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

Start with the point/slope equation: linearization of f at a is the standard linear approximation of f at a. The linearization is the equation of the tangent line, and you can use the old formulas if you like.

Important linearizations for x near zero: This formula also leads to non-linear approximations:

dy can be considered a very small change in y. 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.

Let be a differentiable function. The differential is an independent variable. The differential is:

Example: Consider a circle of radius 10. If the radius increases by 0 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)

(approximate change in area) Compare to actual change: New area: Old area:

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

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

Guess: (new guess)

Guess: (new guess)

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.

Guess: Newton’s Method: 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.

Find where crosses .

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

Newton’s method is built into your calculator 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.

Now let’s do one on the TI-84: Approximate the positive root of: Enter f(x) into the y= window Enter f’(x) and unselect it Graph the function and make a guess Go back to home screen and store your guess as x Then type x – y1/y2 and store your answer Re-Type x – y1/y2 sto X and your calculator will store Each answer automatically Continue to hit enter until your answers converge **Repeat for others zeros if the function has more! Answers: 1.347826087