x to x + a is

C.) Ex.- Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in A very small change in r (approximate change in area)

Compare to actual change: New area: Old area:

I. Examples The side of a square is measured with a possible percentage error of ±5%. Use differentials to estimate the possible percentage error in the area of the square.

Therefore, the possible percentage error in the measurement of the area of the square will be between ±10%

The area of a circle is to be computed from a measured value of its diameter. Estimate the maximum permissible percentage error in the measurement if the percentage error in the area must be kept within 1%.

Therefore, the percentage error in the measurement of the diameter must be within ±.5%

IV. Newton’s Method A.) Newton’s Method is an algorithm for finding roots. Newton’s Method: 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.

We will use Newton’s Method to find the root between 2 and 3. B.) Example: Finding a root for:

Guess: (not drawn to scale) (new guess)

Guess: (new guess)

Guess: (new guess)

Guess: Amazingly close to zero!

Find where crosses.

There are some limitations to Newton’s method: Wrong root found Looking for this root. Bad guess. Failure to converge